Mimetic Finite Difference methods - An introduction · Compatible Spatial Discretizations We de ne...

Mimetic Finite Difference methodsAn introduction

Andrea Cangiani

Universita di Roma La Sapienza

Seminario di Modellistica Differenziale Numerica – 2 dicembre 2008

Andrea Cangiani (IAC–CNR) mimetic finite difference methods 1 / 30

Some cronology...

RECENTLY (1995–2001)

Shashkov-Steinberg, JCP 1995 (support-operator methods)

(Hyman)-Shashkov-Steinberg, JCP 1996-(7) (rough diffusion)

Hyman-Shashkov, SINUM 1999 (Maxwell)

Campbell-Shashkov, JCP 2001 (gas dynamics)

MORE RECENTLY (2004–2008)

Kuznetsov-Lipnikov-Shashkov, Comp. Geos. 2004 (polygons)

Brezzi-Lipnikov-Shashkov, SINUM 2005 (error analysis)

Brezzi-Lipnikov-Simoncini, M3AS 2005 (a new family of MFD)

Brezzi-Lipnikov-Shashkov, M3AS 2006 (curved faces)


Some cronology (continued)

MORE RECENTLY (2004–2008) – continued

Beirao Da Veiga, NM 2007 (residual error estimator)

C.-Manzini, CMAME 2008 (post-processing)

C.-Manzini-Russo, SINUM accepted (convection-reaction-diffusion)

IN PROGRESS

Convection-dominated diffusion

Higher-order MFD

Nodal MFD

Mimetic curl operator

Mimetic discretization of Stokes


Features

RECENTLY

Generalisation of finite differences to hexahedral meshes.

Discrete differential operators defined so as to mimic the properties of theunderlying continuum operators (e.g. vector calculus identities, conservationlaws, solution symmetries)

Applied to wide range of problems.

MORE RECENTLY

Generalisation of (low order) Mixed Finite Elements/finite volumes togeneral polyhedral meshes. Constructs a family of methods.

Discrete differential operators defined so as to mimic the properties of theunderlying continuum operators. May gain from extra freedom given bymethod construction

Currently limited to linear diffusion problems


Compatible Spatial Discretizations

We define compatible spatial discretizations as those that inherit or mimicfundamental properties of the PDE such as topology, conservation,symmetries, and positivity structures and maximum principles.(IMA ”Hot Topics” Workshop Compatible Spatial Discretizations forPartial Differential Equations, May 11-15, 2004)

Advantages:

Conserve crucial features of physical, geometrical, and mathematicalmodel:

I Conservation lawsI SymmetryI Positivity and monotonicityI Duality properties of differential operators

Provide reliability and accuracy


Compatible Spatial Discretizations

The following can be cast as compatible spatial discretizations:

mixed finite element methods, mimetic finite differences, support operatormethods, control volume methods, discrete differential forms, Whitneyforms, conservative differencing, discrete Hodge operators, discreteHelmholtz decomposition, finite integration techniques, staggered grid anddual grid methods, etc.

Support operator method:

Most PDEs are written in terms of invariant differential operators(div, grad, and curl)

Define discrete analogues of these invariant operators that satisfyexacly the discrete analogs of the identities satisfied by the continuumoperators (e.g. Gauss’, Stokes’, Hodge orthogonal decomposition).


Linear diffusion in mixed form

Consider the linear diffusion equation bvp

−div(Kgrad p) = b in Ω ⊂ IR2,3

p = 0 on ∂Ω

with K strongly elliptic full symmetric tensor.

Mixed formulation:

F = −K grad p (Constitutive Equation)

div F = b (Conservation Equation)

We also introduce the flux operator

Gp = −Kgrad p.



