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Mathematical analysis and approximation of a multiscale elliptic-parabolic system

Omar Richardson

Om

ar Richardson | M

athematical analysis and approxim

ation of a multiscale elliptic-parabolic system

| 2018:33


We study a two-scale coupled system consisting of a macroscopic elliptic equation and a microscopic parabolic equation. This system models the interplay between a gas and liquid close to equilibrium within a porous medium with distributed microstructures. We use formal homogenization arguments to derive the target system. We start by proving well-posedness and inverse estimates for the two-scale system. We follow up by proposing a Galerkin scheme which is continuous in time and discrete in space, for which we obtain well-posedness, a priori error estimates and convergence rates. Finally, we propose a numerical error reduction strategy by refining the grid based on residual error estimators.

LICENTIATE THESIS | Karlstad University Studies | 2018:33

Faculty of Health, Science and Technology

Mathematics


ISSN 1403-8099

ISBN 978-91-7063-962-3 (pdf)

ISBN 978-91-7063-867-1 (print)



Omar Richardson

Print: Universitetstryckeriet, Karlstad 2018

Distribution:Karlstad University Faculty of Health, Science and TechnologyDepartment of Mathematics and Computer ScienceSE-651 88 Karlstad, Sweden+46 54 700 10 00

© The author

ISSN 1403-8099

urn:nbn:se:kau:diva-68686

Karlstad University Studies | 2018:33

LICENTIATE THESIS

Omar Richardson


WWW.KAU.SE

ISBN 978-91-7063-962-3 (pdf)

ISBN 978-91-7063-867-1 (print)

Mathematical analysis and approximation ofa multiscale elliptic-parabolic system

Omar Malcolm Richardson

Supervisor: Prof. dr. habil. Adrian Muntean

Co-supervisor: Dr. Martin Lind

Examiner: Asoc. prof. dr. Niclas Bernhoff

Opponent: Prof. dr. Alfred Schmidt

i

Abstract

We study a two-scale coupled system consisting of a macroscopic el-liptic equation and a microscopic parabolic equation. This system modelsthe interplay between a gas and liquid close to equilibrium within a porousmedium with distributed microstructures. We use formal homogenizationarguments to derive the target system. We start by proving well-posednessand inverse estimates for the two-scale system. We follow up by proposinga Galerkin scheme which is continuous in time and discrete in space, forwhich we obtain well-posedness, a priori error estimates and convergencerates. Finally, we propose a numerical error reduction strategy by refiningthe grid based on residual error estimators.

Keywords: Two-scale modeling, two-scale Galerkin approximation, in-verse Robin estimates, elliptic-parabolic system, a priori analysis, a posteri-ori error estimators

MSC Subject Classification (2010): 35K58, 65N30, 65N15, 76S05, 35B27,35R10

ii

Acknowledgements

First of all, I would like to thank my supervisor prof. dr. habil. Adrian Munteanfor the unlimited support, freedom and faith he has given me during the com-pletion of this work, always going above and beyond what I could ask for.

Secondly, I would like to thank dr. Martin Lind for his comraderie and wis-dom, not only in mathematics, but in the ’real world’ as well. I’d like to thankprof. dr. Alfred Schmidt for taking the responsibility to be my opponent, dr. DaisukeTagami for valuable feedback, dr. Niclas Bernhoff for examining my work, andStina Rojder Berglund and Birgitta Landin for helping me with all the issues Ichallenged them with.

Finally, I would like to thank Irene Man and my parents for all their indis-pensible love and support.

Contents

1 Introduction 11.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Model derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Technical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Well-posedness and inverse Robin estimates 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Existence and uniqueness of the solution . . . . . . . . . . . . . . . 152.4 Energy and stability estimates . . . . . . . . . . . . . . . . . . . . . 202.5 Local stability for the inverse Robin problem . . . . . . . . . . . . 242.6 Proof of inverse Robin estimate . . . . . . . . . . . . . . . . . . . . 28

3 A well-posed semidiscrete Galerkin scheme 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Well-posedness of (P1) . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Convergence rates for semidiscrete Galerkin approximations . . . 513.5 A posteriori mesh refinement strategy . . . . . . . . . . . . . . . . 563.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Discussion 654.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

iii

iv CONTENTS

Bibliography 67

List of Figures 74

CHAPTER 1Introduction

1.1 Preface

The goal of a mathematical model is to describe a certain process in mathemat-ical concepts, so that the process can be analyzed and perhaps even predicted.For complex processes, the activity often happens on different time or lengthscales. The mathematics used to describe these processes should reflect that inorder not to ignore relevant effects. One way to do this is by designing a multi-scale model that is defined on each of the relevant scales. A concise introductioninto the principles of multiscale modeling is provided in e.g. [15]. This thesisdeals with the mathematical analysis of certain transport and interactive pro-cesses that act on two length scales.

Multiscale modeling is applicable to a range of processes. Popular exam-ples include modeling groundwater flow, transport in cells and the dynamics ofcrowds. These examples have shown that, if one wants to take phenomena intoaccount that reveal themselves on several scales, the demands on the resolutionof a single scale model quickly leads to computational infeasibility.

This thesis concerns itself with a specific multiscale model, derived from aRichards’-like equation for liquid flow in unsaturated porous media. The modelgives rise to a system of equations that interacts on the two chosen spatial scales.

In this thesis, we derive the aforementioned system and show that it is well-posed. We continue by proving inverse estimates that can help to determinesome of the model parameters of the system in experiments, should measure-ments be available. Secondly, we propose a finite element approximation scheme,show that it converges to the weak solution of the original system, and derive

1

2 CHAPTER 1. INTRODUCTION

a priori two-scale rates of convergence. We propose a macroscopic mesh refine-ment strategy based on a posteriori error estimates. Finally, we give an outline ofthe future work planned in the rest of this project.

With this work, we hope to increase the understanding of multiscale mod-els, as well as the approximation techniques one can use and the theoreticalestimates that one can obtain when dealing with such models. As such, thiswork fits partly in the field of mathematical modeling and partly in the field ofnumerical analysis.

1.2 Model derivation

In this section, we propose a formal derivation1 of a Richards’-like equationposed in a two-scale setting. For a more detailed explanation we refer the readerto [8], where this line of arguments is based on.

Let Ω denote a partially saturated locally periodic porous medium. The time-dependent microscopic pore space is denoted by Bε(t), where ε is the ratio be-tween the characteristic size of the pore and the size of the sample and t repre-sents the time variable. Bε(t) is partitioned in Bεf (t), occupied by the fluid, andBεg(t), occupied by the gas. An incompressible viscous fluid flows through Ω,interacting with a gas which is present in the remaining part of the pore space.We write ρf , vεf , and pεf to denote the density, velocity and pressure of the fluid,respectively. We denote the density of the gas with ρεg.

Starting from the Stokes equations coupled with the diffusion equation forthe gas, we obtain the following system:

∇pεf − µε2∆vεf = 0 in Bεf (t),div vεf = 0 in Bεf (t),∂ρεg∂t− div

(Dε2∇ρεg

)= 0 in Bεg(t),

(1.1)

where µ is the dynamic viscosity of the fluid and D is the diffusion coefficient ofthe gas, both scaled with a factor ε2. The system in (1.1) satisfies the followingboundary conditions:

vεf = 0 on Γεf ,

Dε2∇ρεg · n = 0 on Γεg,

Dε2∇ρεg · nε = β(pεf −Rρεg

)on Γεfg(t),

(1.2)

1This discussion is inspired by a private (unpublished) communication between G. Sciarra(Nantes), E. N. M. Cirillo (Rome) and A. Muntean (Karlstad).

1.2. MODEL DERIVATION 3

Figure 1.1: Example of a locally periodic two-dimensional macroscopic domainΩ. The solid space is in white, the pore space in gray. In three dimensions, boththe pore space and the solid space should be seen as connected.

where Γεf and Γεg represent the interface between the fluid and gas, respectively,in Bε. Γεfg represents the free boundary in the pore space, governed by the mo-tion of the fluid particles.

To describe the motion of the free boundary, we use the level set equationf ε(t, x) = 0. This way we can evaluate the outward unit normal nε to the fluiddomain Bεf , having the following evolution equation for the interface Γεfg:

nε =∇f ε

||∇f ε||(1.3)

∂f ε

∂t−∇f ε · vεf = 0. (1.4)

We introduce a periodic structure on Ω by covering Ω with replicates of the cellY := (0, 1)N which consists of the solid part S and the pore space B, definedsuch that S ∪ B = Y and S ∩ B = ∅. Now, Bε can be interpreted as the locallyperiodic intersection between Ω and the union of porous subsets of each cell B,rescaled by ε. Similarly, Bεf (t), Bεg(t), Γεfg, Γεf , and Γεg are constructed from theintersections between Ω and unions of Bf (t), Bg(t), Γfg, Γf , and Γg, respectively.A schematic overview of Ω is shown in Figure 1.1, with an illustration of itsmicrostructure Y shown in Figure 1.2. We start our derivation by postulatingtwo-scale expansions for the quantities vεf , pεf , ρεg and f ε. Let gε = g(x, x

ε) be a


Γfg(t)

Γf

Γg

Bf(t)

Bg(t)Y

S

Figure 1.2: Close up on a microstructure Y . The pore is partly filled with fluid(Bf ), partly with gas (Bg), the boundaries of which are indicated with Γf and Γg,respectively. The fluid-gas interface is indicated with Γfg(t). Outside of theseboundaries lies the solid skeleton of the porous medium S.

function depending on ε. Then we assume that gε has the following asymptoticexpansion:

gε(x) = g0(x, y) + εg1(x, y) +O(ε2), (1.5)

with y = x/ε being the micro-scale variable and g0, g1, . . . being periodic withrespect to y.

The asymptotic expansion of the level set equation yields the following ex-pression for the normal vector nε(x, x/ε) of the free boundary:

nε = n0 + εn1 +O(ε2)

=∇yf0

||∇yf0||+ ε

(∇xf0 −∇yf0

||∇yf0||

)+O

(ε2). (1.6)

Expanding (1.4), which denotes the interface evolution, gives us the followingexpression:

∂f ε

∂t= −1

ε∇yf0 · v0 − (∇xf0 · v0 +∇yf0 · v1) +O (ε) , (1.7)

which, by separating the terms at order 1/ε and at order 1, yields the followingrelations:

∇yf0 · v0 = 0,∂f0

∂t= ∇xf0 · v0 +∇yf0 · v1. (1.8)

In the rest of this discussion, we assume that f0 does not depend on y, eliminat-ing the last term of (1.8). Substituting (1.6) in (1.8) results in:

v0 · n0 = 0,∂f0

∂t||∇yf0||−1 = (v0 · n1)(y, t) + (v1 · n0)(y, t). (1.9)

1.2. MODEL DERIVATION 5

Consider the asymptotic expansion for interface Γfg(t) (determined by f ε(x, x/ε)).Γ0(x, t) denotes the order 1 boundary, with normal vector n0, which is orthogo-nal to the fluid velocity on the interface.

Expanding the pressure gradient and the Laplacian of the fluid velocity re-sults in:

∇pεf =1

ε∇yp0 + (∇xp0 +∇yp1) +O (ε) ,

ε2∆vεf = ∆yv0 +O (ε) ,(1.10)

which, by substituting in (1.1) and (1.2) and by applying a scale separation ar-gument, leads to the following identities which hold for all y ∈ Bf (x, t):

∇yp0 = 0 leads to p0 = p0(x),

µ∆yv0 = ∇yp1 +∇xp0,

divy v0 = 0, divy v1 + divx v0 = 0,

v0 · n0 = 0 = p0 −Rρ0,

−D∇yρ0 · n0 = ρf (v0 · n1 + v1 · n0) = β(p1 −Rρ1).

(1.11)

According to (1.11), p0 does not depend on y, yielding it to be a representationof the fluid pressure averaged over the unit cell Y .

Let ejNj=1 denote the canonical basis in RN . Writing∇xp0 =∑N

j=1 ej(−∂jp0(x)),we can denote the cell problems corresponding to (1.1) as follows

∆ywj = ∇yπj = ej in Bf (x, t),divy wj = 0 in Bf (x, t),∂ρ0∂t− divy(D∇yρ0) = 0 in Bg(x, t),

n0 · wj = p0 −Rρ0 = 0 on Γ0(x, t),

wj = 0 on Γf (x, t),

∇yρ0 · n0 = 0 on Γg(x, t),

(1.12)

where wj and πj are the Y -periodic order 1 velocity and order ε pressure of thefluid, respectively. They are determined by the relations:

v0(x, y) =1

µ

N∑j=1

wj(y)(−∂jp0(x)),

p1(x, y) =N∑j=1

πj(y)(−∂jp0(x)).

(1.13)


Let the averaged fluid velocity v(x) be defined as

v(x) =

∫Bf (x,t)

v0(x, y)dy. (1.14)

We postulate a relation between the ith component of the averaged velocity andthe averaged pressure p0 for i ∈ 1, ..., N based on Darcy’s law as follows:

v = −A∇p0, (1.15)

where the permeability tensor A = [Aij] is defined component-wise as:

Aij =1

µ

(∫Bf (x,t)

∇ywi · ∇ywjdy +

∫Γ0(x,t)

∂ywj∂n0

· widy

), (1.16)

for i, j ∈ 1, . . . , N. More details regarding the structure of (1.16) can be foundin [21]. Finally, in the absence of free boundaries, we can reformulate (1.12) interms of a ’distributed pressures’-model:

Find the macroscopic fluid pressure p0(t, x) and the microscopic gas densityρ0(t, x, y) that satisfy the following system of equations:

−Aρf∆xp0 = C(t)p0 in Ω,

p0 = p on ∂Ω,∂ρ0∂t− divy(D∇yρ0) = 0 in Bg(x),

p0 −Rρ0 = 0 on Γ0(x),

∇yρ0 · n = 0 on Γg(x),

(1.17)

for some atmospheric pressure p.For more details on models with distributed microstructures, we refer the

reader to the chapter on microstructure models of porous media by R. E. Showal-ter (see [21], Chapter 9), as well as to the PhD thesis [29].

In this framework, our attention is focused on models of type (1.17) withuniform (i.e., x-independent) distributed microstructures.

1.3 Technical preliminaries

This section contains some mathematical concepts that will be used throughoutthis thesis. Let D ⊂ Rd for some d ∈ 1, 2, 3, and let f, g : D → R. Then theLebesgue and Sobolev norms are defined as follows:

||f ||Lp(D) :=

(∫

D|f(x)|pdx

)1/p for 1 ≤ p <∞,

ess sup |f | : x ∈ D for p =∞,(1.18)

1.3. TECHNICAL PRELIMINARIES 7

||f ||Hk(D) :=

∑|α|≤k

∫D

|∂αf |2 dx

1/2

for k ∈ N, (1.19)

with ∂αf denoting derivatives in the weak sense.Furthermore, for L2(D) and Hk(D), for some positive k ∈ N, we use the

following inner products:

〈f, g〉L2(D) :=

∫D

f(x)g(x)dx, (1.20)

〈f, g〉Hk(D) :=∑|α|≤k

〈∂αf, ∂αg〉L2(D). (1.21)

Let B be a Banach space with norm || · ||B. Then u belongs to the Bochnerspace L2(S;B) if u is Bochner-integrable and its norm, defined as

||u||L2(S,B) :=

(∫

S||u(t)||pBdt

)1/p for 1 ≤ p <∞,

ess supt∈S ||u(t)||B for p =∞,(1.22)

is finite.An introduction to the concepts of Lebesgue and Bochner spaces, inner prod-

ucts and norms can be found in any functional analysis textbook (e.g. [1]).Furthermore, we introduce the following well-known inequalities and lem-

mata.

Lemma 1 (Poincare’s inequality). Let D ⊂ Rd be a bounded, open subset of Rd forsome d ≥ 1 and denote by cP (D) the smallest constant such that

‖u‖2L2(D) ≤ cP (D)‖∇xu‖2

L2(D),

hold for all u ∈ H10 (D).

The constant cP (D) is referred to as the Poincare constant of the domain D.

Lemma 2 (Interpolation-trace inequality). Assume that Y ⊂ Rd is a Lipschitz do-main and u ∈ L2(D;H1(Y )). For any a > 0 we have∫

D

∫∂Y

u2dσydx ≤ a

∫D

∫Y

|∇yu|2dydx+ ca

∫D

∫Y

|u|2dydx,

where ca ∼ 1/a. In particular,

‖u‖L2(D;L2(∂Y )) ≤ c‖u‖L2(D;H1(Y )).


The next result provides a useful equivalent norm on H1(Y ).

Lemma 3. Let Y ⊂ Rd be a simply connected domain and Γ ⊂ ∂Y where Γ has positive(d− 1)-dimensional surface measure. Then there are constants c1, c2 such that

c1‖u‖2H1(Y ) ≤

∫Γ

u2dσy + ‖∇xu‖2L2(Y ) ≤ c2‖u‖2

H1(Y ).

