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IMA Journal of Numerical Analysis (2014) of 27doi:10.1093/imanum/drn000

L2 and pointwise a posteriori error estimates for FEM for elliptic PDEson surfaces

Fernando Camacho† and Alan Demlow‡, Department of Mathematics, University ofKentucky, Lexington, KY 40506

[Received on 27 June 2014]

Surface Finite Element Methods (SFEM) are widely used to solve surface partial differential equationsarising in applications including crystal growth, fluid mechanics and computer graphics. A posteriori er-ror estimators are computable measures of the error and are used to implement adaptive mesh refinement.Previous studies of a posteriori error estimation in SFEM have mainly focused on bounding energy normerrors. In this work we derive a posteriori L2 and pointwise error estimates for piecewise linear SFEMfor the Laplace-Beltrami equation on implicitly defined surfaces. There are two main error sources inSFEM, a “Galerkin error” arising in the usual way for finite element methods, and a “geometric error”arising from replacing the continuous surface by a discrete approximation when writing the finite ele-ment equations. Our work includes numerical estimation of the dependence of the error bounds on thegeometric properties of the surface. We provide also numerical experiments where the estimators havebeen used to implement an adaptive FEM over surfaces with different curvatures.

Keywords: Surface partial differential equations, a posteriori error analysis, adaptivity

1. Introduction

In this paper we consider the model elliptic surface PDE

−∆Γ u = f on Γ . (1.1)

Here −∆Γ is the Laplace-Beltrami operator on a sufficiently smooth, closed, two-dimensional surfaceΓ of class C3 embedded in R3; extension to higher-dimensional surfaces of codimension one is mostlyimmediate. A canonical surface finite element method (SFEM) was defined in Dziuk (1988). In thismethod Γ is approximated by a polyhedral surface Γh having triangular faces and the finite elementequations using piecewise linear discrete spaces are then solved over Γh. This is the method we considerthroughout, though extension to higher order FEM and surface approximations could also be consideredDemlow (2009); Mekchay et al. (2011). The application of SFEM in diverse areas including imageand surface processing, fluid mechanics, and spectral geometry has become widespread over the pastdecade; cf. Clarenz et al. (2004); Groß et al. (2006); Gross & Reusken (2007); Olshanskii et al. (2009);Reuter et al. (2009) among many others. Adaptive finite element methods are also now a standard toolin computational science and engineering because they are computationally more efficient than uniformrefinement when solving problems with singularities or other strong local variations in the solution.

Residual a posteriori estimates for energy errors were proved for SFEM in Demlow & Dziuk (2007).Subsequent works including Ju et al. (2009); Wei et al. (2010); Mekchay et al. (2011); Demlow &

†Corresponding author. E-mail: [email protected]‡E-mail: [email protected]

c© The author 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 of 27 F. CAMACHO AND A. DEMLOW

Olshanskii (2012); Bonito et al. (2013) have considered variations on such estimates and related adaptiveFEM. A posteriori estimates for SFEM include standard residual a posteriori terms along with geometriccomponents arising from the discretization of Γ . It was shown in Demlow & Dziuk (2007) that if thediscrete approximation fh to f used to construct the finite element solution is defined properly, then

‖∇Γ (u−uh)‖L2(Γ ) 6C

(∑

T∈T‖Ah‖`2,L∞(ωT )η1(T )2

)1/2

+C‖(P−A`h)M∇Γ u`h‖L2(Γ ). (1.2)

Here uh is the discrete solution, ` denotes a lift of a function from Γh to Γ , η1 is a standard energyresidual indicator, ωT is the patch of elements in the mesh T surrounding a triangle T ∈ T , and Pis the projection onto the tangent plane to Γ . Also, M is a matrix of unit size. A`

h depends on thedistance function d (of which Γ is the zero level set), the normal~ν to Γ , the Weingarten map H whoseeigenvalues are the principal curvatures of Γ , and normal ~νh to Γh. A`

h satisfies ‖P−A`h‖L∞(T ) . h2

T ,where hT = diam(T ). In addition, ‖Ah‖`2,L∞(ωT ) → 1 as hT → 0. Thus the first residual term in (1.2)is of order h (in an a priori sense) and the geometric error contribution term ‖(P−A`

h)M∇Γ u`h‖L2(Γ ) isof order h2. Geometric errors caused by approximation of Γ may thus still drive adaptivity on coarsemeshes when implemented in adaptive algorithms (cf. Demlow & Dziuk (2007); Mekchay et al. (2011)),but under our assumptions they generally lose importance as the mesh is refined. Also, in (1.2) theconstant C does not depend on geometric properties of Γ .

In this paper we present efficient a posteriori L2 and pointwise error estimates for SFEM using piece-wise linear finite element spaces. As in (1.2) our estimates include geometric and Galerkin (residual)components, but as in Lp a priori estimates these contributions are now roughly speaking of the sameorder as the Galerkin error; cf. Dziuk (1988); Demlow (2009). For the L2 norm we prove that

‖u−u`h‖L2(Γ ) 6CΓ

(∑

T∈Tθ2(ωT )

2η0(T )2

)1/2

+C‖[(P−A`h)∇Γ u`h]‖L2(a(T ))

+C‖(1−µh)u`h‖L2(a(T )).

(1.3)

Here η0 is an L2 residual indicator which in an a priori sense is of order O(h2). θ2 is a geometricconstant which approaches 1 as hT → 0, but it depends on the derivatives Hxi of H as well as d, ν , andH and thus incorporates higher-order geometric information than does (1.2). Also, µh is the ratio of themetrics on Γh and Γ , and 1−µh is of order h2. Thus both the first residual term and the second and thirdgeometric terms in (1.3) are of order h2 in an a priori sense, and in contrast to the energy-norm casegeometric contributions may drive refinement on fine as well as on coarse meshes. Finally, CΓ dependson Γ through an H2 regularity constant which we do not explicitly measure, so in contrast to (1.2) wedo not computably measure all relevant geometric information in (1.3). A more detailed version of (1.3)and a similar maximum-norm a posteriori estimate are the main theoretical results of this paper.

We next discuss literature background and motivation for our results. On convex Euclidean do-mains L2 a posteriori estimates may be proved by a standard duality argument; cf. Verfurth (1996).Maximum-norm a posteriori bounds on nonconvex Euclidean polygonal domains were first proved inEriksson (1994); Nochetto (1995), with subsequent extensions and improvements in Dari et al. (2000);Nochetto et al. (2006); Demlow & Georgoulis (2012). Technicalities associated with nonconvex poly-hedral domains makes these analyses more challenging; our proofs are simpler because we assume thatΓ is closed and smooth. We instead demonstrate via a numerical test that our L∞ estimators are effectiveon a surface with reentrant corner. Applications where L2 or L∞ results are of interest include certain

A POSTERIORI ESTIMATES IN L2 AND L∞ FOR SFEM 3 of 27

nonlinear elliptic problems Nochetto et al. (2005, 2006). L2 a posteriori bounds for the (Euclidean)parabolic Allen-Cahn equation are contained in Bartels & Muller (2011), and these employ both ellipticL2 and elliptic L∞ a posteriori estimates as building blocks. The Allen-Cahn equation may also be con-sidered on surfaces; cf. Dziuk & Elliott (2007b). Spectral properties of the Laplace-Beltrami operatorhave been employed as a “shape DNA” fingerprint which can be helpful in deciding whether two givensurfaces are isometric; cf. Reuter et al. (2005, 2006); Reuter (2010). Maximum-norm error control isnatural for some spectral quantities considered in these works.