The operators div and grad satisfy the integral identity∫Ω

p divF dV +

∫Ω

F · gradp dV = 0

which we rewrite: ∫Ω

p divF dV −∫

ΩK−1F · Gp dV = 0 (1)

Introducing the inner-products:

(p, q)W =∫

Ω p q dV

(F ,G )V =∫

Ω K−1F · G dV

the identity (1) becomes:

(divF , p)W − (F ,Gp)V = 0

expressing the fact that divergence and flux are adjoint: G = grad∗Andrea Cangiani (IAC–CNR) mimetic finite difference methods 8 / 30


Consider the linear diffusion equation bvp

−div(Kgrad p) = b in Ω ⊂ IR2,3

p = 0 on ∂Ω

with K strongly elliptic full symmetric tensor.

Mixed formulation:

F = −K grad p (Constitutive Equation)

div F = b (Conservation Equation)

Mixed variational formulation: find (p,F ) ∈W × V s.t.

(K−1F ,G )− (p,divG ) = 0 ∀G ∈ V

(divF , q) = (b, q) ∀q ∈W

W = L2(Ω), V = H(div; Ω)Andrea Cangiani (IAC–CNR) mimetic finite difference methods 9 / 30

Mixed Finite Element RT0-P0 Discretisation

Ω

E

E


Mixed Finite Element RT0-P0 Discretisation

Let Ωh be a triangularization of Ω ⊂ IR2. For E ∈ Ωh, let

P0(E ) = Polynomials of degree zero on E

RT 0(E ) = a + bx : a ∈ (P0(E ))2, b ∈ P0(E )

from which we define the conforming FE spaces:

Wh = qh : qh|E ∈ P0(E ), ∀E ∈ Ωh

Vh = Gh : Gh|E ∈ RT 0(E ) and Gh · n cont. on ∂Ωh

MFE RT0-P0 discretization: find (ph,F h) ∈Wh × Vh s.t.

(K−1F h,Gh)− (ph, divGh) = 0 ∀Gh ∈ Vh

(divF h, qh) = (b, qh) ∀qh ∈Wh


RT0-P0 nodal variables

Finite Element Nodal variables Lifting(E ,P0,N0) N0(qh) = qE LE

0 qE ∈ P0

qE := 1|E |∫E

q LE0 qE ≡ qE

(E ,RT 0,NRT ) NRT (G h) = (G eE )e = GE LE

RT GE ∈ RT 0

G eE := 1

|e|∫e

G · neE LE

RT GE |e · neE = G e

E

Global FE space Global nodal spaces Global lifting

Wh Qh := q = (qE )E∈Ωh L0 : Qh →Wh

Vh Xh :=

G = (GE )E∈Ωh

:

G eE−

+ G eE+

= 0 ∀e ∈ ∂Ωh

LRT : Xh → Vh.

Where E− and E+ are the two elements sharing the edge.


Discrete divergence

Let LRT G = Gh. Then,

|E | div Gh|E =

∫E

divGhGauss

=

∫∂E

G · nE =∑e∈∂E

|e|G eE

⇒ divLERT G = divGh|E =

1

|E |∑e∈∂E

|e|G eE =: divhGE

We may now define the discrete divergence operator

divh : Xh → Qh

G→ divhG = (divhGE )E

satisfying divLRT G = divhG.

⇒ (ph,div Gh) =∑E

|E |pEdivhGE = [p,divhG]Qh


Scalar product in Xh

(K−1F h,Gh) =

∫Ω

K−1LRT F · LRT G

=∑E

∫E

K−1LERT FE · LE

RT GE

=∑E

[FE ,GE ]E =: [F,G]Xh

MFE in terms of the spaces Qh and Xh: find (p,F) ∈ Qh × Xh:

[F,G]Xh− [p, divhG]Qh

= 0 ∀G ∈ Xh

[divhF,q]Qh= [b,q]Qh

∀q ∈ Qh.

with b = bI, I =interpolation operator defined for q ∈ V by

(qI)E :=1

|E |

∫E

q ∀E ∈ Ωh


Mimetic finite difference discretization

Ωh partition of Ω into polygonal (polyhedral) elements.

Ω

e

E

E

qE

GE

Why polyhedral meshes?