Finally, we state the following classical compactness results, two fixed-pointtheorems and an indispensable inequality. For details, see e.g. [44] or [18].

Lemma 4 (Aubin-Lions theorem [2]). Let B0 → B ⊂⊂ B1 (i.e. B0 is compactlyembedded in B and B is continuously embedded in B1) be Banach spaces. Let

W =u ∈ L2 (0, T ;B0) |∂tu ∈ L2 (0, T ;B1)

. (1.23)

Then the embedding of W into L2 (0, T ;B) is compact.

Lemma 5 (Schauder’s fixed point theorem). Let B be a nonempty, closed, convex,bounded set and F : B → B a compact operator. Then there exists at least one r ∈ Bsuch that F (r) = r.

Lemma 6 (Banach fixed point theorem). Let B be a Banach space, and let F : B →B be a nonlinear operator. Suppose that

||F (u1)− F (u2)||B ≤ γ ||u1 − u2||B ,

for some γ < 1. Then there exists exactly one u ∈ B such that F (u) = u.

Lemma 7 (Gronwall’s inequality). Let η(t) be a non-negative, absolutely continuousfunction defined on [0, T ], satisfying

η′(t) ≤ ϕ(t)η(t) + ψ(t),

for a.e. t, with ϕ(t), ψ(t) : [0, T ]→ R nonnegative and integrable. Then

η(t) ≤(∫ t

0

ψ(s)ds

)exp

(∫ t

0

ϕ(s)ds

).

1.4 Structure of this thesis

The rest of this thesis is structured as follows.

1.4. STRUCTURE OF THIS THESIS 9

In Chapter 2, we analyze a generalization of the model derived in (1.17).Well-posedness is proven, as well as inverse estimates for measurements on theRobin boundary.

In Chapter 3, we derive a Galerkin approximation of (1.17). We show thatthis approximation is well-posed as well, and that its approximation convergesto the initial system. We provide rates of convergence and design an mesh re-finement strategy based on error estimators.

In Chapter 4, we summarize the work and provide an outlook for the rest ofthis project.

CHAPTER 2Well-posedness and inverse Robin

estimates

In this chapter1 we establish the well-posedness of a coupled two-scale parabolic-elliptic system modeling the interplay between two pressures in a gas-liquidmixture close to equilibrium that is filling a porous media with distributed mi-crostructures. Additionally, we prove a local stability estimate for the inversemicro-macro Robin problem.

2.1 Introduction

We are interested in developing evolution equations able to describe multiscalespatial interactions in gas-liquid mixtures, targeting a rigorous mathematicaljustification of Richards-like equations - upscaled model equations generallychosen in a rather ad hoc manner by the engineering communities to describethe motion of flow in unsaturated porous media. The main issue is that onelacks a rigorous derivation of the Darcy’s law for such flow (see [21] (chapter1) for a derivation via periodic homogenization techniques of the Darcy law forthe saturated case).

If air-water interfaces can be assume to be stagnant for a reasonable timespan, then averaging techniques for materials with locally periodic microstruc-tures (compare e.g. [8]) lead in suitable scaling regimes to what we refer hereas two-pressure evolution systems. These are normally coupled parabolic-elliptic

1This chapter has been published as an article under the name Well-posedness and inverseRobin estimate for a multiscale elliptic/parabolic system in Applicable Analysis, 2018, vol. 97, nr. 1.

11

12 CHAPTER 2. WELL-POSEDNESS AND INVERSE ROBIN ESTIMATES

systems responsible for the joint evolution in time t ∈ (0, T ) (T < +∞) ofa parameter-dependent microscopic pressure Rρ(t, x, y) evolving with respect toy ∈ Y ⊂ Rd for any given macroscopic spatial position x ∈ Ω and a macroscopicpressure π(t, x) with x ∈ Ω for any given t. HereR denotes the universal constantof gases. The two-scale geometry we have in mind is depicted in Figure 1 below.

Figure 2.1: The macroscopic domain Ω and microscopic pore Y at x ∈ Ω.

To cast the physical problem in mathematical terms as stated in (2.1), weneed a number of dimensional constant parameters (A (gas permeability), D(diffusion coefficient for the gaseous species), pF (atmospheric pressure), ρF (gasdensity)) and dimensional functions (k (Robin coefficient) and ρI (initial liquiddensity)). It is worth noting that excepting the Robin coefficient k, all the modelparameters and functions are either known or can be accessed directly via mea-surements. Getting grip on a priori values of k is more intricate simply becausethis coefficient is defined on the Robin part of the boundary of ∂Y , say ΓR, wherethe micro-macro information transfer takes actively place. The Neumann partof the boundary ΓN := ∂Y − ΓR is assumed to be accessible via measurements,while ΓR is thought here as inaccessible.

Our aim here is twofold:

(1) ensure the well-posedness in a suitable sense of our two-pressure systemwith k taken to be known;

(2) prove stability estimates with respect to k for the inverse micro-macroRobin problem (k is now unknown, but measured values of the micro-scopic pressure are available on ΓN ).

The main results reported in this chapter are Theorem 1 (the weak solvabilityof (2.1)) and Theorem 4 (the local stability for the inverse micro-macro Robinproblem).

The choice of problem and approach is in line with other investigations run-ning for two-scale systems, or systems with distributed microstructures, like

2.2. PROBLEM FORMULATION 13

[26, 29, 36]. As far as we are aware, this is for the first time that an inverse Robinproblem is treated in a two-scale setting. A remotely connected single-scale in-verse Robin problem is treated in [34].

2.2 Problem formulation

We consider the following parabolic-elliptic problem posed on two spatial scalesx ∈ Ω and y ∈ Y .

−AρF∆xπ = f(π, ρ) in Ω,

∂tρ−D∆yρ = 0 in Ω× Y,

D∇yρ · ny = k(π + pF −Rρ) on Ω× ΓR,

D∇yρ · ny = 0 on Ω× ΓN ,

π = 0 at ∂Ω,

ρ(t = 0) = ρI(x, y) in Ω× Y,

(2.1)

where the parameters, coefficients and the nonlinear function f satisfy the as-sumptions discussed below (see Section 2.1). The initial condition for π followsfrom the coupling between ρ and p.

A prominent role in this chapter is played by the micro-macro Robin transfercoefficient k, which is selected from the following set:

K := k ∈ L2(ΓR) : 0 < k ≤ k(y) ≤ k for y ∈ ΓR,

for some positive constants k, k.

Assumptions

(A1) The domains Ω, Y have Lipschitz continuous boundaries.

(A2) The parameters satisfy A,D, ρF , R ∈ (0,∞).

(A3) The microscopic initial value satisfies ρI ∈ H1(Ω× Y ).

(A4) f(λu, µv) = λαµβf(u, v) where α + β = 1, α, β > 0.

(A5) There is a structural constant C∗ > 0 such that∫Ω

|f(u1, v)− f(u2, v)|2dx ≤ C∗‖u1 − u2‖2L2(Ω)

uniformly in v ∈ L2(Ω;L2(Y )).


(A6) There is a constant C > 0 such that∫Ω

f(u, v)2dx ≤ C‖v‖2L2(Ω;H1(Y )).

(A7) The constant C∗ in (A5) satisfies

C∗cP (Ω) < 1,

where cP (Ω) is the Poincare constant of the domain Ω.

Assumptions (A1)-(A3) have clear geometrical or physical meanings, while(A4)-(A7) are technical. Assumption (A7) is only used when deriving uniquenessof the weak solution to (2.1). Note also that for some special classes of domains,the Poincare constant can be quantitatively estimated, see e.g. [30].

Auxiliary results

In this section, we state some auxiliary results that will be useful in this context.We start with a Sobolev-type embedding and a simple trace theorem.

Proposition 1. Assume that U ⊂ Rd, then Hd(U) ⊂ L∞(U) and

‖u‖L∞(U) ≤ C‖u‖Hd(U).

Proposition 2. Assume that U ⊂ Rd and Γ ⊂ ∂U is Lipschitz continuous. Then

Hd+1(U) ⊂ Hd(Γ),

and‖u‖Hd(Γ) ≤ c‖u‖Hd+1(U).

We have the following existence and regularity results.

Proposition 3 (see e.g. [18]). Consider the problem∂tv −D∆yv = 0 on Ω× Y,

D∇yv · ny = k(g −Rv) on Ω× ΓR,

D∇yv · ny = 0 on Ω× ΓN ,

v(t = 0) = vI on Ω× Y.

(2.2)

If g ∈ L2(0, T ;H1(Ω)) and vI ∈ H1(Ω;Hm(Y )) (m ∈ N), then the problem (2.2) has aunique weak solution v ∈ L2(0, T ;H1(Ω;Hm+1(Y ))).

2.3. EXISTENCE AND UNIQUENESS OF THE SOLUTION 15

Proposition 4 (see e.g [6]). Let v(t, x, y) be in L2(0, T ;H1(Ω;Hm+1(Y ))) (m ∈ N)

and consider the problem−∆xu = f(u, v) on Ω× Y

u(t, x) = 0 on ∂Ω, t > 0.(2.3)

where the nonlinear function satisfies (A4)-(A6). Then the problem (2.3) has a uniqueweak solution u ∈ L2(0, T ;H1(Ω)).

2.3 Existence and uniqueness of the solution

Existence of weak solution

The main result of this subsection is the following theorem.

Theorem 1. Assume that (A2)-(A6) hold. Then the problem (2.1) has at least a weaksolution solution (π, ρ) ∈ L2(0, T ;H1

0 (Ω))× L2(0, T ;L2(Ω;H1(Y ))).

Proof. We decouple the problem. The first sub-problem is as follows: given π ∈L2(0, T ;H1

0 (Ω)) and ρI ∈ H1(Ω, H1(Y )), we let ξ be the weak solution to∂tξ −D∆yξ = 0 on Ω× Y,

D∇yξ · ny = k(π + pF −Rξ) on Ω× ΓR,

D∇yξ · ny = 0 on Ω× ΓN ,

ξ(t = 0) = ρI on Ω× Y.

(2.4)

The weak formulation of (2.4) is: find ξ such that for a.e. t ∈ [0, T ] and everyψ ∈ L2(Ω, H1(Y )) there holds∫

Ω

∫Y

∂tξψdydx+

∫Ω

∫Y

D∇yξ∇yψdydx =

∫Ω

∫ΓR

k(π + pF −Rξ)ψdσydx, (2.5)

and ξ(t = 0) = ρI . Existence and regularity of ξ is provided by Proposition 3(recall that (A3) states that ρI ∈ H1(Ω× Y )).

The second sub-problem is: given data ξ, consider the problem−∆xπ = f(π, ξ) on Ω,

π = 0 on ∂Ω.(2.6)

Let λ > 0 be a free parameter. By the scaling properties of f and uniqueness ofweak solution, we have that if π is the weak solution of (2.6) with data ξ, then


π = λπ is the weak solution to (2.6) with data λξ. Hence, if π is the weak solutionto −∆xπ = λβf(π, ξ) on Ω

π = 0 on ∂Ω,(2.7)

then π = λπ, again by the scaling properties of f . The weak form of (2.7) is asfollows: find π such that for a.e. t ∈ [0, T ] and all ϕ ∈ H1

0 (Ω), there holds∫Ω

∇xπ · ∇xϕdx = λβ∫

Ω

f(π, ξ)ϕdx.

Existence and regularity of π is guaranteed by Proposition 4.We now use a fixed point argument a la Schauder (see Lemma 5) to show that

there exists a λ > 0 for which the functions of the pair (π, ξ) are weak solutionsto the sub-problems (2.4) and (2.7). Then we recover (π, ρ), a weak solution to(2.1), by taking π = π/λ and ρ = ξ.

Define the operators

T1 : L2(0, T ;L2(Ω))→ L2(0, T ;L2(Ω;L2(Y )))

by T1(π) = ξ (the weak solution of (2.4)) and

T λ2 : L2(0, T ;L2(Ω;H1(Y )))→ L2(0, T ;H1(Ω))

by T λ2 (ξ) = π (the weak solution of (2.7)). Finally, consider the operator Aλ onthe space L2(0, T ;L2(Ω)) into itself defined by

Aλ(π) = T λ2 (T1(π)) = π. (2.8)

To obtain existence of solution, we prove that the operator Aλ has a fixed point.This π will provide us with ξ. The idea of the proof is to first use Schauder’sfixed point theorem (Lemma 5).

We prove that there exist a λ > 0 and a set B such that

1. Aλ is a compact operator;

2. B is convex, closed, bounded and satisfies Aλ(B) ⊂ B.

To obtain compactness of Aλ = T λ2 T1, it is sufficient to demonstrate that T1 iscompact and that T λ2 is continuous. Recall that we have

T1 : L2(0, T ;H1(Ω))→ L2(0, T ;L2(Ω;L2(Y ))).


However, since we assume that ξI ∈ H1(Ω, H1(Y )) we get that T1(π) = ξ ∈L2(0, T ;H1(Ω× Y )) and ∂tξ ∈ L2(0, T ;L2(Ω× Y )). Whence,

T1(L2(0, T ;H1(Ω))) ⊂ V,

where

V = u : u ∈ L2(0, T ;H1(Ω× Y )), ∂tu ∈ L2(0, T ;L2(Ω× Y )).

By Lemma 4,V ⊂⊂ L2(0, T ;L2(Ω;L2(Y ))).

Thus, for any bounded set M ⊂ L2(0, T ;L2(Ω;L2(Y ))) × L2(0, T ;L2(Ω)), thereholds T1(M) ⊂ V and since V is compactly contained inL2(0, T ;L2(Ω;L2(Y )) wehave that T1(M) is precompact in L2(0, T ;L2(Ω;L2(Y ))). Hence, T1 is compact.

We continue to prove that T λ2 is continuous. Assume we have two solutionsπ1 = T λ2 (ξ1) and π2 = T λ2 (ξ2). Substituting these both in (2.8) and subtracting,we obtain ∫

Ω

∇x(π1 − π2) · ∇xϕdx = λβ∫

Ω

[f(π1, ξ1)− f(π2, ξ2)]ϕdx,

and for ϕ = π1 − π2, we get

‖∇x(π1 − π2)‖2L2(Ω) = λβ

∫Ω

[f(π1, ξ1)− f(π2, ξ2)][π1 − π2]dx

= λβ∫

Ω

[f(π1, ξ1)− f(π2, ξ1)][π1 − π2]dx

+ λβ∫

Ω

[f(π2, ξ1)− f(π2, ξ2)][π1 − π2]dx.

Using (A5) and (A6), we obtain that

‖∇x(π1 − π2)‖2L2(Ω) ≤ Cλβ‖π1 − π2‖L2(Ω)‖ξ2 − ξ1‖L2(Ω;H1(Y )).

By Poincare’s inequality, we obtain

‖∇x(π1 − π2)‖L2(Ω) ≤ Cλβ‖ξ1 − ξ2‖L2(Ω;H1(Y ))

and we conclude the mapping T λ2 is continuous.LetK > 0 be a fixed number that we specify later and letBK be the collection

of functions u ∈ L2(0, T ;H1(Ω)) such that

max‖u‖L2(0,T ;L2(Ω)), ‖∇xu‖L2(0,T ;L2(Ω)) ≤ K.


For each K > 0, the set

BK ⊂ L2(0, T ;H1(Ω))

is a convex, closed and bounded. We show that we may select K > 0 and λ > 0

such that

Aλ(BK) ⊂ BK . (2.9)

Note that T1(BK) is a bounded subset of L2(0, T ;L2(Ω;L2(Y ))), with a bounddepending only on K. In other words,

‖ξ‖L2(0,T ;L2(Ω;L2(Y )))) ≤ CK. (2.10)

Indeed, this follows from the fact that T1 is a compact operator.We proceed by observing that we may choose λ > 0 such that if u ∈ BK is

arbitrary and v = T λ2 (T1(u)), then

max‖v‖L2(0,T ;L2(Ω)), ‖v‖L2(0,T ;H1(Ω))

≤ K. (2.11)

Let ξ = T1(u) so that v = T λ2 (ξ). Testing the weak formulation of (2.7) with ϕ = u

and using Cauchy-Schwarz’ inequality and Poincare’s inequality, we get

‖∇xu‖2L2(Ω) ≤ Cλβ‖u‖L2(Ω)‖ξ‖L2(Ω;L2(Y )) ≤ CKλβ‖ξ‖L2(Ω;L2(Y )).

Integrating over [0, T ] and using (2.10), we obtain after using Poincare’s inequal-ity

‖u‖2L2(0,T ;H1(Ω)) ≤ C ′K2λβ.

By taking λ small enough (depending on K), we obtain (2.11) whence (2.9) fol-lows.

We point out that, instead of using scaling arguments and Schauder’s fixedpoint theorem, we could have used alternatively the Schaefer/Leray-Schauderfixed point theorem.

Uniqueness of weak solutions

We proceed to prove the following uniqueness theorem.

Theorem 2. Assume that in addition to the assumptions of Theorem 1 condition (A7)also holds. Then the weak solution to (2.1) is unique.