Our results are presented in the following structure. In Section 2 we introduce notation, definitions,and prove a lemma detailing geometric information needed to relate W 2

p norms on Γh and Γ . In Section3, we prove that Scott-Zhang interpolants yield good approximation properties on Γh, taking into accountthe fact that we must consider broken Sobolev (W 2

p ) spaces on Γh instead of globally defined spaces. InSection 4 we prove that an L2 error estimator is efficient and reliable. In Section 5 we prove similarpointwise results, and in Section 6 we present numerical tests.

2. Geometric preliminaries

Throughout we consider (1.1) while assuming∫

Γu dσ =

∫Γ

f dσ = 0 in order to ensure existence anduniqueness of solutions. As above we approximate Γ by a polyhedral surface Γh having triangular facesand nodes lying on Γ . Sh denotes a space of continuous functions that are affine on each face of Γh.

2.1 Continuous and discrete surfaces

Let ~ν(x) for all x ∈ Γ and ~νh(x) for almost all x ∈ Γh be the unit outward normal vectors to Γ and Γh,respectively. Let d(x) be the signed distance function from Γ to x where d < 0 inside of Γ and d > 0in the exterior of Γ and distance(x,Γ ) = |d(x)|. Following Chapter 14 of Gilbarg & Trudinger (1998),because Γ is a C3 surface there exists a tubular region U containing Γ such that the projection

a(x) : U → Γ , a(x) = x−d(x)~ν(x) for x ∈U, (2.1)

is unique for all x ∈ U (c.f. Demlow & Dziuk (2007) equation (2.4) for sufficient conditions on thewidth of U). The unit normal~ν is given by

~ν(x) = ∇d(x). (2.2)

Let µh(x) be the Jacobian of the transformation a(x)|Γh : Γh→ Γ , so that

µhdσh = dσ . (2.3)

We let T be a shape regular conforming triangular mesh whose elements T are the faces of Γh. ThusΓh =

⋃T∈T T . We also assume that a : Γh → Γ is a bijection, and that ~ν ·~νh > 0 everywhere on Γh.

Let hT denote the diameter of T for any T ∈ T . Typical refinement algorithms in Rn preserve shaperegularity. This also seems to be the case over surfaces, but we are unaware of a proof (c.f. Demlow &Dziuk (2007) Section 2.2). We assume that the number of simplices in the patch ωT = ∪T ′∈T :T∩T ′ 6= /0T ′

is bounded by a fixed constant for any T ∈T . This is always true for shape regular meshes over Rn, butdoes not necessarily hold for arbitrary surface meshes. However if the number of elements in each ωT isbounded in the initial mesh in an adaptive FEM, standard algorithms maintain the bound for subsequentmeshes (Demlow & Dziuk (2007) section 2.2). Shape regularity implies that there exist fixed constantsc1 and c2 such that for any T ∈T the following inequality holds

c1hTi 6 hT 6 c2hTi ∀ Ti ∈ ωT . (2.4)


In addition, for T ∈T

‖d‖L∞(T )+hT‖~ν−~νh‖L∞(T ) . h2T . (2.5)

By a . b we mean a 6 Cb, where C depends on properties of Γ via PDE regularity constants and onshape regularity of the mesh but not other essential quantities.

Let P(x) and Ph(x) be the projection matrices onto the tangent spaces of Γ and Γh respectively, i.e.

P(x) = I−~ν⊗~ν , Ph(x) = I−~νh⊗~νh. (2.6)

The Weingarten map is given by H(x) = D2d(x) = ∇~ν(x). Since |~ν(x)|= 1, ~νT H =~0T and H~ν =~0.

2.2 Surface derivatives

For any function η defined on an open set of R3 containing Γ we define its surface gradient by

∇Γ η = ∇η−~ν(~ν ·∇η). (2.7)

For a scalar function η , we regard ∇η as a column vector and write ∇Γ η = P∇η . For a vector functionv regarded as a column vector, P postmultiplies the gradient so that ∇Γ v = ∇vP. The Laplace-Beltramioperator is defined by

∆Γ η = ∇Γ ·∇Γ η = ∇ ·∇Γ η−~ν [∇(∇Γ η)]~νT . (2.8)

In the above equation~ν is regarded as a row vector and ∇(∇Γ η) is the matrix corresponding to the totalderivative of ∇Γ η . For more details see Demlow & Dziuk (2007) and Dziuk & Elliott (2007a).

2.3 Finite element approximation

The weak form of (1.1) is: find u ∈ H1(Γ ) such that∫

Γu dσ = 0 and∫

Γ

∇Γ u∇Γ v dσ =∫

Γ

f v dσ ∀v ∈ H1(Γ ). (2.9)

Denote by fh(x) an approximation of f over Γh satisfying∫

Γhfh = 0, for example, fh(x) = µh f (a(x)),x∈

Γh. Recalling that Sh is the space of affine Lagrange finite elements on Γh, our finite element methodproduces uh ∈ Sh that solves the problem∫

Γh

∇Γhuh∇Γhvh dσh =∫

Γh

fhvh dσh ∀vh ∈ Sh. (2.10)

2.4 Lifts and extensions

See Dziuk (1988); Demlow & Dziuk (2007) for more details. We extend v defined on Γ to U by

v`(x) = v(a(x)), x ∈U. (2.11)

For vh defined on Γh we define the lift vh by vh(x) = vh(a(x)), where a(x) ∈ Γ is as in (2.1). For vhdefined on Γh and x ∈U we extend vh to U by the equation v`h(x) = vh(a(x)). The relationship between∇Γ u`h(a(x)) and ∇Γhuh(x) is given by

∇Γ u`h(a(x)) = [(I−dH)(x)]−1[

I−~νh⊗~ν~νh ·~ν

]∇Γh uh(x). (2.12)


Following Demlow & Dziuk (2007) we define:

Ah(x) = A`h(a(x)) =

1µh(x)

[P(x)][(I−dH)(x)][Ph(x)][(I−dH)(x)][P(x)]. (2.13)

Equation 2.22 of Demlow & Dziuk (2007) yields∫Γh

∇Γhvh ·∇Γhψh dσh =∫

Γ

[∇Γ v`h(a(x))]T [A`

h][∇Γ ψ`h(a(x))] dσ . (2.14)

2.5 Comparison of Sobolev norms on discrete and continuous surfaces

Our main results are proved by duality arguments involving dual functions lying in W 2p Sobolev spaces.