They naturally arise in the treatment of complex solution domainsand heterogeneous materials (e.g. reservoir models)

They facilitate adaptive mesh refinement/de-refinement


Discrete MFD pressure and flux spaces

Define the discrete pressure space Qh and flux space Xh.

q ∈ Qh → q = (qE )E∈Ωh(⇒ piecewise constant)

G ∈ Xh → G =(G e

E

)e∈∂E

E∈Ωh(⇒ constant normal component)

e ∈ ∂E− ∩ ∂E+ ⇒ F eE− + F e

E+= 0

We have the interpolation operators:

For q ∈W , (qI)E = 1|E |∫E q ∀E ∈ Ωh,

For G ∈ V ,(G

I)eE

= 1|e|∫e G · ne

E ∀E ∈ Ωh ∀e ∈ ∂E

And the discrete divergence operator:

divh : Xh → Qh

G→ divhG = (divhGE )E divhGE =1

|E |∑e∈∂E

|e|G eE


Discretisation of the conservation equation

Let b = bI. The mimetic discrete conservation equation reads:

divhF = b

i.e.1

|E |∑e∈∂E

|e|FeE = (b)E ∀E (2)

Equivalently,[divhF,q]Qh

= [b,q]Qh∀q ∈ Qh

where the scalar product for the pressure space Qh is given by

[p,q]Qh=∑

E∈Ωh

|E |pE qE

This MFD discrete conservation equation (2) coincides with the MFERT0-P0 and also with many finite volume discretisation of theconservation equation.


Finite volume discretisation of the constitutive equation

Directly discretize the constitutive equation defining the flux:

F = −K grad p (with K = k I)

LnAB

B

A

K

L

B

A

K

−nAB · F ≈ kpK − pL

|K− L|− F ≈ k

(pK − pL

|K− L|nAB

γAB+

pB − pA

|B− A|nKL

γKL

)


Mimetic discretisation of the constitutive equation

Introduce a scalar product for the flux space Xh

[F,G]Xh=∑

E∈Ωh

[F,G]E . ([·, ·]E to be defined!).

Moreover, define a discrete flux operator Gh : Qh → Xh as the adjoint ofdivh:

[G,Ghq]Xh= [q, divhG]Qh

, ∀q ∈ Qh ∀G ∈ Xh.

This definition naturally establishes a discrete Green formula with respectto the discrete scalar products. We say that the discrete operators mimicthe continuous ones.


Discretisation of the constitutive equation (cont.)

The mimetic discretisation of the constitutive equation reads:

F = Ghp

or, equivalently,

[F,G]Xh= [Ghp,G]Xh

=[p,divhG]Qh∀G ∈ Xh

The (family of) Mimetic Finite Difference (MFD) schemes:

[F,G]Xh− [p,divhG]Qh

= 0 ∀G ∈ Xh

[divhF,q]Qh= [b,q]Qh

∀q ∈ Qh

or, equivalently,[A Bt

B 0

] [Fp

]=

[0b

]Andrea Cangiani (IAC–CNR) mimetic finite difference methods 20 / 30

MFE and MFD

As we already saw, if Ωh is made of triangles (tetrahedrons),

The MFD conservation equation discretisation is equivalent toRT0-P0

If we define the Xh-scalar product to be

[F,G]Xh=

∫Ω

K−1LRT F · LRT G

the constitutive equation disretisation is equivalent.

...and the two methods coincide!

Which properties define LRT ? Locally, on each E ∈ Ωh,

1 LERT G|e · ne

E = G eE ∀e ∈ ∂E ,

2 divLERT G = divhG|E

3 LERT G

I= G if G is constant


The flux scalar product: P0-compatible liftings

IDEA: any reasonable lifting can be used to define a scalar productyielding a reasonable MFD formulation.

Set XE = Xh|E . We define P0-compatible lifting a linear mapLE : XE → L2(E ) such that for all G ∈ XE ,

1. LEG|e · neE = G e

E , ∀e ∈ ∂E ,

2. divLEG = divhG|E ,

and, for all constant vector C ,

3. LE CI

= C .