Proof. The weak formulation of the uncoupled problem can be expressed as fol-lows: find (π, ρ) ∈ L2(0, T ;H1

0 (Ω))× L2(0, T ;L2(Ω;H1(Y ))) where ρ(t = 0) = ρI

and for a.e. t ∈ [0, T ] it holds that

AρF

∫Ω

∇xπ · ∇xϕdx =

∫Ω

f(π, ρ)ϕdx (2.12)

and∫Ω

∫Y

∂tρψdydx+

∫Ω

∫Y

D∇yρ · ∇yψdydx =

∫Ω

∫ΓR

k(π+ pF −Rρ)ψdσydx (2.13)

for all ϕ ∈ H10 (Ω) and all ψ ∈ L2(Ω;H1(Y )).

Assume that two pairs of solutions exist: (π1, ρ1) and (π2, ρ2). Let q := π1−π2

and z := ρ1 − ρ2. If we substitute the two solutions in (2.12) and (2.13) andsubtract, we obtain that

AρF

∫Ω

∇xq · ∇xϕdx =

∫Ω

(f(π1, ρ1)− f(π2, ρ2))ϕdx (2.14)

and∫Ω

∫Y

∂tzψdydx+

∫Ω

∫Y

D∇yz · ∇yψdydx =

∫Ω

∫ΓR

k(q −Rz)ψdσydx (2.15)

for all ϕ ∈ H10 (Ω) and all ψ ∈ L2(Ω;H1(Y )).

Choosing specific test function ϕ = q, using Young’s inequality with param-eter ε1 > 0 and (A5), we obtain from (2.14) the first key estimate

AρF‖∇xq‖2L2(Ω) ≤ (C∗ + ε1)‖q‖2

L2(Ω) + cε1‖z‖2L2(Ω,L2(Y )). (2.16)

We focus on (2.15), which, using test function ψ = z, yields

1

2

d

dt||z||2L2(Ω;L2(Y )) +D||∇yz||2L2(Ω;L2(Y )) =

∫Ω

∫ΓR

k(q −Rz)z. (2.17)

Now, we estimate the right hand side of (2.17) by using the trace inequality andthe fact that k ≤ k on ΓR. We have∫

Ω

∫ΓR

|k(q −Rz)z|dσydx ≤ k

∫Ω

∫ΓR

|qz|dσydx+Rk

∫Ω

∫ΓR

z2dσydx

≤ k|ΓR|2‖q‖2

L2(Ω) +

(R +

1

2

)k

∫Ω

∫ΓR

z2dσydx.

The second term at the right-hand side of the previous inequality can be es-timated by using the trace inequality and Young’s inequality with parameterε > 0: ∫

Ω

∫ΓR

z2dσydx ≤ ε‖∇yz‖2L2(Ω;L2(Y )) +

c0

ε‖z‖2

L2(Ω;L2(Y )),


for some absolute constant c0 > 0. Using the previous estimates and rearranging(2.17), we obtain

1

2

d

dt||z||2L2(Ω;L2(Y )) + (D − ε)||∇yz||2L2(Ω;L2(Y ))

≤ k|ΓR|2‖q‖2

L2(Ω) +c0k(R + 1/2)

ε‖z‖2

L2(Ω;L2(Y )). (2.18)

By Poincare’s inequality, we have

‖q‖2L2(Ω) ≤ cP (Ω)‖∇xq‖2

L2(Ω),

where cP (Ω) is the Poincare constant of the domain Ω. Using this in (2.16), weobtain

AρF‖∇xq‖2L2(Ω) ≤ (C∗ + ε1)cP (Ω)‖∇xq‖2

L2(Ω) + C‖z‖2L2(Ω,L2(Y )).

By (A7), we may take ε1 > 0 small enough such that (C∗+ε1)cP (Ω) < AρF . Thenwe obtain

‖q‖2L2(Ω) ≤ cP (Ω)‖∇xq‖2

L2(Ω) ≤CcP (Ω)

AρF − (C∗ + ε1)cP (Ω)‖z‖2

L2(Ω,L2(Y )). (2.19)

Whence, it follows from the previous estimate and (2.18) with ε = D/2 that

1

2

d

dt||z||2L2(Ω;L2(Y )) +

D

2||∇yz||2L2(Ω;L2(Y )) ≤ C‖z‖2

L2(Ω;L2(Y )).

By using Gronwall’s inequality and the fact that z(0, x, y) = 0, it follows that z =

0. From (2.19), we obtain q = 0 as well. This demonstrates the uniqueness.

2.4 Energy and stability estimates

We start this section by stating the following energy estimates for our problem.

Proposition 5. Assume (A2)-(A6) and let (u, v) be a weak solution to

−∆xu = f(u, v) in Ω

∂tv −D∆yv = 0 in Ω× Y

D∇yv · ny + k(Rv − u) = g on Ω× ΓR

D∇yv · ny = 0 on Ω× ΓN

u = 0 at ∂Ω

v(t = 0) = vI in Ω× Y.

(2.20)

2.4. ENERGY AND STABILITY ESTIMATES 21

Then the following energy estimate holds

‖u‖2L2(0,T ;H1(Ω)) + ‖v‖2

L2(0,T ;L2(Ω,H1(Y )))

≤ C(‖g‖2

L2(0,T ;L2(Ω;L2(ΓR))) + ‖vI‖2L2(Ω;L2(Y ))

). (2.21)

The proof of Proposition 5 follows by similar arguments as the proof of The-orem 3 below, therefore we omit it.

We proceed to study the stability of solutions with respect to some of theparameters involved. Some preliminary remarks:

• We do not need to study the stability of the solution with respect to ρF , pFand R. Recall that R is an universal physical constant, while ρF , pF fix thetype of fluid and gas we are considering.

• We could investigate the stability of (π, ρ) with respect to structural changesinto the non-linearity f(·, ·). We omit to do so mainly because our main in-tent lies in understanding the role of the micro-macro Robin coefficient k.

• For this stability proof, we decide to use a direct method which relies es-sentially on energy estimates; see e.g. [31].

For i ∈ 1, 2, let (πi, ρi) be two weak solutions corresponding to the sets ofdata (ρIi, Ai, Di, ki), where ρIi, Ai, Di, ki denote the initial data, diffusion coeffi-cients and mass-transfer coefficients of the solution (πi, ρi). Denote

δu := u2 − u1 where u ∈ π, ρ, ρI , A,D, k .

Theorem 3. Assume that for i = 1, 2, (Ai, Di) belongs to a fixed compact subset of R2,that ki ∈ K and that ‖ρIi‖L2(Ω;L2(Y )) ≤ C. Let (πi, ρi) (i = 1, 2) be weak solutions to(2.1) corresponding to the choices of data above. Then the following estimate holds:

‖δπ‖2L2(0,T ;H1

0 (Ω)) + ‖δρ‖2L2(0,T ;L2(Ω;H1(Y )))

≤ c(‖δk‖2

L2(ΓR) + |δA|+ |δD|+ ‖δρI‖2L2(Ω;L2(Y ))

), (2.22)

Proof. We have for i = 1, 2

AiρF

∫Ω

∇xπi · ∇xϕdx =

∫Ω

f(πi, ρi)ϕdx

and∫Ω

∫Y

∂tρiψdydx+

∫Ω

∫Y

Di∇yρi · ∇yψdydx =

∫Ω

∫ΓR

ki (πi + pF −Rρi)ψdσydx,


for all ϕ ∈ H10 (Ω) and ψ ∈ L2(Ω;H1(Y )).

Subtracting the corresponding equations and then testing with ϕ := π2 − π1

and ψ := ρ2 − ρ1 gives:

ρF

(A2

∫Ω

∇xπ2 · ∇xϕdx− A1

∫Ω

∇xπ1 · ∇xϕdx

)=

∫Ω

(f(π2, ρ2)− f(π1, ρ1))ϕdx,

(2.23)and

d

2dt‖ψ‖2

L2(Ω;L2(Y )) +

∫Ω

∫Y

(D2∇yρ2 −D1∇yρ1) · ∇yψdydx

=

∫Ω

∫ΓR

(k2(π2 + pF −Rρ2)− k1(π1 + pF −Rρ1))ψdσydx.(2.24)

Regarding (2.23), note that

A2

∫Ω

∇xπ2 · ∇xϕdx− A1

∫Ω

∇xπ1 · ∇xϕdx

= A2‖∇xϕ‖2L2(Ω) + (A2 − A1)

∫∇xπ1 · ∇xϕdx.

Using (A5) and (A6), we may estimate the right-hand side of (2.23) and obtain

A2ρF‖∇xϕ‖2L2(Ω) ≤ C∗‖ϕ‖2

L2(Ω) + c

(‖ψ‖2

L2(Ω;L2(Y )) + |δA|∫

Ω

|∇xπ1||∇xϕ|dx).

Using Poincare’s inequality, assumptions on f and Young’s inequality with pa-rameter ε > 0, we get

‖∇xϕ‖2L2(Ω) ≤ c

(‖ψ‖2

L2(Ω;L2(Y )) + |δA|∫

Ω

|∇xπ1||∇xϕ|dx)

≤ cε‖∇xϕ‖2L2(Ω) + c

(‖ψ‖2

L2(Ω;L2(Y )) + |δA|‖∇xπ1‖2L2(Ω)

).

Choosing ε = 1/(2c), rearranging and using energy estimates for π1, we obtain

‖∇xϕ‖2L2(Ω) ≤ c

(‖ψ‖2

L2(Ω;L2(Y )) + |δA|). (2.25)

We proceed to estimate ‖ψ‖2L2(Ω;L2(Y )), using (2.24). Note that∫

Ω

∫Y

(D2∇yρ2 −D1∇yρ1) · ∇yψdydx

= (D2 −D1)

∫Ω

∫Y

∇yρ2 · ∇yψ +D1‖∇yψ‖2L2(Ω;L2(Y )).

2.4. ENERGY AND STABILITY ESTIMATES 23

Hence, it follows that

d

2dt‖ψ‖2

L2(Ω;L2(Y )) +D1‖∇yψ‖2L2(Ω;L2(Y )) ≤ |δD|

∫Ω

∫Y

|∇yξ1||∇yψ‖dydx

+

∫Ω

∫ΓR

|(k2 − k1)ψ|+ |(k2π2 − k1π1)ψ|+ |R(k2ρ2 − k1ρ1)ψ|dσydx. (2.26)

We have∫Ω

∫ΓR

|k2 − k1||ψ|dσydx ≤ ε

∫Ω

∫ΓR

ψ2dσydx+ cε|Ω|‖k2 − k1‖2L2(ΓR)

≤ cε(‖ψ‖2

L2(Ω;L2(Y )) + ‖∇yψ‖2L2(Ω;L2(Y ))

)+c‖k2 − k1‖2

L2(ΓR). (2.27)

Furthermore,∫Ω

∫ΓR

|(k2π2 − k1π1)ψ|dσydx ≤ c‖k2 − k1‖2L2(ΓR) +

∫Ω

∫ΓR

|k1||π2 − π1||ψ|dσydx

≤ c‖k2 − k1‖2L2(ΓR) + εk|ΓR|‖π2 − π1‖2

L2(Ω) + cε

∫Ω

∫ΓR

ψ2dσydx

≤ c‖k2 − k1‖2L2(ΓR) + ε

(‖∇xϕ‖2

L2(Ω) + ‖∇yψ‖2L2(Ω;L2(Y ))

)+ cε‖ψ‖2

L2(Ω;L2(Y )).

Finally,

R

∫Ω

∫ΓR

|(k2ρ2 − k1ρ1)ψ|dσydx ≤ R

∫Ω

∫ΓR

k22ψ

2 + (k2 − k1)2ρ21dσydx.

We assume that for all y ∈ ΓR, we have∫Ω

ρ21dx ≤ K,

this can be ensured by taking ρI1 smooth enough. Hence,∫Ω

∫ΓR

k22ψ

2 + (k2 − k1)2ρ21dσydx ≤ k

∫Ω

∫ΓR

ψ2dσydx+K‖k2 − k1‖2L2(Ω)

≤ε‖∇yψ‖2L2(Ω;L2(Y )) + cε‖ψ‖2

L2(Ω;L2(Y )) +K‖k2 − k1‖2L2(ΓR).

Taking all the estimates above into consideration, and compensating terms byselecting small ε > 0, we finally obtain

d

2dt‖ψ‖2

L2(Ω;L2(Y )) +D1‖∇yψ‖2L2(Ω;L2(Y ))

≤ c(‖ψ‖2

L2(Ω;L2(Y )) + |δD|+ ‖k2 − k1‖2L2(ΓR)

).


Applying Gronwall’s inequality leads to

‖ψ(t)‖2L2(Ω;L2(Y )) ≤ C

[‖δρI‖2

L2(Ω;L2(Y )) + |δD|+ ‖k2 − k1‖2L2(ΓR)

]and, by integration over [0, T ], we obtain:

‖ψ‖2L2(0,T ;L2(Ω;L2(Y ))) ≤ CT

[‖δρI‖2

L2(Ω;L2(Y )) + |δD|+ ‖k2 − k1‖2L2(ΓR)

].

It also follows that

‖∇yψ‖2L2(0,T ;L2(Ω;L2(Y ))) ≤ CT 2

[‖δρI‖2

L2(Ω;L2(Y )) + |δD|+ ‖k2 − k1‖2L2(ΓR)

].

Furthermore, by (2.25) and Poincare’s inequality, we have

‖ϕ‖2L2(0,T ;H1

0 (Ω) ≤ C(‖ψ‖2

L2(0,T ;L2(Ω;L2(Y ))) + |δA|).

Taking all the above estimates together, we obtain

‖ϕ‖2L2(0,T ;H1

0 (Ω) + ‖ψ‖2L2(0,T ;L2(Ω;H1(Y )))

≤ C(|δA|+ |δD|+ ‖δρI‖2

L2(Ω;L2(Y )) + ‖k2 − k1‖2L2(ΓR)

),

which concludes the proof.

2.5 Local stability for the inverse Robin problem

In this section, we study the inverse problem of recovering the micro-macroRobin coefficient k ∈ L2(ΓR) from measurement on ΓN ; the Neumann part ofthe boundary. (Usually, one thinks of ΓR as the inaccessible part of ∂Y , whileΓN is the accessible part.) Our discussion is influenced by the work in [12]. Analternative approach could be to follow the abstract result in [4].

Recall that we denote the set of admissible Robin coefficients as

K = k ∈ L2(ΓR) : 0 < k ≤ k(y) ≤ k for y ∈ ΓR,

Denote by k∗ the true Robin coefficient of our problem and define the setV(k∗, a) as

V(k∗, a) =k ∈ K : ‖k − k∗‖L2(ΓR) ≤ a

.

In the rest of this discussion, (π(k), ρ(k)) denotes the solution to (2.1) corre-sponding to the coefficient k ∈ K. Our main result is the following theorem.

2.5. LOCAL STABILITY FOR THE INVERSE ROBIN PROBLEM 25

Theorem 4. Assume that ρI ∈ H1(Ω, Hd(Y )) and ρ(k∗) ≥ c0 > 0 on [0, T ]×Ω×ΓR.Then there exists a > 0 such that

‖ρ(k2)− ρ(k1)‖L2(0,T ;L2(Ω;L2(ΓN ))) ≥ c‖k2 − k1‖L2(ΓR) (2.28)

for every k1, k2 ∈ V(k∗, a).

The discussion around Theorem 4 can be extended to the case of recov-ering micro-macro Robin coefficient with a genuine two-scale structure, e.g.k ∈ L2(Ω;L2(ΓR)) or k ∈ L2(0, T ;L2(Ω;L2(ΓR))). In this case, two-scale mea-surements are needed. To keep the presentation as simple as possible, we focusour attention on k ∈ K.

In the rest of this section, we prove establish several lemmata. The proof ofTheorem 4 is given in Section 2.6.

Lemma 8. For any k ∈ K and d ∈ L∞(ΓR) let (π(k), ρ(k)) be the solution to (2.1) and(u, v) = (u(k), v(k)) the solution to

−∆xu = F (u, v) in Ω

∂tv −D∆yv = 0 in Ω× Y

−D∇yv · ny + k(Rv − u) = d(π(k)− pF −Rρ(k)) on Ω× ΓR

−D∇yv · ny = 0 on Ω× ΓN

u = 0 at ∂Ω

v(0, x, y) = 0 in Ω× Y,

(2.29)

where F (u, v) is specified below. Then ρ(k) is continuously Frechet differentiable andits derivative ρ′(k)d at d ∈ L∞(ΓR) is given by v(k).

Proof. One can observe that the well-posedness of (2.29) follows by similar ar-guments as in the previous sections. Take k ∈ K and d ∈ L∞(ΓR) such thatk + d ∈ K.