Dziuk (1988) contains a brief comparison of W 2p Sobolev norms of functions on Γ and their extensions

to Γh. We give a more precise statement about the geometric dependencies of these relationships.

LEMMA 2.1 Let T ∈T and v ∈W 2p (a(T )) for some 16 p6 ∞. Then

|v`|W 2p (T )6

∥∥∥∥ 1µh

∥∥∥∥1/p

L∞(T )

(‖Ph[I−dH]‖2

L∞(a(T ))|v|W 2p (a(T ))

+

[‖PhH‖L∞(a(T ))‖~ν− (~ν ·~νh)~νh‖L∞(a(T ))+ max

i=1,2,3‖dPhHxi‖L∞(a(T ))

]|v|W 1

p (a(T ))

).

(2.15)

Before beginning the proof, we mention a couple of notational conventions. First, for vectorsa,b,c,d we have (a⊗b)(c⊗d) = (b · c)a⊗d. Second, we regard ∇v(a(x)) as a column vector.Proof. By equation (2.11) and the change of variable formula (2.3) we have

|v`|W 2p (T )

=

∑|α|=2

∫T[Dα

h (v`(x))]pdσh

1/p=

∑|α|=2

∫a(T )

1µh

[Dαh (v(a(x)))]pdσ

1/p. (2.16)

(2.7), (2.1), the chain rule and the fact that the projection matrix Ph is constant in each triangle yield

D2h(v

`(x)) = ∇ΓhPh[P−dH]∇v(a(x))= Ph∇Γh[P−dH]∇v(a(x))= Ph∇[I−dH]∇Γ v(a(x))Ph.

Next we expand the right hand side of the previous equation:

D2h(v

`(x)) =Ph

∇∇Γ v(a(x))[P−dH]−H∇Γ v(a(x))⊗~ν−d[Hxi∇Γ v(a(x))]3i=1

−dH∇∇Γ v(a(x))[P−dH]

Ph.(2.17)

Here Hxi denotes the derivative of H with respect to xi, and [Hxi∇Γ v(a(x))]3i=1 is a matrix whose i-thcolumn is given by Hxi∇Γ v(a(x)). Regrouping terms then yields

∇2Γ v(a(x))[I−dH]−dH∇

2Γ v(a(x))[I−dH] = [I−dH]∇2

Γ v(a(x))[I−dH].

Hence we write

D2h(v

`(x)) = Ph

[I−dH]∇2

Γ v(a(x))[I−dH]−H∇Γ v(a(x))⊗~ν−d[Hxi∇Γ v(a(x))]3i=1

Ph. (2.18)


Using (2.16), (2.18) and Holder’s inequality we obtain (2.15). For p = 2 Lemma 2.1 has the same form as Lemma 3 in Dziuk (1988), which states that |v`|H2(T ) 6

C

hT |v|H1(a(T ))+ |v|H2(a(T ))

. The difference is that Lemma 2.1 includes explicit geometric informa-

tion about the constant. We quickly verify that using equation (2.15) we get Lemma 3 of Dziuk (1988).Because Γ ∈ C3, ‖Ph[I− dH]‖L∞(a(T )) . 1 in (2.15). From (2.5) and Γ ∈ C3 it follows that the termmultiplying |v|W 1

p (a(T ))is of order hT , reducing to the statement of Dziuk (1988) Lemma 3.

3. Approximation results.

Proofs of residual a posteriori error estimates typically employ quasi-interpolants of Clement or Scott-Zhang type. In Demlow & Dziuk (2007) the authors defined and proved approximation properties forsuch an operator on Γh, but their operator only yields the first-order approximation properties neededfor energy estimates. Such estimates are simpler because u ∈W 1

p (Γ ) implies u` ∈W 1p (Γh). We must

instead consider broken spaces, since u ∈W 2p (Γ ) implies only u` ∈W 2

p (T ) for T ∈ T . Typical proofsof higher-order approximation properties for Scott-Zhang type interpolants employ a Bramble-Hilbertlemma which in our context would require u` ∈W 2

p (ωT ) for patches ωT , so the standard proof doesnot apply. The main technical ideas in this section are essentially contained in Theorem 3.1 of Veeser(2014), though they are applied there in a somewhat different context.

3.1 Interpolant (Scott-Zhang)

Let T and T be as defined previously. We consider finite element spaces Sh of arbitrary degree n overmeshes T of arbitrary space dimension, since the proof is no more difficult and more general results areof interest when considering SFEM in higher space dimensions; cf. Demlow (2009). Demlow (2009)also considers higher-degree surface approximations. Appropriate Scott-Zhang interpolants on suchsurfaces may also be obtained from the corresponding operator on Γh by simple composition; we do notgive the details. Let N be the set of Lagrange nodes of degree n on T . For all nodes z ∈N define:

Fz :=

T if z is an interior node of T ,eTz if z is an interior point of eTz. Here eTz is a face of the simplex T,

eTz for some arbitrary face containing z, if z is contained in more than one face.(3.1)

Let ϕzz∈N be the nodal basis for Sh, i.e., ϕzi(z j) = δi j and deg(ϕz) = n. Let ψzz∈N be thebasis dual to ϕzz∈N , i.e.

∫Fz

ψziϕz j = δi j, where zi,z j are nodes associated with the simplex Fz andψz : Fz→ R is a polynomial of degree deg(ϕz) = n.

Following Scott & Zhang (1990), we define the interpolant Ihv` of v` as

Ihv` = ∑z∈N

ϕz

∫Fz

ψz(ζ )v`(ζ ) dζ , v ∈W 11 (Γ ). (3.2)

Ih is a projection, that is, Ihs = s ∀s ∈ Sh. For any node z ∈N

‖ψz‖Lq(Fz) . h−dim(Fz)(1−1/q)T . (3.3)

Let dimT= d, 0 < hT < 1. Since |ϕz|6 1, it follows that

‖ϕz‖W kp (T ). h−k+d/p

T . (3.4)


Finally, for any element T ∈ T and Φ ∈W 1p (T ), 16 p < ∞, a standard scaled trace inequality (cf.

Brenner & Scott (2008) Theorem 1.6.6) yields

‖Φ‖Lp(∂T ) . h−1/pT ‖Φ‖Lp(T )+h1−1/p

T ‖∇Φ‖Lp(T ). (3.5)

3.2 Approximation properties

Equation (4.3) of Scott & Zhang (1990) and Lemma 1.130 of Ern & Guermond (2004) give approxima-tion properties for the Scott-Zhang interpolator of the form

‖IhΦ−Φ‖W mp (T ) . h`−m

T |Φ |W `p(ωT )

, 06 m6 `6 n+1, `> 1.