The following defines a MFD scalar product for the flux space:

[F,G]E =

∫Ω

K−1LEF · LEG dS .


The flux scalar product: P0-compatible liftings

Assume that K is constant on E .A crucial property. For any linear function q1 and G ∈ XE

[(K∇q1)I,G]E =

∫E

K−1LE((K∇q1)I

)· LEG dS

3=

∫E

∇q1 · LEG dS

Green= −

∫E

q1divLEG dS +

∫∂E

q1(LEG · next) dl

2,1= −

∫E

q1divhG dS +∑e∈∂E

G eE

∫e

q1 dl

(local consistency)

the scalar product is independent from the lifting if one of the factors is theinterpolant of a constant vector field.

the scalar product is exact on the interpolant of constant vector fields:

[(ei )I, (ej)

I]E = (K−1)i,j |E |


A general locally consistent scalar product

IDEA: Use local consistency to construct a scalar product without a lifting!

Let kE be the number of edges (faces) of E . Set:

[F,G]E =

kE∑s,t=1

Ms,tE F es

E G etE

Imposing local consistency and symmetry we get a set of positivesemi-definite matrices

Imposing stability: exists s∗, s∗ > 0 independent of h s.t.

s∗

kE∑i=1

(G eiE )2|E | ≤ [G,G]E ≤ s∗

kE∑i=1

(G eiE )2|E | ∀G ∈ XE .

we restrict to a set of acceptable (symmetric & pos. def.) matricesdefining a family of reasonable (e.g. exact on constants) scalarproducts.


A family of consistent and stable MFD scalar products[Brezzi-Lipnikov-Simoncini, Math. Models Methods Appl. Sci. (2005)]

Imposing:

local consistency ⇒ M0E symmetric & semidef.

stability ⇒ ME = M0E +CUC t symmetric & pos. def., where

C is a given (computable) full-rank kE × (kE − d) matrix,U is any symmetric & pos. def. (kE − d)× (kE − d) matrix

Proposition: Under a reasonable assumption on the minimum eigenvalueof U, the matrix ME is induced by a lifting.

In practice (so far) we use:

U = u (|E |Trace(K−1)) IIkE−d , u ∈ IR

and the proposition requires that u is sufficiently large.


MFD error in function of u

10-3

10-2

10-1

100

10110

-3

10-2

10-1

100

101

102

pressure |||.|||-errorpressure L2-errorlinear pressure L2-errorflux |||.|||-error

Quadrilaterals 20x20BLS Example 1


Back to Mixed finite elementsC. & Manzini, CMAME, 2008

Example. If Ωh is made of triangles,

ME = M0E + uE

[|ei | |ej |

]i ,j.

with M0E symmetric and positive semidefinite.

xexB

uE =

∑kEi=1

(xei − xB

)TK−1

(xei − xB

)12|E |

⇐⇒ RT 0− P0

Example Rectangles → central finite differences


Assumptions on the partition

The number of edges per face and the number of faces per element isunif. bounded.

The length l of each edge is such that l ≤ Ch (shape regularity).

Every element is uniformly strictly starshaped

(if Ω ∈ IR3) Every face is uniformly strictly starshaped


Error estimates[Brezzi-Lipnikov-Shashkov, SIAM J. Num. Anal. (2005)]

If

Ω is a convex polyhedron (polygon) with Lipschitz continuousboundary,Ωh is a partition satisfying the previous slide assumptions,the scalar product satisfies local consistency and stability.p ∈ H2(Ω)

Then,

‖|pI − p‖|Qh+ ‖|F I − F‖|Xh

≤ C h,

⇒ ‖p − L0p‖L2 ≤ C h

where‖| · ‖|2Qh= [·, ·]Qh

and ‖| · ‖|2Xh= [·, ·]Xh

.

If, moreover,

The scalar product is induced by (some) lifting.

Then,

‖|pI − p‖|Qh≤ C h2 (pressure super convergence)


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