Note first that (u1(k), v1(k)) = (π(k + d)− π(k), ρ(k + d)− ρ(k)) solves

−∆xu1 = f1(u1, v1) in Ω

∂tv1 −D∆yv1 = 0 in Ω× Y

−D∇yv1 · ny + k(Rv1 − u1) = d(π(k + d)− pF −Rρ(k + d)) on Ω× ΓR

−D∇yv1 · ny = 0 on Ω× ΓN

u1 = 0 at ∂Ω

v1(0, x, y) = 0 in Ω× Y,


where f1(u1, v1) = f(π(k + d), ρ(k + d))− f(π(k), ρ(k)). Denote by F = f1 and

U = u1 − u = π(k + d)− π(k)− u, V = v1 − v = ρ(k + d)− ρ(k)− v,

then (U, V ) solves the problem

−∆xU = f2(U, V ) in Ω

∂tV −D∆yV = 0 in Ω× Y

−D∇yV · ny + k(RV − U) = d(u1 −Rv1) on Ω× ΓR

−D∇yV · ny = 0 on Ω× ΓN

U = 0 at ∂Ω

V (0, x, y) = 0 in Ω× Y,

where f2(U, V ) = f1(u1, v1) − f1(u, v). Note that the nonlinearities f1, f2 satisfythe conditions of the energy estimate Proposition 5. Thus,

‖V ‖2L2(0,T ;L2(Ω,H1(Y ))) ≤ C‖d‖2

L∞(ΓR)‖u1 −Rv1‖2L2(L2(0,T ;L2(Ω;L2(ΓR))).

Whence,

‖ρ(k + d)− ρ(k)− v‖L2(0,T ;L2(Ω,H1(Y )))

‖d‖L∞(ΓR)

≤ C(‖u1‖L2(L2(0,T ;L2(Ω;L2(ΓR))) + ‖v1‖L2(L2(0,T ;L2(Ω;L2(ΓR)))

),

and it is sufficient to show that the right-hand side above tends to 0 as ‖d‖L∞(ΓR) →0. Using the interpolation-trace inequality, we obtain

‖v1‖L(0,T ;L2(Ω;L2(ΓR))) ≤ C‖v1‖L2(0,T ;L2(Ω,H1(Y ))).

Furthermore, using Proposition 5 and the interpolation-trace inequality again,we obtain

‖u1‖2L2(0,T ;H1(Ω)) + ‖v1‖2

L2(0,T ;L2(Ω,H1(Y )))

≤ C‖d‖2L∞(ΓR)‖π(k + d)− pF −Rρ(k + d)‖2

L2(0,T ;L2(Ω,L2(ΓR))) ≤ C ′‖d‖2L∞(ΓR)

from which follows that

limd→0

‖ρ(k + d)− ρ(k)− v‖L2(0,T ;L2(Ω,H1(Y )))

‖d‖L∞(ΓR)

= 0.

The proof of continuity follows by a similar argument, we refer to the discussionin [12].

2.5. LOCAL STABILITY FOR THE INVERSE ROBIN PROBLEM 27

We proceed now in a similar fashion as in e.g. [9, 12].Let g ∈ L2(ΓR) and (θ, ω) = (θ(g), ω(g)) be the weak solution to the system

−∆xθ = f(θ, ω) in Ω

∂tω −D∆yω = 0 in Ω× Y

−D∇yω · ny + k∗Rω = −gρ(k∗) on Ω× ΓR

−D∇yω · ny = 0 on Ω× ΓN

θ = 0 at ∂Ω

ω(0, x, y) = 0 in Ω× Y

(2.30)

For g ∈ L2(ΓR), define the operator

N : L2(ΓR)→ L2(0, T ;L2(Ω;L2(ΓR)))

byN(g) = −D∇yω(g) · ny.

Then N is a bounded linear operator (boundedness follow from energy esti-mates).

Lemma 9. The operator N is bijective and ‖N−1‖ is finite.

Proof. First, we prove the surjectivity ofN . Assume thatϕ ∈ L2(0, T ;L2(Ω;L2(ΓR))),we must prove that there exists some g ∈ L2(ΓR) such that N(g) = ϕ. Using(2.30), we obtain

ϕ+ k∗Rω(g) = −gρ(k∗),

or, equivalently,ϕ

ρ(k∗)+ g = −k

∗Rω(g)

ρ(k∗). (2.31)

DefineO : L2(ΓR)→ L2(0, T ;L2(Ω;L2(ΓR))).

by

O(g) = −k∗Rω(g)

ρ(k∗).

Then we haveϕ

ρ(k∗)= (O − I)(g).

Note further that O = BA, where

A : g 7→ ω(g), B : q 7→ −k∗Rq

ρ(k∗).


We have seen that A is compact and B is clearly continuous. Hence, O is com-pact.

We claim now that 1 is not an eigenvalue of O. Then, by the Fredholm alter-native theorem, O − I is invertible and

g = (O − I)−1

(ϕ

ρ(k∗)

).

To prove that 1 is not an eigenvalue of O, assume that O(g) = g for some g ∈L2(ΓR). It follows from (2.31) that ϕ/ρ(k∗) = 0, so ϕ = N(g) = 0. Hence,−D∇yω(g) · ny = 0 on [0, T ]× Ω× ΓR. Since ω(g) solves (2.30) it also solves

−∆xθ = f(θ, ω) in Ω

∂tω −D∆yω = 0 in Ω× Y

−D∇yω · ny = 0 on Ω× ∂Y

θ = 0 at ∂Ω

ω(0, x, y) = 0 in Ω× Y.

(2.32)

Hence, ω(g) = 0, but then −gρ(k∗) = 0 from the Robin boundary condition of(2.30), and since ρ(k∗) ≥ c0 > 0, we get g = 0. In other words, 1 is not aneigenvalue of O. In conclusion, N is invertible.

Since N is bounded, bijective and linear, the open mapping theorem ensuresthat N−1 exists and is bounded.

2.6 Proof of inverse Robin estimate

We are now ready to prove our main result.

Proof of Theorem 4. Let ε > 0 and consider the scaled problem

−∆xξ = f(ξ, ζ) in Ω

∂tζ −D∆yζ = 0 in Ω× Y

−D∇yζ · ny + kRζ = k(ξ + εpF ) on Ω× ΓR

−D∇yζ · ny = 0 on Ω× ΓN

ξ = 0 at ∂Ω

ζ(0, x, y) = ερI in Ω× Y.

(2.33)

2.6. PROOF OF INVERSE ROBIN ESTIMATE 29

Recall that we have f(εu, εv) = εf(u, v). From this it follows that the solu-tion (ξε, ζε) to the above problem satisfies (ξε, ζε) = (επ, ερ). Note that ζε(k) =

εζ(k) ≥ εc0 > 0 on [0, T ]× Ω× ΓR. Define the norm ‖ · ‖ε on L2(ΓR) by

‖g‖ε =

∥∥∥∥ 1

ζε(k∗)g

∥∥∥∥L2(0,T ;L2(Ω,L2(ΓR)))

.

Furthermore, define the mapping

σε : K → L2(0, T ;L2(Ω;L2(ΓR))),

by σε(k) = D∇y(ζε)·ny. It follows from the fact that ζε(k) is Frechet differentiable

with continuous derivative that σε is a C1-diffeomorphism. We have

σ′ε(k)g = D∇yζε1(k∗, g) · ny (2.34)

where ζε1(k∗, g) is the solution to

−∆xξε1 = f(ξε1, ζ

ε1(k∗, g)) in Ω

∂tζε1(k∗, g)−D∆yζ

ε1(k∗, g) = 0 in Ω× Y

−D∇yζε1(k∗, g) · ny + kRζε1(k∗, g) = −gζε(k∗) on Ω× ΓR

−D∇yζε1(k∗, g) · ny = 0 on Ω× ΓN

ξε1 = 0 at ∂Ω

ζε1(k∗, g)(0, x, y) = 0 in Ω× Y.

Since ζε(k∗) = ερ(k∗), it follows from (2.30) that ζε1(k∗, g) = εω(g) and thatσ′ε(k

∗)g = εN by (2.34). It follows that (σ′ε(k∗)g)−1 = N−1/ε.

Since σ′ε(k∗)g is a C1-diffeomorphism, there exists a neighbourhood N(k∗, a)

such that for any k1, k2 ∈ N(k∗, a) we have

‖k2 − k1‖ε ≤ 2‖(σε(k∗)g−1)′‖‖σ′ε(k2)g − σ′ε(k1)g‖L2(0,T ;L2(ΓR))

(see the discussion in [12]). We have

‖k2 − k1‖ε ≤ 2‖(σε(k∗)g−1)′‖‖D∇yζε(k2) · ny −D∇yζ

ε(k1) · ny‖L2(0,T ;L(Ω;L2(ΓR)))

≤ C

ε‖D∇yζ

ε(k2) · ny −D∇yζε(k1) · ny‖L2(0,T ;L(Ω;L2(ΓR))).

Using (2.33), one obtains the estimate

‖D∇yζε(k2) · ny −D∇yζ

ε(k1) · ny‖L2(0,T ;L(Ω;L2(ΓR)))

≤ C(‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;L2(ΓR))) + ‖ξε(k2)− ξε(k1)‖L2(0,T ;L2(Ω))

)+ C

2∑j=1

‖ζε(kj)(k2 − k1)‖L2(0,T ;L2(Ω;L2(ΓR))).


By the interpolation-trace inequality, we have

‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;L2(ΓR))) ≤ C‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;H1(Y ))).

Furthermore, by Poincare’s inequality, it holds that:

‖ξε(k2)− ξε(k1)‖L2(0,T ;L2(Ω)) ≤ C‖ξε(k2)− ξε(k1)‖L2(0,T ;H1(Ω)).

Hence, we obtain

‖k2 − k1‖ε ≤C

ε

[‖ξε(k2)− ξε(k1)‖L2(0,T ;H1(Ω))

+ ‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;H1(Y ))) (2.35)

+2∑j=1

‖ζε(kj)(k2 − k1)‖L2(0,T ;L2(Ω;L2(ΓR)))

].

We have2∑j=1

‖ζε(kj)(k2 − k1)‖L2(0,T ;L2(Ω;L2(ΓR)))

≤ C‖k2 − k1‖ε2∑j=1

‖ζε(kj)ζε(k∗)‖L2(0,T ;L2(Ω;L∞(ΓR)))

Since ΓR is (d− 1)-dimensional, we have

‖ζε(k)‖L∞(ΓR) ≤ c‖ζε(k)‖Hd−1(ΓR) ≤ c‖ζε(k)‖Hd(Y )

by Proposition 2 and Proposition 1. Furthermore, by Proposition 3 and the as-sumption ρI ∈ L2(Ω;Hd−1(Y )), we have ζε(k) ∈ L2(0, T ;H1(Ω;Hd(Y ))). Hence,

‖ζε(k∗)ζε(k1)‖L2(0,T ;L2(Ω;L∞(ΓR)))

≤ c‖ζε(k∗)‖L2(0,T ;L2(Ω;Hd−1(ΓR)))‖ζε(kj)‖L2(0,T ;L2(Ω;Hd−1(ΓR))) (2.36)

≤ c‖ζε(k∗)‖L2(0,T ;L2(Ω;Hd(Y )))‖ζε(kj)‖L2(0,T ;L2(Ω;Hd(Y )))

≤ cε2‖ρ(k∗)‖L2(0,T ;L2(Ω;Hd(Y )))‖ρ(kj)‖L2(0,T ;L2(Ω;Hd(Y )))

≤ C ′ε2.

By the above estimates, we can show

(1− Cε)‖k2 − k1‖ε

≤ C

ε

(‖ξε(k2)− ξε(k1)‖L2(0,T ;H1(Ω)) + ‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;H1(Y )))

). (2.37)

2.6. PROOF OF INVERSE ROBIN ESTIMATE 31

Using the equivalent norm on L2(0, T ;L2(Ω;H1(Y ))) given by Lemma 3, weobtain

(1− Cε)‖k2 − k1‖ε ≤C

ε‖ξε(k2)− ξε(k1)‖L2(0,T ;H1(Ω))

+C

ε‖∇y(ζ

ε(k2)− ζε(k1))‖L2(0,T ;L2(Ω;H1(Y ))) (2.38)

+C

ε‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω,L2(ΓN ))).

Set Xε := ξε(k2)− ξε(k1) and Zε := ζε(k2)− ζε(k1), then (Xε, Zε) solves

−∆xXε = f1(Xε, Zε) in Ω

∂tZε −D∆yZ

ε = 0 in Ω× Y

−D∇yZε · ny + k1RZ

ε = (k2 − k1)ζε(k2) on Ω× ΓR

−D∇yZε · ny = 0 on Ω× ΓN

Xε = 0 at ∂Ω

Zε(0, x, y) = 0 in Ω× Y,

where f1(Xε, Zε) = f(ξε(k2), ζε(k2))− f(ξε(k1), ζε(k1)).Using Proposition 5 and similar estimates as above, we obtain

‖ξε(k2)− ξε(k1)‖L2(0,T ;H1(Ω)) + ‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω;H1(Y )))

≤ C‖(k2 − k1)ζε(k2)‖L2(0,T ;L2(Ω;L2(ΓR))) ≤ Cε2‖k2 − k1‖ε.

This finally yields the crucial estimate

(1− C ′ε)‖k2 − k1‖ε ≤C

ε‖ζε(k2)− ζε(k1)‖L2(0,T ;L2(Ω,L2(ΓN ))).

Choose ε∗ > 0 such that 1− C ′ε∗ = 1/2 and use that ζε = ερ, then we obtain

‖k2 − k1‖ε∗ ≤ C‖ρ(k2)− ρ(k1)‖L2(0,T ;L2(Ω,L2(ΓN ))).

Finally, since ζ∗(k) ≥ εc0, we obtain that

‖k2 − k1‖L2(ΓR) ≤ C‖k2 − k1‖ε∗ ≤ C‖ρ(k2)− ρ(k1)‖L2(0,T ;L2(Ω,L2(ΓN ))).

CHAPTER 3A well-posed semidiscrete Galerkinscheme: a priori convergence rates

and two-scale a posteriori errorestimators

In this chapter1, we study a coupled elliptic-parabolic system of equations posedon two separated spatial scales, describing the interplay between macroscopicand microscopic pressures in an unsaturated heterogeneous medium with dis-tributed microstructures. Besides ensuring the well-posedness of our two-scalemodel, we design two-scale convergent numerical approximations and provea priori error estimates and propose an a posteriori error estimator. Finally, wepropose a macroscopic mesh refinement strategy that ensures a redistributionof the local macroscopic errors until an overall error reduction is achieved.

3.1 Introduction

This work concerns itself with the design and approximation of evolution equa-tions able to describe multiscale spatial interactions in gas-liquid mixtures, tar-geting a rigorous mathematical justification of Richards-like equations.

Generally, upscaled equations are chosen in a rather ad hoc manner by theengineering communities when describing the motion of flow in unsaturated

1This chapter has been submitted for publication under the title A semidiscrete Galerkin schemefor a coupled two-scale elliptic-parabolic system: well-posedness and convergence approximation rates.

33

34 CHAPTER 3. A WELL-POSED SEMIDISCRETE GALERKIN SCHEME

porous media. Often, one lacks a rigorous derivation of Darcy’s law for suchflows altering between compressibility and incompressibility. We refer the readerto Chapter 1 of [21] for a derivation of Darcy’s law in the saturated case usingperiodic homogenization arguments. Regarding the context of Darcy’s law forunsaturated flows, related work is reported, for instance, in [37], [28] and [13].

Moreover, we are concerned with the two-scale computability issue – com-plex systems of evolution equations acting on two spatial scales are notoriouslyhard to approximate, especially if moving boundaries or stochastic dynamicsare involved within e.g. the distributed microstructures.

In this chapter, we consider a coupled system of partial differential equationsdescribing the evolution of the pressure of a compressible air-liquid mixture ontwo spatial scales, when the amount of liquid is low and trapped in the internalstructure of a porous medium.

If we assume the interface between air and liquid to remain fixed for a rea-sonable time span, then using homogenization techniques for locally periodicmicrostructures (compare e.g. [8]) leads in suitable scaling regimes to a so-calledtwo-pressure evolution systems. This system can be expressed as a coupled elliptic-parabolic systems that describe the joint evolution in time t ∈ (0, T ) (T < +∞)of a parameter-dependent microscopic pressureRρ(t, x, y) (whereR represents theuniversal gas constant) evolving with respect to y ∈ Y ⊂ Rd for any givenmacroscopic spatial position x ∈ Ω and a macroscopic pressure π(t, x) with x ∈ Ω

for any given t. An illustration of the two-scale geometry we have in mind isdepicted in Figure 3.1.

We consider the following problem, posed on two spatial scales Ω ⊂ Rd1 andY ⊂ Rd2 with d1, d2 ∈ 1, 2, 3 in time interval t ∈ S := (0, T ) for some T > 0.Find the two pressures π : S × Ω→ R and ρ : S × Ω× Y → R that satisfy:

− A∆xπ = f(π, ρ) in S × Ω, (3.1)

∂tρ−D∆yρ = 0 in S × Ω× Y, (3.2)

D∇yρ · ny = k(π + pF −Rρ) in S × Ω× ΓR, (3.3)

D∇yρ · ny = 0 in S × Ω× ΓN , (3.4)

π = 0 in S × ∂Ω, (3.5)

ρ(0, x, y) = ρI(x, y) in Ω× Y , (3.6)

where ΓR ∪ΓN = ∂Y , ΓR ∩ΓN = ∅ and f : S×Ω×Y → R is a function. We referto (3.1)-(3.6) as (P1).