For our purpose we consider functions Φ that lie in W `p(K) for all K ∈ωT but that may not be in W `

p(ωT ).Assuming Φ ∈W 1

1 (ωT ) in order to guarantee continuity of traces, our goal is to prove that

‖IhΦ−Φ‖W mp (T ) . h`−m

T ∑K∈ωT

|Φ |W `p(K).

Lemma 3.1 and Theorem 3.1 below were inspired by Veeser (2014).

LEMMA 3.1 Let z ∈N and let Fz be a simplex of dimension d−1 defined as in (3.1) and let T ∈T bea simplex of dimension d such that z ∈ T ∩Fz. Define ωT to be the set of all simplices in T that touchT . For p > 1 let Φ ∈W `

p(T ) for some 1 6 ` 6 n+ 1 and for all T ∈ T . Let pT ∈ Sh be the `− 1-stdegree average Taylor polynomial of Φ over the simplex T , as defined in Lemma 4.3.8 of Brenner &

Scott (2008). Pick q such that1p+

1q= 1 and assume that T satisfies the assumptions of §2.1. Then

∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣. h−(d−1)(1−1/q)+(`−1/p)

T ∑T∈ωT

|Φ |W `p(T )

. (3.6)

Proof. If Fz is a face of T , then the claim follows from the trace inequality (3.5) and the Bramble HilbertLemma. Hence assume that T ∩Fz is a simplex of dimension at most d−2, and let TjM

j=1 be a chain ofd-dimensional simplices inside of ωT such that Tj+1∩Tj = Fj are (d−1)-dimensional simplices, z ∈ Fjfor 1 6 j 6M, T1 = T and Fz is a face of TM but Fz 6= FM−1. M is uniformly bounded over T , by ourassumptions. (3.2) yields ∫

Fz

ψz(Φ− pT ) ds = Ih(Φ− pT )(z).

Let ψ j be the dual to ϕz on Fj as in (3.2) so that∫

Fjψ jϕz = 1 and

∫Fj

ψ jvh = vh(z), vh ∈ Sh. Notethat in general ψz 6= ψ j since Fz 6= Fj. Then (pTj+1 − pTj)(z) =

∫Fj

ψTj(pTj+1 − pTj) dσ , so that using atelescoping sum along with Holders and triangle inequalities we obtain

∫Fz

ψz(Φ− pT ) ds =∫

Fz

ψz(Φ− pTM ) ds+M−1

∑j=1

∫Fj

ψα j(pTj+1 − pTj) ds

6‖ψz‖Lq(Fz)‖Φ− pTM‖Lp(Fz)+M−1

∑j=1‖ψ j‖Lq(Fj)

(‖Φ− pTj+1‖Lp(Fj)+‖Φ− pTj‖Lp(Fj)

).

(3.7)


It then follows from (2.4) and (3.3) that∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣.h−(d−1)(1−1/q)

T

[‖Φ− pTM‖Lp(Fz)

+M−1

∑j=1

(‖Φ− pTj+1‖Lp(Fj)+‖Φ− pTj‖Lp(Fj)

)].

(3.8)

Then by the Trace Inequality (3.5), shape regularity, and the Bramble Hilbert Lemma (c.f. Brenner &Scott (2008) 4.3.8.) we obtain∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣. h−(d−1)(1−1/q)+(`−1/p)

T

[|Φ |W `

p(TM)+M−1

∑j=1

(|Φ |W `

p(Tj+1)+ |Φ |W `

p(Tj)

)]. h−(d−1)(1−1/q)+(`−1/p)

T ∑T∈ωT

|Φ |W `p(T )

,

thus finishing the proof.

THEOREM 3.1 Let n denote the degree of the finite element space Sh, and let 16 `6 n+1, 16 p < ∞,and 06 k 6 `. For Φ ∈W 1

p (ωT ) satisfying Φ ∈W `p(T ) for all T ∈ ωT ,

‖IhΦ−Φ‖W kp (T ). h`−k

T ∑T∈ωT

|Φ |W `p(T )

. (3.9)

Proof. Let pT be as in Lemma 3.1. Then the triangle inequality and (3.4) yield

‖IhΦ−Φ‖W kp (T )6 ‖Φ− pT‖W k

p (T )+‖Ih(Φ− pT )‖W k

p (T ).

Applying the Bramble Hilbert Lemma to the first term in the right hand side and using (3.2), we obtain


T |Φ |W `p(T )

+ ∑z∈T

∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣ ‖ϕz‖W k

p (T ).

Let ˚NT denote the set of interior nodes of T and let ∂NT be the set of boundary nodes of T . Then


T |Φ |W `p(T )

+ ∑z∈ ˚NT

∣∣∣∣∫Fz


p (T )

+ ∑z∈∂NT

∣∣∣∣∫Fz


p (T ).

(3.10)

Let q be such that 1p +

1q = 1 and let d be the dimension of T . Observe that Fz = T for z ∈ ˚NT and the

number of nodes z ∈ T is bounded by a fixed constant C(n) depending on n. We use Holders inequality,(3.3), (3.4), the Bramble Hilbert Lemma and −d(1−1/q)− k+d/p =−k to obtain

∑z∈ ˚NT

∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣‖ϕz‖W k

p (T ). ‖ψz‖Lq(T )‖Φ− pT‖Lp(T )‖ϕz‖W k

p (T )

. h−d(1−1/q)−k+d/pT ‖Φ− pT‖Lp(T ) . h`−k

T |Φ |W `p(T )

.

(3.11)


Lemma (3.1) and the fact that the number of nodes of any T ∈T is bounded by C(n) imply that

∑z∈∂NT

∣∣∣∣∫Fz


p (T ). h−(d−1)(1−1/q)+(`−1/p)

T ∑T∈ωT

|Φ |W `p(T )‖ϕz‖W k

p (T ).

(3.4) applied to ‖ϕz‖W kp (T )

, −(d−1)(1−1/q)+(`−1/p)−k+d/p = `−k, and shape regularity yield

∑z∈∂NT

∣∣∣∣∫Fz

ψz(Φ− pT ) ds∣∣∣∣ ‖ϕz‖W K

p (T ) . h`−kT ∑

T∈ωT

|Φ |W `p(T )

. (3.12)

We substitute (3.11) and (3.12) into (3.10) to obtain (3.9), finishing the proof.

The following are scaled versions of standard Sobolev embedding theorems; cf. Adams & Fournier(2003) Theorem 4.12. We only consider ranges of indices used in our proofs below.

LEMMA 3.2 Assume that 1 6 p1 < d−1d−2 , 1 6 p2 < d

d−2 and either m = 2 and s = 1 or m = 1 andd p j

d+p j6 s < d

d−1 , s6 p j for j = 1,2. Then

‖Φ‖Lp1 (∂T ) .m

∑j=0

h j−d/s+(d−1)/p1T |Φ |W j

s (T ), for Φ ∈W j

s (T ), (3.13a)

‖Φ‖Lp2 (T ) .m

∑j=0

h j−d/s+d/p2T |Φ |W j

s (T ), for Φ ∈W j

s (T ). (3.13b)

In the following sections we apply our approximation results in the following form.