3.1. INTRODUCTION 35

ΓR

ΓN

∂Ω

Y

Ωx

Figure 3.1: The macroscopic domain Ω and microscopic pore Y at x ∈ Ω.

Note that (P1) describes the interaction between a compressible viscous fluid(with density ρ) in a porous domain Ω, where the pores are partly filled with agas that exerts an average (macroscopic) pressure π. The interaction betweenthe fluid and the gas is determined by the right hand side of (3.1) and the micro-scopic boundary condition in (3.3), through the fluid-gas interface representedby ΓR. The mathematical problem stated in (3.1)-(3.6) (referred to as (P1)), con-tains a number of dimensional constant parameters: A (gas permeability), D(diffusion coefficient for the gaseous species), pF (atmospheric pressure) and ρF

(gas density). In addition, we need the dimensional functions k (Robin coeffi-cient) and ρI (initial liquid density). Except for the Robin coefficient k, all themodel parameters and functions are either known or can be accessed directlyvia measurements.

If the Neumann part of the boundary ΓN := ∂Y \ ΓR is accessible via mea-surements, then the inaccessibility of the boundary ΓR can be compensated forin such a way that parameters like k entering two-scale transmission conditionscan be identified (compare [27]).

In this context, we prove the existence and uniqueness of a discrete-in-space,continuous-in-time finite element element approximation and prove conver-gence of this approximation of (P1). The main results of this contribution are the


well-posedness of the Galerkin approximation (Theorem 5), convergence ratesfor the approximation (Theorem 6), and controlled error estimators which canthen be used to refine the grid (Theorem 7).

The choice of problem and approach is in line with other investigations run-ning for two-scale systems, or systems with distributed microstructures, like[26, 29, 36]. The reader is also referred to the FEM2 strategies developed bythe engineering community to describe the evolution of mechanical deforma-tions in structured heterogeneous materials; see e.g. [24] and references citedtherein. Other classes of computationally challenging two-scale problems arementioned, for instance, in [38], where the pore scale model has a priori un-known boundaries, and in [23] for a smoldering combustion scenario.

This chapter continues an investigation started in related works. In [27], westudy the solvability issue and derive inverse Robin estimates for a variant ofthis model problem.

Two-scale Galerkin approximations have been derived previously for relatedproblem settings; see e.g. our previous attempts [33], [32], [7], and [26]. Ref. [26]stands out since it is for the first time that the question of feedback estimates isput in the context of computational efficiency of PDEs posed on multiple scales.Unfortunately, the obtained theoretical estimates in loc. cit. are not computable.This aspect is addressed here in Theorem 7.

The rest of this chapter is structured as follows. In Section 3.2, we discussthe technical concepts and requirements we need before starting our analysis.Then, in Section 3.3, we show the Galerkin approximation is well-posed andconverges to the weak solution of the original system. In Section 3.4, we provea priori convergence rates for the Galerkin approximation. Next, in Section 3.5,we design an error estimator, prove upper and lower bounds with respect tothe true error, and propose a macroscopic mesh refinement strategy. Finally, inSection 3.6, we conclude this chapter and provide an outlook into future steps.

3.2 Preliminary concepts

Weak solutions

We are interested in solutions to (P1) in the weak sense. This is motivated by thefact that the underlying structured media can be of composite type, allowingfor discontinuities in the model parameters. However, it is worth mentioning

3.2. PRELIMINARY CONCEPTS 37

that the solutions to (P1) are actually more regular than stated, i.e. with mini-mal adaptions of the working assumptions, the regularity of the solutions canbe lifted so that they turn out to be strong or even classical. We will lift theirregularity only when needed.

Definition 1 (Weak solution). A weak solution of (P1) is a pair

(π, ρ) ∈ L2(S;H10(Ω))× L2(S;L2(Ω;H1(Y )))

that satisfies for all test functions (ϕ, ψ) ∈ H10(Ω)× L2(Ω;H1(Y )) the identities

A

∫Ω

∇xπ · ∇xϕdx =

∫Ω

f(π, ρ)ϕdx, (3.7)

and∫Ω

∫Y

∂tρψdydx+D

∫Ω

∫Y

∇yρ · ∇yψdydx = κ

∫Ω

∫ΓR

(π+ pF −Rρ)ψdσydx, (3.8)

for almost every t ∈ S .

Assumptions

We introduce a set of assumptions that allows us to ensure the weak solvabilityand approximation of (P1).

(A1) The domains Ω and Y are convex polygons.

(A2) All model parameters are positive; in particular D,R, pF , κ.

(A3) A > max (Cf , Cπ). The value of Cf and Cπ is given in Section 3.3.

(A4) ρI ∈ L2(Ω;H1(Y )).

(A5) f : S × Ω× Y in (3.1) satisfies the following structural conditions:

(i) f is once continuously differentiable in π and ρ.

(ii) f(s, r) is a contraction in s for all r.

(iii) f(0, r) = 0 for all r.

(iv) There exists a θ > 0 such that f(s, r) = 0 for all s > θ.

(A2) and (A4) are straightforward assumptions related to the physical setting.(A1) is a condition to ease the interaction with the finite element mesh. (A3) and(A5) are technical conditions required to prove well-posedness of the solution.

Before moving on, we remark that the formal shape of the right hand side in(3.1) must be of the form f(π, g(ρ)) for some g : S × Ω× Y → S × Ω.


Technical preliminaries

The rest of the section introduces the notation of the functional spaces andnorms used in this chapter. In addition to the preliminaries discussed in Sec-tion 1.3, in this chapter we require the following concepts.

We useH10(D) to denote the following function space:

H10(D) :=

u ∈ H1(D) : u|∂D = 0

, (3.9)

andH−1(D) to denote the dual space of (3.9), equipped with the norm

||T ||H−1(D) = sup〈T, u〉 : u ∈ H1

0(D), ||u||H10(D) ≤ 1

. (3.10)

By πI ∈ H10(Ω) we denote the solution of

−A∆xπ = f(π, ρI) in Ω,

π = 0 in ∂Ω.(3.11)

(3.11) is a stationary elliptic equation giving access to the value of π from (3.1)at time t = 0.

Finally, we introduce several constants: ci refers to constants from the interpolation-trace theorem (see Lemma 13) and cp to constants arising from Poincare’s in-equality. Moreover, we define the following two constants:

cπ := maxr|∂rf(r, s)|,

cρ := maxs|∂sf(r, s)|.

3.3 Well-posedness of (P1)

In this section we prove that (P1) has a weak solution by approximating it witha Galerkin projection. We show the projection exists and is unique, and proceedby proving it converges to the weak solution of (P1). First, we introduce thenecessary tools for defining the Galerkin approximation.

We use one mesh partition for each of the two spatial scales. LetBH be a meshpartition for Ω consisting of simplices. We denote the diameter of an elementB ∈ BH with HB, and the global mesh size with H := maxB∈BH HB. We intro-duce a similar mesh partition Kh for Y with global mesh size h := maxK∈Kh hK .

Our macroscopic and microscopic finite element spaces VH and Wh are, re-spectively:

VH :=v ∈ L2(Ω)

∣∣ v|B ∈ H1(B) for all B ∈ BH,

Wh :=w ∈ L2(Y )

∣∣ w|K ∈ H1(K) for all K ∈ Kh.

3.3. WELL-POSEDNESS OF (P1) 39

Let 〈ξB〉BH := span(VH) and 〈ηK〉Kh := span(Wh), and let αB, βBK : S → Rdenote the Galerkin projection coefficient for a patch B and B×K, respectively.We introduce the following finite-dimensional Galerkin approximations of thefunctions π and ρ:

πH(t, x) :=∑B∈BH

αB(t)ξB(x),

ρH,h(t, x, y) :=∑

B∈BH ,K∈Kh

βBK(t)ξB(x)ηK(y),(3.12)

where we clamp αB(t) = 0 for all B ∈ BH with ∂B ∩ Ω 6= ∅ to represent themacroscopic Dirichlet boundary condition.

Reducing the space of test functions to V H and W h, we obtain the followingdiscrete weak formulation: find a solution pair

(πH(t, x), ρH,h(t, x, y)) ∈ L2(S;V H)× L2(S;V H ×W h)

that are solutions to

A

∫Ω

∇xπH · ∇xϕdx =

∫Ω

f(πH , ρH,h)ϕdx, (3.13)

and ∫Ω

∫Y

∂tρH,hψdydx+D

∫Ω

∫Y

∇yρH,h · ∇yψdydx

= κ

∫Ω

∫ΓR

(πH + pF −RρH,h)ψdσydx,(3.14)

for any ϕ ∈ VH and ψ ∈ VH ×Wh and almost every t ∈ S.These concepts lead us to the first proposition.

Proposition 6 (Existence and uniqueness of the Galerkin approximation). Thereexists a unique solution (πH , ρH,h) to the system in (3.13)-(3.14).

Proof. The proof is divided in three steps. In step 1, the local existence in timeis proven. In step 2, global existence in time is proven. Step 3 is concerned withproving the uniqueness of the system.

We introduce an integer index for αB(t) and βBK(t) to increase the legibilityof arguments in this proof. Let N1 := 1, . . . , |BH | and N2 := 1, . . . , |Kh|. Weintroduce bijective mappings n1 : N1 → BH and n2 : N2 → Kh, so that each index


j ∈ N1 corresponds to an element B ∈ BH and each index k ∈ N2 correspondsto a K ∈ Kh.

Step 1: local existence of solutions to (3.13) - (3.14): By substituting ϕ = ξi andψ = ξiηk for i ∈ N1 and k ∈ N2 in (3.13)-(3.14) we obtain the following system ofordinary differential equations coupled with algebraic equations.

∑j∈N1

Pijαj(t) = Fi(α, β) for i ∈ N1, (3.15)

β′ik(t) +∑

j∈N1,l∈N2

Qijklβjl(t) = cik +∑j∈N1

Eijkαj for i ∈ N1 and k ∈ N2, (3.16)

with

Pij := A

∫Ω

∇xξi · ∇xξj dx,

Fi :=

∫Ω

f

(∑j∈N1

αj(t)ξj,∑

j∈N1,l∈N2

βjl(t)ξjηl

)ξi dx,

Qijkl := D

∫Ω

ξiξjdx

∫Y

∇yηk · ∇yηldy + κR

∫Ω

ξiξjdx

∫ΓR

ηkηldσy,

Eijk := κ

∫Ω

ξiξjdx

∫ΓR

ηkdσy,

cik := κpF

∫Ω

ξidx

∫ΓR

ηkdσy

(3.17)

Applying (3.6) to (3.13) and (3.14) yields:

αi(0) =

∫Ω

ξiπI dx,

βik(0) =

∫Ω

∫Y

ξiηkρIdydx.

(3.18)

For all t ∈ S, the coefficients αi(t), βik(t) of (3.12) are determined by (3.15),(3.16) and (3.18).

Since the system of ordinary differential equations in (3.16) is linear, we areable to explicitly formulate the solution representation for βik with respect to αi.Let αi be given, and let Q and E denote matrices given by:

Qβ =∑

j∈N1,l∈N2

Qijklβjl, (3.19)

Eα =∑j∈N1

Eijkαj. (3.20)


Then βik can be expressed as

βik(t) = βik(0)e−Qt +Q−1(c+ Eαi)(I − e−Qt). (3.21)

Substituting (3.21) in (3.16) results in the expression:

−Qβik(0)e−Qt + (c+ Eαj)e−Qt +Qβik(0)e−Qt + (c+ Eαj)(I − e−Qt) = (c+ Eαj).

(3.22)(A5) implies that for all i ∈ N1, Fi are contractions. Pick β1(t), β2(t) that satisfy(3.21) given some α1(t), α2(t). Then it holds that:

|Fi(α1, β1)− Fi(α2, β2)|,≤ |Fi(α1, β1)− Fi(α1, β2) + Fi(α1, β2)− Fi(α2, β2)|,

=

∣∣∣∣∣∫

Ω

f

(∑j∈N1

αj,1(t)ξj,∑

j∈N1,l∈N2

βjl,1(t)ξjηl

)ξi

− f

(∑j∈N1

αj,1(t)ξj,∑

j∈N1,l∈N2

βjl,2(t)ξjηl

)ξi dx

∣∣∣∣∣+

∣∣∣∣∣∫

Ω

f

(∑j∈N1

αj,1(t)ξj,∑

j∈N1,l∈N2

βjl,2(t)ξjηl

)ξi

− f

(∑j∈N1

αj,2(t)ξj,∑

j∈N1,l∈N2

βjl,2(t)ξjηl

)ξi dx

∣∣∣∣∣ ,≤

∣∣∣∣∣∫

Ω

cρ∑

j∈N1,l∈N2

(βjl,1(t)− βjl,2(t))ξjηl + cπ∑j∈N1

(αj,1(t)− αj,2(t))ξj dx

∣∣∣∣∣ ,≤ cβ

∣∣∣∣∣∑j∈N1

βjl,1(t)− βjl,2(t)

∣∣∣∣∣+ cα

∣∣∣∣∣∑j∈N1

αj,1(t)− αj,2(t)

∣∣∣∣∣ ,

(3.23)

with cα, cβ defined as

cα := cπ maxj∈N1

∫Ω

ξj dx ≤ cπ, cβ := cρ maxj∈N1,l∈N2

∫Ω

ξjηldx ≤ cρ. (3.24)

Now, we derive a time-dependent continuity estimate for sufficiently small t.Again picking β1(t) and β2(t) (not necessarily the same as in (3.23)):

||β1(t)− β2(t)|| = ||I − eQt|| · ||Q−1D|| · ||α1(t)− α2(t)||,= ||Qt+O

(t2)|| · ||Q−1D|| · ||α1(t)− α2(t)||,

≤ tC||α1 − α2|| for small t.

(3.25)


Using (3.25) we obtain a Lipschitz bound on all Fi in the interval [0, τ ] for anychoice of τ < T :

||Fi(α1(t), β1(t))− Fi(α2(t), β2(t))||≤ ||Fi(α1(t), β1(t))− Fi(α1(t), β2(t))||+ ||Fi(α1(t), β2(t))− Fi(α2(t), β2(t))||,≤ cα||α1(t)− α2(t)||+ cβ||β1(t)− β2(t)||,≤ (cα + cβCτ) ||α1(t)− α2(t)||.

(3.26)

Choosing τ small enough to satisfy cα + cβCτ < 1 makes F a contraction on[0, τ ]. By Banach’s fixed point theorem, it follows that the equationF (α(t), β(t)) =

α(t) has a solution for α in L2(S). Substitution of α(t) into (3.21) leads to the cor-responding β. Existence of πH and ρH,h follows directly.

Step 2: global existence of solutions to (3.13) - (3.14): We cover time intervalS into N intervals of length at most τ such that S ⊆

⋃n((n − 1)τ, nτ ]. From

the arguments in the previous paragraph it is clear a solution exists on the firstinterval [0, τ ]. This allows us to provide an induction argument for the existenceof a solution on interval n:

Given that interval n has local solution β ([(n− 1)τ, nτ ]), we can obtain val-ues β(nτ), β′(nτ), α(nτ) as initial values to the local system on interval n + 1,and show existence of a solution. This way, we are able to construct a solutionsatisfying (3.13) - (3.14) everywhere on S.

Step 3: uniqueness of solutions to (3.13) - (3.14): We decouple the system anduse a fixed point argument to show that this system has a globally unique solu-tion in time.

Let (α1, β1) and (α2, β2) be two solutions satisfying (3.13) - (3.14) with thesame initial data. Let β(t) := β1(t) − β2(t) and α(t) := α1(t) − α2(t). By startingfrom (3.16) and multiplying both equations with β(t), we obtain

〈β(t), β′(t)〉 = 〈Qβ(t), β(t)〉+ 〈Eα(t), β(t)〉,1

2

d

dt||β(t)||2 ≤ ||Q||||β(t)||2 + ||E||||α(t)|| ||β(t)||.

(3.27)

Since β(0) = 0, by applying Gronwall’s differential inequality, we know thatβ(t) ≡ 0. Combined with (3.21), it immediately follows that α(t) ≡ 0, andtherefore, (α1, β1) = (α2, β2).


Note that showing the stability of the finite element approximation with re-spect to data and initial conditions follows an analogous argument. The proofis omitted here.

The remaining part of this section is devoted to proving that the system in(3.13)-(3.14) converges to the weak solution of (P1) (as stated in Proposition 7).To this end, we first formulate the lemmata that help us prove this statement.

Lemma 10 (Aubin-Lions lemma). Let B0 → B ⊂⊂ B1 be Banach spaces, i.e. B0 becompactly embedded in B and B be continuously embedded in B1. Let

W :=u ∈ L2 (S;B0) |∂tu ∈ L2 (S;B1)

. (3.28)

Then the embedding of W into L2 (S;B) is compact.

Lemma 11 (Weak maximum principle). Assume (A1) – (A5): Then πH(t, ·) ∈L∞(Ω) and ρH,h(t, ·) ∈ L∞(Ω× Y ) for all t ∈ (0, T ).

We refer the reader to [2] for the original proof of the statement.

Proof. We use a weak maximum principle according to Stampacchia ([40]). Con-sequently, we test the weak formulation with ϕ = (πH −M1)+ and ψ = (ρH,h −M2)+ for suitable M1 and M2. Assumptions (A3) and (A5) are used in this con-text. From (3.13) we obtain

A

∫Ω

∣∣∇x · πH∇x(πH −M1)+

∣∣ dx =

∫Ω

∣∣f(πH , ρH,h)(πH −M1)+∣∣ dx, (3.29)

The left hand side of (3.29) can be manipulated as:

A

∫Ω

∣∣∇xπH · ∇x(π

H −M1)+∣∣ dx = A

∫Ω

∣∣∇x(πH −M1) · ∇x(π

H −M1)+∣∣ dx,

= A

∫Ω

∣∣∇x(πH −M1)+

∣∣2 ,(3.30)

which can be bounded with the right hand side of (3.29):

A

∫Ω

∣∣∇x(πH −M1)+

∣∣2≤∫

Ω

∣∣f(πH , ρH,h)− f(M1, ρH,h) + f(M1, ρ

H,h)∣∣ (πH −M1)+ dx,

≤∫

Ω

cπ|πH −M1|(πH −M1)+ + |f(M1, ρH,h)|(πH −M1)+ dx,

= cπ

∫Ω

((πH −M1)+

)2dx ≤ cpcπ

∫Ω

(∇x(π

H −M1)+)2dx.

(3.31)


Proceeding similarly with (3.14):∫Ω

∫Y

∂tρH,h(ρH,h −M2)+dydx+

∫Ω

∫Y

D∇yρH,h · ∇y(ρ

H,h −M2)+dydx,

=

∫Ω

∫ΓR

k(πH + pF −RρH,h)(ρH,h −M2)+dσydx,

= κ

∫Ω

∫ΓR

(πH −M1 + pF +M1 −RM2 −R(ρH,h −M2)

)(ρH,h −M2)+dσydx,

= κ

∫Ω

∫ΓR

((πH −M1)+ − (πH −M1)−(ρH,h −M2)+

)(ρH,h −M2)+dσydx

+ κ

∫Ω

∫ΓR

(pF +M1 −RM2)(ρH,h −M2)+dσydx

+ κR

∫Ω

∫ΓR

((ρH,h −M2)+ − (ρH,h −M2)−

)(ρH,h −M2)+dσydx,

≤ κ

∫Ω

∫ΓR

(πH −M1)+(ρH,h −M2)+ − (πH −M1)−(ρH,h −M2)+dσydx

+ κ (pF +M1 −RM2)

∫Ω

∫ΓR

(ρH,h −M2)+dσydx+ κR

∫Ω

∫ΓR

(ρH,h −M2)+2dσydx

− κR∫

Ω

∫ΓR

(ρH,h −M2)−(ρH,h −M2)+dσydx

≤ ε

∫Ω

∫ΓR

(πH −M1)+2dσydx+ κ(R + cε)

∫Ω

∫ΓR

(ρH,h −M2)+2dσydx.

(3.32)

Add (3.31) and (3.32) with ε > 0 small. Applying the trace inequality twice in(3.32) ensures that πH and ρH,h are uniformly bounded if the pair (M1,M2) ischosen such that

M2 = θ,

M1 < Rθ − pF ,

M1 ≥ ||ρI ||L2(Ω×Y ),

θ >pfR.

(3.33)

Lemma 12 (Regularity lift). Let (πH , ρH,h) be a solution to (3.13)-(3.14). Then itmust hold thatπH ∈ L∞((0, T )× Ω) ∩ L∞((0, T );H1

0 (Ω)),

ρH,h ∈ L2((0, T );L2(Ω;H1(Y ))) ∩ L∞((0, T );L∞(Ω;L∞(Y )).(3.34)


Proof. Testing (3.13) with ϕ = πH and (3.14) with ψ = ρH,h yields identities

A∣∣∣∣∇xπ

H∣∣∣∣2L2(Ω)

=

∫Ω

f(πH , ρH,h)πH dx, (3.35)

and

1

2

d

dt

∣∣∣∣ρH,h∣∣∣∣2L2(Ω×Y )+D

∣∣∣∣∇yρH,h∣∣∣∣2L2(Ω×Y )

=

∫Ω

∫ΓR

κ(πH + pF )ρH,h dσy dx− κR∣∣∣∣ρH,h∣∣∣∣2L2(Ω×Y )

.(3.36)

We recall the embeddingH1(Y ) → L2(ΓR), (3.37)

which implies that there exists a cE such that

||u||L2(Ω;L2(ΓR)) ≤ cE||u||L2(Ω;H1(Y )), (3.38)

for all u ∈ L2(Ω;H1(Y )). Using Cauchy-Schwarz’ inequality and (3.38), we canbound the right hand side of (3.36) as∫

Ω

∫ΓR

κ(πH + pF )ρH,h dσy dx ≤ κ|ΓR|(||πH ||L2(Ω) + |Ω|pF

)||ρH,h||L2(Ω×ΓR).

≤ κcE|ΓR|(|πH ||L2(Ω) + pF |Ω|

)||ρH,h||L2(Ω;H1(Y )).

(3.39)

Then, we add to both sides of (3.36) a term D||ρH,h||2L2(Ω×Y ) to get

1

2

d

dt||ρH,h||2L2(Ω×Y ) +D||ρH,h||2L2(Ω;H1(Y )),

≤ D||ρH,h||2L2(Ω×Y ) + κcE|ΓR|(|πH ||L2(Ω) + pF |Ω|

)||ρH,h||L2(Ω;H1(Y )).

(3.40)

After applying Young’s inequality with the small parameter ε > 0, we get

1

2

d

dt||ρH,h||2L2(Ω×Y ) + (D − ε)||ρH,h||2L2(Ω;H1(Y )),

≤ D||ρH,h||2L2(Ω×Y ) + κ2c2Ecε|ΓR|2

(||πH ||2L2(Ω) + p2

F |Ω|2).

(3.41)

By applying Gronwall’s inequality we obtain the desired estimates:

||ρH,h||2L2(Ω×Y ) ≤ CρeDt, (3.42)

||∇ρH,h||2L2(Ω×Y ) ≤ Cρ + ε||ρH,h||L2(Ω×Y ), (3.43)


with

Cρ = κ2c2Ecε|ΓR|2

(||πH ||2L2(Ω) + p2

F |Ω|2).

To obtain a bound on πH , we test (3.13) with πH and then use (A3) and (A5):

A||∇xπH ||2 =

∫Ω

f(πH , ρH,h)πH dx

=

∫Ω

(f(πH , ρH,h)− f(0, ρH,h) + f(0, ρH,h)

)πHdx,

≤

√∫Ω

(cρπH)2 dx||πH ||L2(Ω) ≤ cπ||πH ||L∞(Ω)||∇xπH ||L2(Ω).

(3.44)

This yields the upper bound

||∇xπH ||L2(Ω) ≤

cπA||πH ||L∞(Ω). (3.45)

With these lemmata we are ready to state and prove a first convergence re-sult.

Proposition 7 (Convergence of the Galerkin approximation). Let (πH , ρH,h) ∈L2(S;H1

0(Ω)) × L2(S;L2(Ω;H1(Y ))) be a solution to (3.13)-(3.14) and let (π, ρ) bethe weak solution to (P1). Then

πH → π,

ρH,h → ρ,(3.46)

for H, h→ 0.

Proof. To apply Lemma 10, we first need to show

∂tπH ∈ L2((0, T );L2(Ω)). (3.47)

Then, by choosing

B0 = H10(Ω),

B1 = B = L2(Ω),(3.48)

we satisfy the requirements of Aubin-Lions’ lemma to get convergence of πH inL2(Ω).


Concerning ρH,h, again we use Aubin-Lions’ lemma to prove convergence,this time for the following spaces:

B0 = H10(Ω;H1(Y )),

B1 = B = L2(Ω;L2(Y )),(3.49)

Note that ρH,h ∈ L2((0, T );B). To conclude the argument, what remains to showare the following steps:

∂tρH,h ∈ L2((0, T );L2(Ω;L2(Y )), (3.50)

and∇xρ

H,h ∈ L2(((0, T );L2(Ω;H1(Y ))). (3.51)

We start with the estimates that provide us (3.47) and (3.50) and then we handle(3.51).

Let fπ and fρ denote the partial derivatives of f to respectively π and ρ. Weintroduce uH := ∂tπ

H and vh := ∂tρH,h− ρI . We differentiate (P1) with respect to

t and obtain the following system

−A∆xuH = fρv

h + fπuH in S × Ω, (3.52)

∂tvh = D∆yv

h in S × Ω× Y, (3.53)

D∇yvh = κ(uH −Rv) on S × Ω× ΓR, (3.54)

D∇yvh = 0 on S × Ω× ΓN , (3.55)

uH = 0 on S × ∂Ω (3.56)

v(t = 0) = 0 on S × Ω× Y (3.57)

By multiplying (3.53) with vh and integrating the result over Ω × Y , we obtainan equation that can be bounded similarly to (3.39). This gives:

1

2

d

dt||vh||2L2(Ω×Y ) +D||∇yv

h||2L2(Ω×Y )

= κ

∫Ω×ΓR

(uH −Rvh)vh,

≤ κ

∫Ω×ΓR

uHvh − κR∫

Ω×ΓR

(vh)2,

≤ κcE||uH ||L2(Ω;H1(Y ))||vh||L2(Ω;H1(Y )),

≤ κcE

(cε|ΓR|||uH ||2L2(Ω) + ε||vh||2L2(Ω;H1(Y ))

),

≤ κcE

(cε|ΓR|||uH ||2L2(Ω) + ε(cp + 1)||∇yv

h||2L2(Ω×Y )

).

(3.58)


From (3.58), by integration in time we obtain

||∂tρH,h||2L2(Ω×Y ) ≤ cε|ΓR|||∂tπH ||2L2(Ω). (3.59)

By multiplying (3.52) with vh and integrating the result over Ω, we obtain

A||∇xuH ||2L2(Ω) =

∫Ω

(fπu

H + fρvh)uH ≤ c1||uH ||2L2(Ω) + c2||vh||2L2(Ω),

which, by combining this inequality with Poincare’s inequality for uH , yields

||uH ||2L2(Ω) ≤ cp||∇uH ||2L2(Ω). (3.60)

Thus, we obtain an upper bound that holds in the interior of Ω, say Ωδ.(A

cp− c1

)||∂tπH ||2L2(Ωδ)

≤ c2||∂tρH,h||2L2(Ωδ×Y ). (3.61)

Here, Ωδ is defined as any subset of Ω such that d(∂Ω,Ωδ) ≥ δ, (d(·, ·) measuresthe distance between two sets) and where δ > 0.

To handle the integral estimates on the boundary layer Ωδ\Ω, we recall (3.14).For any Ωδ ⊂ Ω, by testing with ψ = ∂tρ

H,h and integrating over the time do-main, we obtain the identity

2

∫ T

0

||∂tρH,h||2L2(Ωδ×Y ) + ||∇yρH,h||2L2(Ωδ×Y ) − ||∇yρ

H,hI ||

2L2(Ωδ×Y )

= 2κ

[∫Ωδ×ΓR

πHρH,h]T

0

− 2κ

∫ T

0

∫Ωδ×ΓR

ρH,h∂tπH

+ κpf

∫Ωδ×ΓR

ρH,hI + κ||ρH,h||2L2(Ωδ×ΓR).

(3.62)

Conveniently rearranging the terms of (3.62) yields:

2

∫ T

0

||∂tρH,h||2L2(Ω\Ωδ) + ||∇yρH,h||2L2(Ω\Ωδ×Y ) (3.63)

≤ 2κ||πH∂tρH,h||L1(Ω\Ωδ×ΓR) + c||ρH,hI ||2L2(Ω\Ωδ×γR) + c||ρH,h||2L2(Ω\Ωδ×γR) (3.64)

+ ||∇yρH,hI ||

2L2(Ω\Ωδ×Y ) + 2κ

∫ T

0

∫Ω\Ωδ×ΓR

|ρH,h∂tπH | ≤ Cδ. (3.65)

Now we can extend the bound in (3.61) to hold on the entire domain Ω, i.e.:(A

cp− c1

)||∂tπH ||2L2(Ω) = sup

δ>0

(A

cp− c1

)||∂tπH ||2L2(Ωδ)

≤ C2||∂tρH,h||2L2(Ω). (3.66)


Finally, to obtain (3.51) we adapt an interior regularity argument from [18],Chapter 6. We let Ωδ ⊂⊂ W ⊂⊂ Ω and define a smooth cutoff function ζ : Ω →[0, 1] satisfying ζ(x) = for x ∈ Ωδ,

ζ(x) = for x ∈ Ω \W.(3.67)

We introduce the directional finite difference

Dλi ρ

H,h :=ρH,h(t, x+ λei, y)− ρH,h(t, x, y)

λfor λ > 0.

for i ∈ 1, ..., d2. We let λ be small and we test (3.14) with

ψ = −D−λi ζ2Dλi ρ,

which gives us:

−∫∂tρD

−λi ζ2Dλ

i ρ−D∫∇yρ · ∇yD

−λi ζ2Dh

i ρ = −κ∫

(π + pF −Rρ)D−λi ζ2Dλi ρ.

(3.68)Because of the properties of the support of ζ , it holds that for any f ∈ Ω∫

Ω

ψD−λi f = −∫

Ω

fDλi ψ. (3.69)

Applying the property in (3.69) to (3.68) yields∫Ω×Y

ζ2Dλi ∂tρ

H,hDλi ρ

H,h +D

∫Ω×Y

ζ2Dλi∇yρ

H,h ·Dλi∇yρ

H,h

= κ

∫Ω×ΓR

ζ2Dλi (πH + pF −RρH,h)Dλ

i ρH,h,

(3.70)

leading to

1

2

d

dt

∫Ω×Y

∣∣ζDλi ρ

H,h∣∣2 +D

∫Ω×Y

∣∣ζDλi∇yρ

H,h∣∣2

= κ

∫Ω×ΓR

ζ2Dλi π

HDλi ρ

H,h − κR∫

Ω×ΓR

∣∣ζDλi ρ

H,h∣∣2 . (3.71)

Using Young’s inequality combined with the interpolation-trace inequality, we


estimate the third term of (3.71) as follows:

κ

∫Ω×ΓR

ζ2Dλi π

HDλi ρ

H,h

≤ κ|ΓR| ||Dλi π

H ||L2(Ω) ||ζDλi ρ

H,h||L2(Ω×ΓR),

≤ cεκ|ΓR| ||Dλi π

H ||2L2(Ω) + ε||ζDλi ρ

H,h||2L2(Ω×ΓR),

≤ Cεκ|ΓR| ||∇xπH ||2L2(Ω) + ε||ζDλ

i ρH,h||Ω×Y ||ζDλ

i∇yρH,h||L2(Ω×Y ),

≤ Cεκ|ΓR| ||∇xπH ||2L2(Ω) +

ε2

2||ζDλ

i ρH,h||Ω×Y

+ε2

2||ζDλ

i∇yρH,h||L2(Ω×Y ).

(3.72)

Now, combining (3.72) with (3.71), we obtain the required estimate for (3.51):

1

2

d

dt

∫Ω×Y

∣∣ζDλi ρ

H,h∣∣2 +

(D − ε2

2

)∫Ω×Y

∣∣ζDλi∇yρ

H,h∣∣2

≤ Cεκ|ΓR| ||∇xπH ||2Ω +

ε2

2||ζDλ

i ρH,h||Ω×Y .

(3.73)

Using Gronwall’s inequality, we conclude that Dλi ρ

H,h ∈ L2(Ω × Y ), and byletting λ→ 0, we obtain

∇xρH,h, ∇x∇yρ

H,h ∈ L2(S × Ω× Y ). (3.74)

With the newly found estimates (3.66), (3.74) and (3.59), we are able to applyLemma 10 and we obtain that

W ⊂⊂ L2(S × Ω× Y ),

which proves:(πH , ρH,h)→ (π, ρ).

for h,H → 0.

The preliminary work allows us to state the first main result of this chapter.

Theorem 5 (Well-posedness of the system). The system in (3.13)-(3.14) has a uniquesolution πH ∈ L2(S;V H) and ρH,h ∈ L2(S;V H ×W h).

Proof. The proof of this theorem is a direct result of Proposition 6 and Proposi-tion 7.