COROLLARY 3.1 Assume that either p1 = p2 = s = 2 and m = 1 or m = 2, or that p1, p2, s, and m arerelated as in Lemma 3.2 above. Then for T ∈T ,

‖IhΦ−Φ‖Lp1 (∂T ) . hm−d/s+(d−1)/p1T ∑

Ti∈ωT

|Φ |W ms (Ti), (3.14a)

‖IhΦ−Φ‖Lp2 (T ) . hm−d/s+d/p2T ∑

Ti∈ωT

|Φ |W ms (Ti). (3.14b)

In addition, for 16 p6 ∞

‖IhΦ‖Lp(T ) . ∑Ti∈ωT

(‖Φ‖Lp(Ti)+hT‖∇Φ‖Lp(Ti)

). (3.15)

Proof. We easily verify (3.14) by combining Theorem 3.1 and (3.13). (3.15) follows from the triangleinequality and Theorem 3.1.

3.3 A generalized Bramble-Hilbert Lemma

In Scott & Zhang (1990) a Bramble-Hilbert Lemma is applied over element patches in order to proveapproximation properties for the Scott-Zhang interpolant. Employing the same notation as above, let


06 j 6 n, 06 `6 n+1, and u ∈W `p(ωT ) with 16 p < ∞. Then

infv∈Sh|u− v|W j

p (ωT )6 inf

pT∈Pn|u− pT |W j

p (ωT ). h`− j

T |u|W `p(ωT )

. (3.16)

Lemma 3.1 and Theorem 3.1 may be rewritten as a Bramble-Hilbert lemma for broken Sobolev spaces.Let 06 j 6 n, k = max j,1, 16 `6 n+1, u ∈W 1

p (ωT ), and u ∈W `p(T

′) for each T ′ ⊂ ωT . Then

infv∈Sh

( ∑T ′⊂ωT

|u− v|pW j

p (T ′))1/p . ( ∑

T ′⊂ωT

infpT ′∈Pn

hk− jT |u− pT ′ |pW k

p (T ′))1/p . h`− j

T ( ∑T ′⊂ωT

|u|pW `

p(T ′))1/p. (3.17)

The two differences between (3.16) and (3.17) are that the former uses standard and the latter brokenSobolev spaces, and that (3.17) requires k, ` > 1. Theorem 3.2 of Veeser (2014) establishes that con-tinuous and discontinuous finite element spaces yield equivalent approximation in the H1 seminormnot only asymptotically but on any mesh satisfying reasonable assumptions; this is essentially the firstinequality in (3.17) with p = 2 and j = k = 1. We thus again emphasize that we apply techniques inVeeser (2014) in a different context but with only modest generalization of the basic ideas.

4. L2 a posteriori estimate

In this section we derive an L2 a posteriori error estimator. We first state a standard regularity result.

LEMMA 4.1 Regularity (c.f. Demlow (2009) Lemma 2.1). Let f ∈ L2(Γ ) satisfy∫

Γ

f dσ = 0, and

assume that Γ is a C2 surface. Then the problem L(u,v) = ( f ,v) ∀v∈H1(Γ ) has a unique weak solution

satisfying∫

Γ

u dσ = 0 and

‖u‖H2(Γ ) 6C‖ f‖L2(Γ ). (4.1)

We next define the error e := u−u`h. From this point on vh will be used to denote the interpolant ofv`, i.e. vh ≡ Ihv`. Our main result is stated in the following theorem.

THEOREM 4.1 Assume that Γ is a C3 surface. Let u(x) be the solution to (1.1) and define

Cp(K) :=∥∥∥∥ 1

µh

∥∥∥∥1/p

L∞(K)

, Cp(ωK) =

∥∥∥∥ 1µh

∥∥∥∥1/p

L∞(ωK)

θp(K) :=Cp(K)(‖Ph[I−dH]‖2

L∞(a(K))+‖PhH‖L∞(a(K))‖~ν− (~ν ·~νh)~νh‖L∞(a(T ))

+ maxi=1,2,3

‖dPhHxi‖L∞(a(K))

),

θp(ωK)p = ∑

K′⊂ωK

θp(K′)p, 16 p < ∞, θ∞(ωK) = maxK′⊂ωK

θ∞(K′)

γ2(K) =C2(K)(1+hT‖P−dH‖L∞(K)), γ2(ωK)2 = ∑

K′⊂ωK

γ2(K′)2.

(4.2)

Then the following bound holds:

‖u−u`h‖L2(Γ ) .[

∑T∈T

(θ2(ωT )

2

h4T‖µh f `+4Γhuh‖2

L2(T )+h3T‖J∇Γh uhK‖2

L2(∂T )

+‖[(P−A`

h)∇Γ u`h]‖2L2(a(T ))+‖u

`h−µhu`h‖2

L2(a(T ))+ γ2(ωT )2‖µh f `− fh‖2

L2(T )

]1/2.

(4.3)


The constant hidden in “.” depends on the regularity constant in (4.1) but not other essential quantities.

The main difference between our estimators and those arising in the case of flat Euclidean domainsis that we include the geometric terms Cp(·), θp(·), θ∞(ωK) and γ2(·). These terms arise naturally whenwe move between the discrete and the continuous surface as discussed in Lemma 2.1. As discussedin the introduction, Θ2(T )→ 1 as hT → 0, while the remaining (additive) geometric terms disappearas hT → 0. Note also that the term Θ2 includes derivatives of H and thus requires C3 regularity of Γ .All other portions of our estimator require only C2 regularity of Γ . The corresponding energy normestimates in Demlow & Dziuk (2007) require C2 regularity of Γ . The higher regularity required hereis due to the fact that we map higher-order derivatives between the discrete and continuous surfaces aspart of our duality argument. It has been observed in the case of energy norm a posteriori estimates thatit is possible to assume only Lipschitz regularity of Γ (cf. Mekchay et al. (2011); Bonito et al. (2013)),but at the expense of a larger geometric error. In particular, the geometric and PDE energy errors areof the same order if a Lipschitz map is used to relate the continuous and discrete surfaces. The higherrate of convergence of the geometric error when a C2 surface is assumed is due to the fact that theorthogonal projection a used in the current work has special orthogonality properties. Translating theseobservations to the case of L2 error estimates, it would be possible to prove a posteriori bounds underthe assumption that Γ is only C1,1. However, the resulting geometric errors would converge at a lowerrate than the residual (PDE) portion of the error measured in L2, and would thus dominate the estimator.We thus assume a C3 surface.

We use the following lemma to prove Theorem 4.1.

LEMMA 4.2 Let u and u`h be the continuous and discrete solutions of equation (1.1). respectively. Let vsolve −4Γ v = u−µhu`h in Γ ,

∫Γ

v dσ = 0. Then the following bound holds:

‖u−u`h‖2L2(Γ ) . I + II + III + IV +‖µhu`h−u`h‖2

L2(Γ ), (4.4)

where

I =∫

Γh

(µh f `+4Γhuh)(v`− vh) dσh, II =−∫

Γ

[(P−A`h)∇Γ u`h] ·∇Γ v dσ ,

III =− 12 ∑

T∈T

∫∂T

J∇Γh uhK(v`− vh) ds, IV =∫

Γh

(µh f `− fh)vh dσh,

and v`(x), x ∈ Γh is the lift of v to Γh defined in (2.11).

Proof. Since∫

Γu = 0 and

∫Γh

uh = 0, it follows that∫

Γ(u− µhu`h) dσ = 0. Now we compute the L2

norm of the error

‖u−u`h‖2L2(Γ ) = (u−u`h,u−µhu`h +µhu`h−u`h)

= (u−u`h,−4Γ v)+(u−u`h,µhu`h−u`h) = A+(u−u`h,µhu`h−u`h),(4.5)

where A = (u−u`h,−4Γ v). By integration by parts we get since ∂Γ = /0:

A = (u−u`h,−4v) =∫

Γ

∇Γ (u−u`h)∇Γ v dσ . (4.6)


The residual identity given in equation (3.5) of Demlow & Dziuk (2007) gives us

A =∫

Γ

∇Γ (u−u`h)∇Γ v dσ =∫

Γh

(µh f `+4Γh uh)(v`− vh) dσh−∫

Γ

[(P−A`h)∇Γ u`h] ·∇Γ v dσ

− 12 ∑

T∈T

∫∂T

J∇ΓhuhK(v`− vh) ds+∫

Γh

(µh f `− fh)vh dσh

= I + II + III + IV.

(4.7)

Combining equations (4.5) and (4.7) we easily get for any ε > 0

‖u−u`h‖2L2(Γ ) 6 I + II + III + IV +‖u−u`h‖L2(Γ )‖µhu`h−u`h‖L2(Γ )

6 I + II + III + IV +ε

2‖u−u`h‖2

L2(Γ )+1

2ε‖µhu`h−u`h‖2

L2(Γ ).(4.8)

Taking ε = 12 and rescaling concludes the proof of the Lemma.

4.1 A posteriori upper bound (Proof of Theorem 4.1)

We now prove bounds for elements I through IV of (4.7).Bound for I. Holder’s inequality yields∫

Γh

|(µh f `+4Γhuh)(v`− vh)| dσh 6 ∑T∈T‖µh f `+4Γhuh‖L2(T )‖v`− vh‖L2(T ). (4.9)

Recall that we defined vh = Ihv`. Then by (3.14b) with p = s = m = 2 we get

‖v`− vh‖L2(T ) . h2T ∑

K∈ωT

|v`|H2(K). (4.10)

Next we apply Lemma 2.1, (4.2), and observe that ‖ · ‖H2(T ) bounds the H1 and H2 semi-norms to get∑K∈ωT |v`|H2(K) . θ2(ωT )‖v‖H2(a(ωT ))

. Finite overlap of the patches ωT then yields

∫Γh

|(µh f `+4Γh uh)(v`− vh)| dσh .

(∑

T∈Th4

T‖µh f `+4Γhuh‖2L2(T )θ2(ωT )

2

)1/2

‖v‖H2(Γ ). (4.11)

Bound for II. The Cauchy-Schwarz inequality and ‖v‖H1(Γ ) 6 ‖v‖H2(Γ ) yields∫Γ

[(P−A`h)∇Γ u`h] ·∇Γ v dσ 6 ‖[(P−A`

h)∇Γ u`h]‖L2(Γ )‖∇Γ v‖L2(Γ )

=

∑

T∈T‖[(P−A`

h)∇Γ u`h]‖2L2(a(T ))

1/2

‖v‖H2(Γ ).

(4.12)

Bound for III. Using Holder’s inequality, (3.14a) with p = s = m = 2, and compute as in (4.12) yields

∑T∈T

∫∂T

J∇ΓhuhK(v`− vh) ds6 ∑T∈τ

‖J∇ΓhuhK‖L2(∂T )‖v`− vh‖L2(∂T )

. ∑T∈T‖J∇ΓhuhK‖L2(∂T )h

3/2T ∑

K∈ωT

|v`|H2(K)

.

(∑

T∈T‖J∇ΓhuhK‖L2(∂T )h

3T θ2(ωT )

2

)1/2

‖v‖H2(Γ ).

(4.13)


Bound for IV. It follows from Holder’s inequality and (3.15) that∫Γh

|(µh f `− fh)vh| dσh 6 ∑T∈T‖(µh f `− fh)‖L2(T )‖vh‖L2(T ),

. ∑T∈T‖µh f `− fh‖L2(T ) ∑

K∈ωT

(‖v`‖L2(K)+hT‖∇Γ v`‖L2(K)).(4.14)

Then we use (2.1), (2.3), (3.15) and Holder’s inequality to deduce that∫Γh

|(µh f `− fh)vh| dσh . ∑T∈T

[‖µh f `− fh‖L2(T ) ∑

K∈ωT

C2(K)(‖v‖L2(a(K))

+hT‖(I−dH)∇v‖L2(a(K)))]

. ∑T∈T

γ2(ωT )‖µh f `− fh‖L2(T )‖v‖H2(a(K))

.

(∑

T∈Tγ2(ωT )

2‖µh f `− fh‖2L2(T )

)1/2

‖v‖H2(Γ )).

(4.15)

Bound for (4.7). Let

η2 = ∑

T∈Th4

T‖µh f `+4Γhuh‖2L2(T )θ2(ωT )

2 +‖[(P−A`h)∇Γ u`h]‖2

L2(Γ )

+ ∑T∈T‖J∇ΓhuhK‖2

L2(∂T )h3T θ2(ωT )

2 + ∑T∈T

γ2(ωT )2‖µh f `− fh‖2

L2(T ).(4.16)

Using (4.11), (4.12), (4.13) and (4.15), and the regularity result (4.1) yields

I + II + III + IV . η‖u−µhu`h‖L2(Γ ). (4.17)

We combine (4.17), (4.7), (4.4) and then use Cauchy and the triangle inequality to get

‖u−u`h‖2L2(Γ ) . η‖u−µhu`h‖L2(Γ )+‖u`h−µhu`h‖2

L2(Γ )

61

4εη

2 + ε‖u−µhu`h‖2L2(Γ )+‖u

`h−µhu`h‖2

L2(Γ )

61

4εη

2 +2ε‖u−u`h‖2L2(Γ )+2ε‖u`h−µhu`h‖2

L2(Γ )+‖u`h−µhu`h‖2

L2(Γ ).

Rearranging terms and taking ε sufficiently small write finalizes the proof of Theorem 4.1. 2

We define the error indicator η(T ) in each triangle T ∈T as follows:

η(T ) :=

h2T‖µh f `+4Γhuh‖L2(T )+h3/2

T ‖J∇ΓhuhK‖L2(∂T )

θ2(ωT )+‖[(P−A`

h)∇Γ u`h]‖L2(a(T ))

+‖u`h−µhu`h‖L2(a(T ))+ γ2(ωT )‖µh f `− fh‖L2(T ).(4.18)

Thus we write ‖u−u`h‖L2(Γ ) .(∑T∈T η2(T )

)1/2.