3.4. CONVERGENCE RATES FOR SEMIDISCRETE GALERKINAPPROXIMATIONS 51

3.4 Convergence rates for semidiscrete Galerkinapproximations

In this section, we obtain convergence rates of the numerical approximations(3.13) – (3.14). The following argument is largely based on standard argumentsfrom [25], adapted to multiscale systems.

Proposition 8 (Regularity lift). Recall (A4) and (A1). If (πH , ρH,h) is a solution to(3.7)-(3.8), then

πH ∈ L2(S;H2(Ω)),

ρH,h ∈ L2(S;H2(Ω;H2(Y ))).

Proof. We omit the proof and refer to [20].

Lemma 13 (Interpolation-trace inequality). Let u ∈ L2(Ω;L2(ΓR)), and let ΓR (∂Y . Then

||u||2L2(Ω;L2(ΓR)) ≤ ε||∇yu||2L2(Ω;L2(Y )) + ci(cε + 1)||u||2L2(Ω;L2(Y )), (3.75)

with trace constant ci independent of ε and cε = (√

2ε)−1.

Proof. The proof follows from applying Young’s inequality with a small param-eter ε to the standard trace inequality.

LetRh andRH be the microscopic and macroscopic Ritz projection operatorrespectively.

Lemma 14 (Projection error estimates). Then there exists strictly positive constantsγl (with l ∈ 1, 2, 3, 4), independent of h and H , such that projections Rhπ and RHρ

that satisfy

||π −RHπ||L2(Ω) ≤ γ1H2||π||H2(Ω), (3.76)

||π −RHπ||H1(Ω) ≤ γ2H||π||H2(Ω), (3.77)

||ρ−RHRhρ||L2(Ω;L2(Y )) ≤ γ3(H2 + h2)||ρ||L2(Ω;H2(Y ))∩L2(Y ;H2(Ω)), (3.78)

for all (π, ρ) ∈ H2(Ω)× [L2(Ω;H2(Y )) ∩ L2(Y ;H2(Ω))].

Proof. (3.76) and (3.77) are standard Ritz projection error estimates. For detailson the proof, see for instance [41] and [25]. Specific to this context, (3.78) is atwo-scale estimate which accounts for the presence of the microscopic Robin


boundary condition (3.3) and therefore requires some tuning. See e.g. [32] forsimilar estimates. Here, we only present the proof of (3.78).

Let ω := Rhρ− ρ. Let ϕ ∈ L2(Ω;H2(Y )) be the weak solution to

(P2)

−∆ϕ = ω in Ω× Y,

−∇ϕ · n = αϕ on Ω× ΓR,

−∇ϕ · n = 0 on Ω× ΓN .

(3.79)

We denote the Ritz projection error of ϕ with eϕ. By testing with ψ and integrat-ing over Ω× Y , we obtain

〈ω, ψ〉L2(Ω;L2(Y )) = 〈∇ϕ,∇ψ〉L2(Ω;L2(Y )) + 〈∇ϕ · n, ψ〉L2(Ω;L2(ΓR)). (3.80)

Testing with ψ = ω specifically, subtracting the Galerkin approximation fromthe weak solution and using (Rh∆ϕ, ω) = 0 we obtain:

||ω||2L2(Ω;L2(Y ))

= 〈∇ϕ,∇ω〉L2(Ω;L2(Y )) + 〈αϕ, ω〉L2(Ω;L2(ΓR)),

= 〈∇eϕ,∇ω〉L2(Ω;L2(Y )) + 〈αeϕ, ω〉L2(Ω;L2(ΓR)),

≤ cε ||∇eϕ||L2(Ω;L2(Y )) ||∇ω||L2(Ω;L2(Y )) + ε ||eϕ||L2(Ω;L2(Y )) ||ω||L2(Ω;L2(Y )) .

Applying the Ritz projection estimates (3.76) and (3.77), we obtain the followingbound:

||ω||2L2(Ω;L2(Y )) ≤ cεh2||ϕ||2L2(Ω;H2(Y )) + εch2||ω||L2(Ω;L2(Y )).

Using Friedrich’s inequality ||ϕ||L2(ΩH2(Y )) ≤ C||∆ϕ||L2(Ω;L2(Y )) = C||ω||L2(Ω;L2(Y ))

for some C and choosing ε < c we obtain

(1− ε)||ω||2L2(Ω;L2(Y )) ≤ Ch2||ω||L2(Ω;L2(Y )). (3.81)

(3.81) yields:||ω||L2(Ω;L2(Y )) = ||Rhρ− ρ||L2(Ω;L2(Y )) ≤ γ3h

2. (3.82)

Finally, we can derive (3.78) as follows:

||ψ −RHRhψ||L2(Ω;L2(Y )) = ||ψ −Rhψ +Rhψ −RHRhψ||L2(Ω;L2(Y )),

≤ ||ψ −Rhψ||L2(Ω;L2(Y )) + ||Rhψ −RHRhψ||L2(Ω;L2(Y )),

≤ γ3h2||ψ||L2(Ω;H2(Y )) + γ4H

2||Rhψ||L2(Y ;H2(Ω)),

≤ γ3(H2 + h2)||ψ||L2(Ω;H2(Y ))∩L2(Y ;H2(Ω)).

(3.83)


By applying Lemma 13 and Lemma 14, we can finally obtain the desiredconvergence rates. Let us denote the errors of the Galerkin projection as

eπ := π − πH ,eρ := ρ− ρH,h.

Theorem 6 (Convergence rates). Let (πH , ρH,h) be a solution to (3.7)-(3.8). Then thefollowing statement holds: there exists constants M1,M2 > 0 independent of h and H ,such that

||eπ||L∞((0,T );L20(Ω)) ≤ C(H2 + h2), (3.84)

||eρ||L∞((0,T );L2(Ω;L2(Y ))) ≤ C(H2 + h2). (3.85)

Proof. We denote the Ritz projection of the solution with

(pH , rH,h) := (RHπ,RHRhρ).

We choose test functions

(ϕ, ψ) = (pH − πH , rH,h − ρH,h).

By testing (3.7) with π and πH and testing (3.13) with πH , we obtain the fol-lowing identities:

A

∫Ω

∇xπ · ∇xπdx =

∫Ω

f(π, ρ)πdx, (3.86)

−A∫

Ω

∇xπ · ∇xπHdx =

∫Ω

f(π, ρ)πHdx, (3.87)

+A

∫Ω

∇xπ · ∇xπHdx =

∫Ω

f(π, ρ)πHdx, (3.88)

−A∫

Ω

∇xπH · ∇xπ

Hdx =

∫Ω

f(πH , ρH)πHdx. (3.89)

which, summed up, results in the following identity:

A

∫Ω

∇xeπ · ∇xeπ =

∫Ω

f(π, ρ)eπ −(f(π, ρ)− f(πH , ρH,h)

)πHdx. (3.90)

Let ϕH ∈ V H be arbitrary. By applying the identity

A

∫Ω

∇xπH · ∇xϕ

H −∇xπ · ∇xϕHdx =

∫Ω

(f(π, ρ)− f(πH , ρH,h)

)ϕHdx, (3.91)


and using (3.90), we obtain the following estimate (omittingL2 norm indicationsin their respective spaces for clarity):

A

cp||eπ||2 ≤ A ||∇xeπ||2

= A

∫Ω

∇x(π − πH) · ∇x(π − πH)dx

= A

∫Ω

∇x(π − πH) · ∇x(π − ϕH)dx

+ A

∫Ω

∇x(π − πH) · ∇x(ϕH − πH)dx,

≤ A ||∇xeπ||∣∣∣∣∇x(π − ϕH)

∣∣∣∣+

∫Ω

(f(π, ρ)− f(πH , ρH,h)(ϕH − πH)dx,

≤ A ||∇xeπ||∣∣∣∣∇x(π − ϕH)

∣∣∣∣+

∣∣∣∣∫Ω


)(ϕH − π)dx

∣∣∣∣+

∣∣∣∣∫Ω


)eπdx

∣∣∣∣ ,≤ A

2

(||∇xeπ||2 +

∣∣∣∣∇x(π − ϕH)∣∣∣∣2)

+ (cπ ||eπ||+ cρ ||eρ||)∣∣∣∣ϕH − π∣∣∣∣+ cρ ||eρ|| ||eπ||+ cπ ||eπ||2 .

(3.92)

Moving the first and last term of (3.92) to the left hand side, we obtain the fol-lowing inequality, which can be further bounded as follows:

(A

2cp− cπ

)||eπ||2 ≤

A

2

∣∣∣∣∇x(π − ϕH)∣∣∣∣2 + (cπ ||eπ||+ cρ ||eρ||)

∣∣∣∣ϕH − π∣∣∣∣+ cρ ||eρ|| ||eπ||

≤ A

2

∣∣∣∣∇x(π − ϕH)∣∣∣∣2 + 2ε ||eπ||2 +

(c2ρ

2+c2ρ

4ε

)||eρ||2

+

(c2ρ

2+c2π

4ε

) ∣∣∣∣ϕH − π∣∣∣∣2(3.93)

Finally, by compensating the small terms and using the finite element approxi-mation property:

minχ∈V H

∣∣∣∣ϕH − χ∣∣∣∣H1(Ω)≤ CH||ϕ||H2(Ω), (3.94)


applied to (3.93) we obtain the inequality

(A

2cp− cπ − 2ε

)||eπ||2L2(Ω) ≤

(c2ρ

2+c2ρ

4ε

)||eρ||2L2(Ω;L2(Y ))

+ max

c2ρ

2+c2π

4ε,A

2

||ϕH − π||2H1(Ω)

≤(c2ρ

2+c2ρ

4ε

)||eρ||2L2(Ω;L2(Y )) + CH2 ||π||2H2(Ω) .

(3.95)

with C a generic constant independent of H .

Continuing, from (3.14), we get

eρ = ρH,h − ρ = (ρH,h −RHRhρ) + (RHRhρ− ρ) =: θ + ψ. (3.96)

We bound ψ by using Lemma 14:

||ψ(t)||L2(Ω;L2(Y )) ≤ γ3(H2 + h2)||ρ||L2(Ω;H2(Y ))∩L2(Y ;H2(Ω)),

= γ3(H2 + h2)

∣∣∣∣∣∣∣∣ρI +

∫ t

0

∂tρds

∣∣∣∣∣∣∣∣L2(Ω;H2(Y ))∩L2(Y ;H2(Ω))

,(3.97)

and bound θ from (3.96) using the formulation: for all ϕ ∈ V h we have that

〈∂tθ, ϕ〉L2(Ω;L2(Y )) +D〈∇θ,∇ϕ〉L2(Ω;L2(Y ))

= −〈Rh∂tρ, ϕ〉L2(Ω;L2(Y )) −D〈∇ρ,∇ϕ〉L2(Ω;L2(Y )),

= 〈∂tρ−Rh∂tρ, ϕ〉L2(Ω;L2(Y )),

= 〈∂tψ, ϕ〉L2(Ω;L2(Y )).

(3.98)

Substituting ϕ = θ in (3.98) yields:

1

2

d

dt||θ||2L2(Ω;L2(Y )) +D||∇θ||2L2(Ω;L2(Y ))

= (∂tρ−RHRh∂tρ, θ) ,

≤ ||∂tρ−RHRh∂tρ||L2(Ω;L2(Y )) ||θ||L2(Ω;L2(Y )) ,

≤ γ3(h2 +H2)||∂tρ||L2(Ω;H2(Y ))||θ||L2(Ω;L2(Y )).

(3.99)


Dividing the left and right hand side of (3.99) by ||θ||, we obtain:

d

dt||θ||L2(Ω;L2(Y )) ≤ γ3(h2 +H2)||∂tρ||L2(Ω;H2(Y )),

||θ(t)||L2(Ω;L2(Y )) ≤ ||θ(0)||L2(Ω;L2(Y )) + γ3(h2 +H2)

∫ t

0

||∂tρ||L2(Ω;H2(Y ))dx,

≤ ||ρH,hI − ρI ||L2(Ω;L2(Y )) + ||ρI −RHRhρI ||L2(Ω;L2(Y ))

+ γ3(h2 +H2)

∫ t

0

||∂tρ||L2(Ω;H2(Y ))dx,

≤ γ3(h2 +H2)

(cI + C +

∫ t

0

||∂tρ||L2(Ω;H2(Y ))dx

).

(3.100)

Because of (A4), the Galerkin projection error of the initial condition satisfies:

||ρI − ρH,hI ||L2(Ω;L2(Y )) ≤ cI(H2 + h2). (3.101)

Combining (3.97) and (3.100) proves the desired estimate in (3.85).

||ρH,h − ρ||L2(Ω;L2(Y )) = ||θ + ψ||L2(Ω;L2(Y )) ≤ C(H2 + h2)||∂tρ||L2(Ω;H2(Y )). (3.102)

Finally, (3.84) follows by combining (3.102) with (3.95).

3.5 A posteriori mesh refinement strategy

In this section we develop a computable error estimator which we will use torefine the finite element grid and obtain a lower overall error for the macro-scopic equation (3.13). In this strategy, we aim for reliable error estimators, i.e.estimators which provide an upper and lower bound on the error.

There is a difference in usability between (a priori) error bounds and (a pos-teriori) error estimators. Where error bounds guarantee the upper bound of theerror, often they are not sharp, not computable, and mainly useful for provingwell-posedness of the numerical approximation. Error estimators, on the otherhand, should be computable quantities approximating the true value of the er-ror, and preferably provide both an upper bound and an lower bound on theerror. An upper bound is required to guarantee the maximum error satisfies acertain tolerance. A lower bound makes sure that the error estimator does notoverestimate the true error too much, ensuring the efficiency of the refinementstrategy.

For a review on the different strategies in error control, we refer to e.g. [5]and [19]. The line of arguments we present, is based on [43].

3.5. A POSTERIORI MESH REFINEMENT STRATEGY 57

In this section, we only describe how to obtain error estimators for the macro-scopic equation. Strategies to obtain error estimators for parabolic equations canbe found in e.g. [3], [35], [17] and [16].

Mesh-related notation

We assume the mesh partition BH with diameter H as defined in Section 3.3.For each element B ∈ BH , we denote the set of vertices with V(B) and the setof edges with E(B). The complete set of edges is denoted with E . Where nec-essary, we differ between vertices (edges) in Ω and on ∂Ω by denoting them asBH,Ω (VH,Ω) and BH,∂Ω (VH,∂Ω), respectively. To denote patches in Ω with certainstructures, we use the following symbols:

• ωB denotes the union of all elements that share an edge with B.

• ωB denotes the union of all elements that share a point with B.

• ωE denotes the union of all elements adjacent to E.

• ωE denotes the union of all elements that share a point with B.

• ωx denotes the union of all elements that have x as a vertex.

Furthermore, for any E ∈ E , nE denotes the unit vector orthogonal to E andJE(v) denotes the jump across E of some piece-wise continuous function v inthe direction of nE .

Finally, for legibility we use the following notation to refer to norms on ele-ments or edges.

||u||B := ||u||L2(B) ,

||u||E := ||u||L2(E) .

Auxiliary results

For any x ∈ V(B), we denote the piece-wise linear basis function that takes value1 in x and 0 in the other nodes by λx. This allows us to define the following cutofffunctions for any B ∈ BH :

ψB := αBΠx∈V(B), (3.103)

and for any E ∈ EψE := αEΠx∈V(E). (3.104)


Here, the coefficients αE , αB are chosen such that:

maxx∈B

ψB(x) = maxx∈E

ψE(x) = 1. (3.105)

With the cutoff functions defined in (3.103) and (3.104), we can obtain thefollowing bounds for any v ∈ H1(B) and ϕ ∈ H1(E):

θ1 ||v||2B ≤∫B

ψBv2dx ≤ ||v||2B ,

||φBv||H1(B)) ≤θ2

HB

||v||B,

θ3 ||ϕ||2E ≤∫E

ψEϕ2dx ≤ ||ϕ||2E ,

||ψEϕ||H1(ωE)) ≤θ4√H||ϕ||E ,

||ψEϕ||L2(ωE)) ≤ θ5

√H ||ϕ||E ,

(3.106)

for any element B ∈ B, edge E ∈ E .In order to obtain error estimates on both elements and edges, we introduce

a quasi-interpolation operator IH defined as

IHϕ :=∑x∈VH

λx

∫ωxϕdx

|ωx|. (3.107)

This allows for the following estimates:

||v − IHv||B ≤ cI1HB ||v||H1(ωB)) ,

||v − IHv||E ≤ cI2√H ||v||H1(ωE)) .

(3.108)

We omit the derivation of (3.106) and (3.108) and refer the reader to [10] andSection 3.1 of [42], respectively.