4.2 Efficiency

Next we verify that the residual part of the estimator (4.18) is bounded above by the true error plusdata oscillation and geometric terms. We use standard techniques developed in Verfurth (1994). We firstclarify the role of geometric terms in mappings between Γh and Γ needed in the course of our arguments.

LEMMA 4.3 Assume that T ∈ T and γ is an edge of T . Let P be the piecewise constants on T . Forx ∈ Γh let x = a(x), vh(a(x)) = vh(x), and T = a(T ). Let φT , φγ be the squares of the interior bubblefunction associated with T and the edge bubble function associated with γ respectively and define

G := [I−dH]−1[

I−~νh⊗~ν~νh ·~ν

]. (4.19)

Then for vh ∈P and 16 p6 ∞

‖4Γ (φT vh)‖Lp(T ) 6(h−2

T ‖G ‖L∞(T )+h−1T ‖DG ‖L∞(T )

)‖v‖Lp(T ), (4.20a)

‖4Γ (φγ vh)‖Lp(T ) 6(

h−2+1/pT ‖G ‖L∞(T )+h−1+1/p

T ‖DG ‖L∞(T )

)‖v‖Lp(γ). (4.20b)

Proof. Let w := φT v. (2.8) yields4Γ w = ∇ ·∇Γ w−~ν [D∇Γ w]~νT . We apply Demlow & Dziuk (2007)equation (2.19) to this equation. Let ei denote the column unit vector with entry-i equal to one and zeroeverywhere else, let [(·)i], 16 i6 3 denote a matrix where the i-th column is given by (·)i and get

4Γ w = ∇ ·(G ∇Γhw

)−~ν

[D(G ∇Γhw

)]~νT

=3

∑i=1

(eT

i G ∂xi(∇Γhw))−~ν

[G ∂xi(∇Γhw)

]3i=1

~νT +3

∑i=1

(eT

i ∂xi (G )∇Γhw)−~ν

[∂xi (G )∇Γhw

]3i=1

~νT .

Taking Lp norm of both sides and using the triangle and Holder’s inequalities yields

‖4Γ w‖Lp(T ) 63

∑i=1

(‖eT

i G ‖L∞(T )‖∂xi∇Γhw‖Lp(T )+‖eTi ∂xiG ‖L∞(T )‖∇Γhw‖Lp(T )

)+‖~νG ‖L∞(T )‖D(∇Γhw)~νT‖Lp(T )+‖DG ‖L∞‖∇Γhw‖Lp(T ),

thus‖4Γ w‖Lp(T ) . ‖G ‖L∞(T )|φT v|W 2

p (T )+‖DG ‖L∞(T )|φT v|W 1

p (T ).

Because v is constant in each T we can use an inverse estimate to deduce (4.20a). The same argument,after observing that we apply a scaling argument to go from T to γ , yields (4.20b).

We now prove our main efficiency result. Note that this bound will also be used in the followingsection to establish efficiency of maximum-norm error estimators.

LEMMA 4.4 Let f be a piecewise constant approximation to f . Let 1 6 q 6 ∞, f ∈ Lq(Γ ). For x ∈ Γhlet x = a(x), vh(a(x)) = vh(x), choose µh f ` = fh, and let G be defined by equation (4.19). Then for anyelementwise constant function f ,

h2T‖µh f `+4Γhuh‖Lq(T )+h1+1/q

T ‖J∇ΓhuhK‖Lq(∂T ) . ‖e‖Lq(ωT )(‖G ‖L∞(ωT )+hT‖DG ‖L∞(ωT ))

+hT‖G ‖L∞(ωT )‖(P−A`h)∇Γ u`h‖Lq(ωT )+h2

T‖µh f `− f‖Lq(ωT ).(4.21)


Before proving Lemma 4.4 we remark on the presence and absence of various geometric factor in(4.21). First, we have not included additive geometric terms such as ‖(P−A`

h)∇Γ u`h‖L2(Γ ) in our effi-ciency analysis. A major reason for this exclusion is that efficiency estimates for the residual componentof the estimator plays an important role in understanding convergence and optimality of surface AFEM,whereas efficiency of the geometric components does not; cf. Bonito et al. (2013). Secondly, if (4.21)were used to derive a global efficiency bound by adding the contributions over Th, then (4.21) shouldfirst be multiplied through by θ2(ωT ) for the sake of consistency with (4.3).

We also comment briefly on the data oscillation term h2T‖µh f `− f‖Lq(ωT ). Because our residual

terms are defined on Γ , this term is defined with respect to the natural mapping µh f ` of f to Γh ratherthan with respect to f . In addition, it is multiplied by h2

T , which is the natural scaling for Lq norms.Proof. We introduce a more compact notation for the residuals:

r := µh f `+4Γhuh, R :=−J∇ΓhuhK. (4.22)

and defineG1 := h−2

T ‖G ‖L∞(T )+h−1T ‖DG ‖L∞(T ). (4.23)

Let r and R denote piecewise constant approximations of r and R respectively. We choose p such that1p +

1q = 1, use the residual equation (4.7) and let v = φT ˜r, vh = 0. Since φT and ∇ΓhφT vanish on ∂T

we obtain ∫T

∇Γ e∇Γ (φT ˜r) dσ =∫

TφT rr dσh−

∫T(P−A`

h)∇Γ u`h∇Γ (φT ˜r) dσ .

Thus after adding and subtracting the appropriate terms and rearranging we get∫T

φT r2 dσh =∫

T(∇Γ e+(P−A`

h)∇Γ u`h)∇Γ (φT ˜r) dσ +∫

TφT r(r− r) dσh. (4.24)

Integration by parts together with Lemma 4.3 and (4.23) gives∫T

∇Γ e∇Γ (φT ˜r) dσ =∫

T−e4Γ (φT ˜r) dσ 6 ‖e‖Lq(T )‖4Γ φT ˜r‖Lp(T )

6 ‖e‖Lq(T )(h−2

T ‖G ‖L∞(T )+h−1T ‖DG ‖L∞(T )

)‖r‖Lp(T ) = G1‖e‖Lq(T )‖r‖Lp(T ).