Macroscopic error estimator

The error estimator is composed from the jump discontinuities on each edgeand the residuals in each patch. Formally defined, for any B ∈ BH and E ∈ E ,we define

RB(πH) := A∆πH + f(πh, ρH,h), B ∈ B,

RE(πH) :=

−JE(nE · A∇uh), E ∈ EH,Ω,

0 , E ∈ EH,∂Ω,

(3.109)


which can be combined into a complete residual operator R : VH → R, definedimplicitly in the following identity:

〈R(πH), ϕ〉 =∑B∈BH

∫B

RB(πH)ϕdx+∑E∈EH

∫E

RE(πH)ϕdx. (3.110)

(3.110) must hold for all ϕ ∈ VH .Recall eπ = π − πH denotes the finite element error of π. There is an equiv-

alence between the norm of residual R and the norm of true error. Subtracting(3.13) from (3.7), we obtain∫

Ω

∇eπ · ∇ϕdx =

∫Ω

f(π, ρ)ϕdx−∫

Ω

A∇πH · ∇ϕdx,

=∑B∈BH

∫B

f(π, ρ)ϕ− A∇πH · ∇ϕdx,

=∑B∈BH

∫B

f(π, ρ)ϕ+ A∆πHϕdx−∫∂B

AϕnB · ∇πHdx,

=∑B∈BH

∫B

(f(π, ρ) + A∆πH

)dx+

∑E∈E

∫E

JE(−nE · ∇πH)ϕdx,

= 〈R(πH), ϕ〉.(3.111)

Picking a suitable c∗ and c1, by applying Poincare’s inequality and Cauchy-Schwarz’ inequality shows that:

c∗ ||eπ||2H1(Ω) ≤ ||∇eπ||2L2(Ω) =

∫Ω

∇eπ · ∇eπdx ≤ ||R(πH)||2H−1(Ω). (3.112)

On the other hand, recalling (3.10) and picking ϕ such that

〈R(πH), φ〉 = ||R(πH)||H−1(Ω),

we obtain

〈R(πH), ϕ〉 ≤ ||∇eπ||L2(Ω) ||∇ϕ||L2(Ω) ,

≤ c∗ ||eπ||H1(Ω) ||ϕ||H1(Ω) ,

= c∗ ||eπ||H1(Ω) ,

(3.113)

showing equivalence of the norms.


Although (3.110) is a reliable estimator, it is not a computable quantity. There-fore, we introduce a new quantity ηR, which is based only on computable values:

η2R,B := H2

B

∣∣∣∣RBπH∣∣∣∣2L2(Ω)

+∑

E∈E(B)

βEH∣∣∣∣RE(πH)

∣∣∣∣2E,

η2R :=

∑B∈B

η2R,B.

(3.114)

Here, βE = 12

if E ∈ EΩ, and βE = 1 if E ∈ E∂Ω.

Theorem 7 (Reliable error estimation). Assume (A1) and (A5) For every t ∈ S, theerror norm ||eπ|| can be approximated by the error estimator ηR. The following boundshold:

c∗ ||eπ||H1(Ω) ≤ ηR ≤ c∗ ||eπ||H1(Ω) +O(H, h2

). (3.115)

Proof. For any ϕ ∈ V H with ||φ||L2(Ω) = 1, the following estimate holds:

〈R(πH), ϕ〉 = 〈R(πH), ϕ− IHϕ〉,

=∑B∈BH

∫B

RB(πH)(ϕ− IHϕ) +∑E∈E

∫E

RE(πH)(ϕ− IHϕ),

=∑B∈BH

∣∣∣∣RB(πH)∣∣∣∣B||ϕ− IHϕ||B +

∑E∈E

∣∣∣∣RE(πH)∣∣∣∣E||ϕ− IHϕ||E ,

≤∑B∈BH

cI1H∣∣∣∣RB(πH)

∣∣∣∣B||ϕ||H1(ωB) +

∑E∈E

cI2√H∣∣∣∣RE(πH)

∣∣∣∣E||ϕ||H1(ωE) ,

≤ max(cI1, cI2)

(∑B∈BH

H2∣∣∣∣RB(πH)

∣∣∣∣2B

+∑E∈E

H∣∣∣∣RE(πH)

∣∣∣∣2E

) 12

,

×

(∑B∈BH

||ϕ||2H1(ωB) +∑E∈E

||ϕ||2H1(ωE)

),

≤ c

(∑B∈BH

H2∣∣∣∣RB(πH)

∣∣∣∣2B

+∑E∈E

H∣∣∣∣RE(πH)

∣∣∣∣2E

) 12

||ϕ||H1(Ω) ,

≤ c∑R

ηR.

(3.116)

This provides us with the first inequality in (3.115).The second inequality requires some auxiliary definitions. Let fH denote the

Galerkin projection of f . We use RB and RE to denote the residuals where f is


replaced with fH . Additionally, we introduce

wB = ψBRB(πH),

wE = ψERE(πH),(3.117)

to conveniently manipulate the residual norm. Finally, let Cf be a constant de-fined as

Cf :=C

θ1

(||π||H2(Ω) + ||ρ||L2(Ω;H2(Y ))

). (3.118)

Here, C is a constant independent of π, ρ, h and H .

Next, we bound each of the terms of ηR to obtain a lower bound for the error.

θ1

∣∣∣∣∣∣RB(πH)∣∣∣∣∣∣2B

≤∫B

RB(πH)2ψB =

∫B

RB(πH)wB

= 〈R(πH), wB〉+

∫B

(fH(πH , ρH,h)− f(π, ρ))wB,

=

∫B

∇eπ · ∇wB +

∫B

(fH(πH , ρH,h)− f(π, ρ))wB,

≤ c∗ ||eπ||H1(B) ||wB||H1(B) +∣∣∣∣fH(πH , ρH,h)− f(π, ρ)

∣∣∣∣B||wB||B ,

≤ c∗θ2

H

∣∣∣∣∣∣RB(πH)∣∣∣∣∣∣B||eπ||H1(B) +

∣∣∣∣fH(π, ρ)− f(π, ρ)∣∣∣∣B

∣∣∣∣∣∣RB(πH)∣∣∣∣∣∣B

+ θ1Cf (H2 + h2)

∣∣∣∣∣∣RB(πH)∣∣∣∣∣∣B.

(3.119)

Dividing (3.119) once by its common factor and rearranging terms results in

H∣∣∣∣∣∣RB(πH)

∣∣∣∣∣∣B≤ c∗θ2

θ1

||eπ||H1(B) +1

θ1

H∣∣∣∣fH(πH , ρH,h)− f(π, ρ)

∣∣∣∣B

+HCf (H2 + h2),

(3.120)

which, after applying the triangle inequality results in

H∣∣∣∣RB(πH)

∣∣∣∣B≤ c∗θ2

θ1

||eπ||H1(B) +

(1 +

1

θ1

)H∣∣∣∣fH(πH , ρH,h)− f(π, ρ)

∣∣∣∣B

+HCf (H2 + h2),

(3.121)

To provide an upper bound on the second term of (3.115), we use the equiv-


alence of the error and residual norms:

θ3

∣∣∣∣RE(πH)∣∣∣∣2E,

≤∫E

RE(πH)wE,

= 〈R(πH), wE〉 −∑B∈ωE

∫B

RB(πH))wE,

=

∫ωE

∇eπ · ∇wE −∑B∈ωE

∫B

RB(πH))wE,

≤ c∗ ||eπ||H1(ωE) ||wE||H1(ωE) +∑B∈ωE

∣∣∣∣RB(πH))∣∣∣∣B||wE||B ,

≤ c∗θ4√H||eπ||H1(ωE)

∣∣∣∣RE(πH)∣∣∣∣E

+∑B∈ωE

θ5

√H∣∣∣∣RB(πH)

∣∣∣∣B

∣∣∣∣RE(πH)∣∣∣∣E.

(3.122)

Dividing both sides of (3.122) by its common factor results in:

√H∣∣∣∣RE(πH)

∣∣∣∣E≤ c∗θ4

θ3

||eπ||H1(ωB) +∑B∈ωE

θ5

√H∣∣∣∣RB(πH)

∣∣∣∣B. (3.123)

Combining (3.122) with (3.123) results in

√H∣∣∣∣RE(πH)

∣∣∣∣E≤(c∗θ4

θ3

+c∗θ5θ2

θ3θ1

)||eπ||H1(ωE)

+θ5

θ3

(1 +

1

θ1

) ∑B∈ωE

∣∣∣∣fH(πH , ρH,h)− f(π, ρ)∣∣∣∣B

+θ5

θ3

Cf (H2 + h2),

(3.124)

which yields the following lower bound on the error:

ηR,B ≤ c

(||eπ||H1(ωE) +H

∑B′∈ωB

∣∣∣∣fH(πH , ρH,h)− f(π, ρ)∣∣∣∣B′

+O(H2, h2

)).

(3.125)

We remark that the presence of H and h2 error terms is a typical feature oftwo-scale models.


Macroscopic mesh refinement strategy

The strong separation of space scales in our setting (microscopic vs. macro-scopic) allows us to propose a macroscopic mesh refinement strategy only weaklybiased by the distribution of microscopic errors, based on local error indicatorηR,B and global error indicator ηR. Inspired by the popular and intuitive ap-proach presented in e.g. [45], the goal of this refinement is to reduce the globalapproximation error and keep the error locally below a prescribed tolerance, i.e.select a refinement strategy satisfying

ηR < η,

where we denote the desired tolerance with η.Our refinement strategy relies on the double sided estimate (3.115) stated in

Theorem 7. This inequality gives us a satisfied upper bound on the global error.We solve (3.13) on BH for some t, compute the error estimator, and evaluate ifrefinement is necessary. If so, we repeat this process until the error estimatorhas reduced to a satisfactory level.

The set of triangles to be refined on each iteration follows directly:

QB =

B ∈ BH

∣∣∣∣ηR,B > η

|BH |

. (3.126)

as well as boundary triangles

Q′B =

B′ ∈

⋃B∈QB

ωB \QB

(3.127)

Each element B ∈ QB is partitioned into 2d1 new elements (with d1 the dimen-sion of Ω), while each element B′ ∈ Q′B is refined 2d1−1 to ensure no verticescollide with edges. An illustration of this process in two dimensions is given inFigure 3.2 and Figure 3.3.

The strategy can be summarized as follows:

(Step 1) Solve (3.13) on BH .

(Step 2) Compute ηR,B and ηR.

(Step 3) Refine the mesh in QB and Q′B.

(Step 4) Repeat (Step 1 – Step 3) until ηR < η.

The convergence estimates from Theorem 7 ensure that this procedure willindeed halt for any fixed η.


Figure 3.2: Subset of BH . The subsetQB is indicated in gray.

Figure 3.3: Subset of BH after meshrefinement.

3.6 Conclusion

We constructed a semidiscrete Galerkin approximation of our elliptic-parabolictwo scale system (P1) and showed that this approximation is well-posed andthat the obtained sequence of Galerkin approximants converges in suitable spacesto the weak solution to the continuous system. Furthermore, we derived a priorirates of convergence and proposed an a posteriori grid refinement strategy at themacroscopic scale.

As natural next steps, future work will address the fully discrete two-scaleGalerkin approximation as well as the numerical implementation of the methodso that the proven convergence rates can be confirmed and the macroscopic re-finement strategy can be tested. Additionally, a mesh refinement strategy on themicroscopic level could also be considered for either this problem setting, or forits elliptic-elliptic variant (obtained by letting t→∞ in (P1)).

At this stage, we would like to remark that the interaction between HB andh2 in the error structure gives an indication on how to choose the mesh sizeh based on the error estimators in HB. Without going into details, it is worthmentioning that in principle, one can choose the microscopic mesh size to cor-respond to the macroscopic grid. This way one can ensure that the macroscopicand microscopic errors are roughly of the same order.

CHAPTER 4Discussion

4.1 Summary

In this thesis, we derived a two-scale distributed-pressures model. We showedthe uniqueness and existence of a class of weak solutions. We also pointed outthe possibility of identifying parameters in two-scale transmission conditionsif measurements are available for at least one of the scales. As a next step, wedeveloped a two-scale Galerkin approximation scheme and showed its well-posedness and convergence to the target solutions. Additionally, we proved ana priori control of the convergence rates as well as the reliability of an a posteriorierror estimator. We used these estimates and the structure of our problem todesign a mesh refinement strategy reducing the overall approximation error.

4.2 Outlook

The work done in this thesis has lead several follow-up questions.Firstly, we plan on continuing the investigation of the two-scale pressure

model by exploring a suitable time discretization. Together with the spatial dis-cretization derived in Chapter 3, this would allow us to obtain a full a priori errorestimate of the numerical scheme. The next step is implementing this scheme,testing if the error estimates hold up in practice, as well as confirming the us-ability of the a posteriori refinement strategy. In doing so, we hope that we canrely on high performance computing techniques to improve the usability of ourmodels, as demonstrated in e.g. [22].

Secondly, we want to provide new applications to our multiscale analysis.

65

66 CHAPTER 4. DISCUSSION

The interaction between different species can be applied to more than just dual-porosity-like porous media. In [39], we investigated interacting populations byrepresenting the different populations on different spatial scales. By applyingthis so-called hybrid model, we are able to mimic different types of phenom-ena and increase computational efficiency. In order to succeed in working withthese kinds of models, we aim to improve and build on the interaction betweenparticles and continuum quantities. An initial step was made in [14], wherewe explored links between stochastic particle dynamics and PDE. The tools wedeveloped there can prove useful to help us continue this line of research.

In addition to the ideas mentioned above, we intend to develop a micro-macro population dynamics model inspired by the way compressible flow movesthrough porous media. This track was started in [11], where we investigatedmultiscale continuum models of particle flow in heterogeneous domains. Thisparticular goal could aid us in creating a unified framework in which to placeour modeling techniques. Such a multiscale modeling strategy could allow usto develop fast and accurate transport and interaction models capable of simu-lating a wide array of scenarios.

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[10] P. G. Ciarlet. The Finite Element Method for Elliptic Problems, volume 40 ofClassics in Applied Mathematics. Society for Industrial and Applied Mathe-matics, 2002.

[11] E. Cirillo, I. de Bonis, A. Muntean, and O.M. Richardson. Driven particleflux through a membrane: two-scale asymptotics of a diffusion equationwith polynomial drift. arXiv preprint arXiv:1804.08392, 2018.

[12] J. Daijun and J. Zou. Local Lipschitz stability for inverse Robin problemsin some elliptic and parabolic systems. arXiv e-prints, 2016.

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List of publications

The main body of this thesis consists of the following papers:

(I) M. Lind, A. Muntean, and O. M. Richardson. Semidiscrete Galerkin ap-proximation of a two-scale coupled problem: well-posedness and conver-gence rates. Submitted for publication, 2018.

(II) M. Lind, A. Muntean, and O. M. Richardson. Well-posedness and inverseRobin estimates for a multiscale elliptic/parabolic system. Applicable Anal-ysis, 97(1):89–106, 2018.

Furthermore, work in related research has lead to the following papers:

(III) H. Duong, A. Muntean, O. M. Richardson. Discrete and continuum linksto a nonlinear coupled transport problem of interacting populations. Eu-ropean Physics Journal: Special Topics, 226(10):2345–2357, 2017.

(IV) O. M. Richardson, A. Jalba, and A. Muntean. Effects of environment knowl-edge in evacuation scenarios involving fire and smoke - a multiscale mod-elling and simulation approach. To appear in Fire Technology, 2018.

(V) M. Colangeli, A. Muntean, O. M. Richardson, T. Thieu. Effects of envi-ronment knowledge in evacuation scenarios involving fire and smoke - amultiscale modelling and simulation approach. To appear in Crowd Dynam-ics vol. 1 - Theory, Models, and Safety Problems, MSSET series, 2018.

73

List of Figures

1.1 Example of a locally periodic two-dimensional macroscopic domainΩ. The solid space is in white, the pore space in gray. In three di-mensions, both the pore space and the solid space should be seen asconnected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Close up on a microstructure Y . The pore is partly filled with fluid(Bf ), partly with gas (Bg), the boundaries of which are indicated withΓf and Γg, respectively. The fluid-gas interface is indicated withΓfg(t). Outside of these boundaries lies the solid skeleton of theporous medium S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 The macroscopic domain Ω and microscopic pore Y at x ∈ Ω. . . . . . 12

3.1 The macroscopic domain Ω and microscopic pore Y at x ∈ Ω. . . . . . 353.2 Subset of BH . The subset QB is indicated in gray. . . . . . . . . . . . . 643.3 Subset of BH after mesh refinement. . . . . . . . . . . . . . . . . . . . 64

74


Omar Richardson

Om

ar Richardson | M

athematical analysis and approxim

ation of a multiscale elliptic-parabolic system

| 2018:33


We study a two-scale coupled system consisting of a macroscopic elliptic equation and a microscopic parabolic equation. This system models the interplay between a gas and liquid close to equilibrium within a porous medium with distributed microstructures. We use formal homogenization arguments to derive the target system. We start by proving well-posedness and inverse estimates for the two-scale system. We follow up by proposing a Galerkin scheme which is continuous in time and discrete in space, for which we obtain well-posedness, a priori error estimates and convergence rates. Finally, we propose a numerical error reduction strategy by refining the grid based on residual error estimators.


Faculty of Health, Science and Technology

Mathematics


ISSN 1403-8099

ISBN 978-91-7063-962-3 (pdf)

ISBN 978-91-7063-867-1 (print)

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