(4.25)

In a similar way we apply Holder’s inequality, use Demlow & Dziuk (2007) equation 2.19 together withthe definition of G , Ainsworth & Oden (2000) Lemma 2.1 and Theorem 2.2 to obtain∫

T(P−A`

h)∇Γ u`h∇Γ (φT ˜r) dσ 6 ‖G ‖L∞(T )‖(P−A`h)∇Γ u`h‖Lq(T )‖∇Γh(φT r)‖Lp(T )

. h−1T ‖G ‖L∞(T )‖(P−A`

h)∇Γ u`h‖Lq(T )‖r‖Lp(T ).(4.26)

We combine (4.24), (4.25) and (4.26) to get∫T

φT r2 dσh 6 ‖r‖Lp(T )

G1‖e‖Lq(T )+h−1

T ‖G ‖L∞(T )‖(P−A`h)∇Γ u`h‖Lq(T )+‖r− r‖Lq(T )

. (4.27)

It follows from Ainsworth & Oden (2000) Lemma 2.1 that ‖r‖2L2(T ) .

∫T

φT r2 dσh. Let d be the dimen-

sion of the simplex T . The equivalence of norms in a finite dimensional space and a scaling argument


gives the bound h−d/p+d/2T ‖r‖Lp(T ) . ‖r‖L2(T ). Thus we have hd−d(1/p+1/q)

T ‖r‖Lp(T )‖r‖Lq(T ) . ‖r‖2L2(T ).

Observe that d−d(1/p+1/q) = 0 and finally apply the triangle inequality to ‖r‖Lq(T ) to obtain

‖r‖Lq(T ) 6 G1‖e‖Lq(T )+h−1T ‖G ‖L∞(T )‖(P−A`

h)∇Γ u`h‖Lq(T )+‖r− r‖Lq(T ). (4.28)

Let γ be an edge of T , v := φγ˜R, vh = 0, and µh f ` = fh. Let ωγ := T ∈T |T ∩ γ 6= /0 and observe

that φγ vanishes outside of ωγ . Then it follows from (4.7) that∫γ

R2 ds.∫

γ

φγ R2 ds =∫

ωγ

(∇Γ e+(P−A`h)∇Γ u`h)∇Γ (φ

`γ

˜R) dσ −∫

ωγ

φγ rR dσh

+∫

γ

φγ R(R−R) dσ .

Following the steps used to derive (4.28) we use Holder’s inequality, Lemma 4.3 of Ainsworth & Oden(2000), Lemma 2.1, (4.23), (4.28), the triangle inequality, and inverse estimates to deduce that

h−1/pT ‖R‖Lq(γ) .G1‖e‖Lqωγ

+h−1T ‖G ‖L∞(ωγ )‖(P−A`

h)∇Γ u`h‖Lq(ωγ )+‖r− r‖Lq(ωγ )+‖R− R‖Lq(γ).

The result follows after multiplying this inequality by h2T , equation (4.28) by h2

T , substituting G1 for theright hand side of (4.23), and choosing R = R and r = f .

5. Pointwise Estimator

Now we proceed to find a pointwise a posteriori estimator for the problem (1.1).

5.1 Regularity properties of the Green’s functions

Following previous works on maximum-norm a posteriori estimation (cf. Eriksson (1994); Nochetto(1995); Demlow & Georgoulis (2012)), we represent the error by writing a weak form of the problemusing the Green’s function as the auxiliary function. In this section we thus establish several propertiesof the Green’s function for Γ . We first cite Demlow (2009) Lemma 2.2.

LEMMA 5.1 There exists a function G(x,y) (unique up to a constant) such that for all functions u(x) ∈C2(Γ ),

u(x)− 1|Γ |

∫Γ

u(x) dσ =∫

Γ

G(x,y)(−4Γ u(y)) dσ =∫

Γ

∇Γ ,yG(x,y)∇Γ ,yu(y) dσ . (5.1)

Let α(x,y) be the surface distance between x,y ∈ Γ , and let d denote the dimension of Γ . Furtherassume that α(x,y)< 1. Then (c.f. Demlow (2009) Lemma 2.2, Aubin (1982) Theorem 4.17)

|G(x,y)|.

ln(

Cα(x,y)

)for d = 2,

α(x,y)2−d d > 2.(5.2)

Let |γ +β |> 0 where γ,β are multiindices. Then

|Dγ

Γ ,yDβ

Γ ,xG(x,y)|. α(x,y)2−d−|γ+β |. (5.3)


LEMMA 5.2 Let d > 2 denote the dimension of the surface Γ . Then

G(x,y) ∈W 1p (Γ ), where p <

dd−1

.

Proof. By (5.3) we have |G|pW 1p (Γ )

=∫

Γ|∇G|p dσ .

∫Γ|α(x,y)1−d |p dσ . ∇G is bounded uniformly

away from the singularity at y = x, so we analyze what happens in a y-neighborhood U of x. There is alocal isomorphism χ that maps U to a disk D contained in a plane of dimension d embedded in Rd+1.We let µ denote the Jacobian of the transformation χ : U → D. Then

|G|pW 1p (U). ‖µ‖L∞(D)

∫D|r(1−d)p| dσD,

where r := |χ(x)− χ(y)|, y⊂U . By a linear scaling we can choose χ such that rD = rU , where rD andrU represent the radii of D and U respectively. Then we use polar coordinates to get:

|G|pW 1p (U). ‖µ‖L∞(D)

∫Sd

∫ rD

0rp(1−d)rd−1 dr dSd . (5.4)

The last integral is finite whenever p(1−d)+d−1 >−1 i.e. p < dd−1 . This completes the proof.

COROLLARY 5.1 Let p satisfy p < dd−1 . Then there is a constant, C(p,d), depending on p and d such

that |G|W 1p (U) .C(p,d) and C(p,d)→ ∞+ as p→

( dd−1

)−.

Proof. From (5.4) we get

|G|W 1p (U) .

1[d− p(d−1)]1/p r1−d+d/p

D , (5.5)

where rD > 0 is a fixed constant and clearly C(p,d) :=(

1d−p(d−1)

)1/p→ ∞+ as p→

( dd−1

)−.

LEMMA 5.3 Let x0 be the singularity of the Green’s function and let U be a neighborhood of x0 suchthat there is a constant c1 for which the disk of radius c1 centered at x0 is contained in the interior of U(i.e Bc1(x0)⊂ U). Then

|G|W 21 (Γ \U) . 1+ ln

(1c1

). (5.6)

Proof. Let c2 denote the diameter of Γ . By equation (5.3), |G|W 21 (Γ \U) .

∫ c2c1

r−1 dr . 1+ ln(

1c1

).

5.2 Estimator

We now state and prove the main result of this section. Define:

e := u(x)−u`h(x)+1|Γ |

∫Γ

u`h(x) dσ . (5.7)


THEOREM 5.1 Let u(x) be the solution to the Laplace-Beltrami equation (1.1), let h = minT∈T

hT,

d > 2, q1 >d2 , q2 > d, q3 > d− 1, and q4 >

d2 . Let θ(T ) be as in (4.2) and define θ∞(T ) := θ∞(T )+

‖ 1µh‖L∞(T )‖I−dH‖L∞(T ) and similarly for θ∞(ωT ). Then for x ∈ Γ , there holds

|e(x)|.maxT∈T

θ∞(ωT )(1+ | lnh|)

(h2−d/q1

T ‖µh f `+4Γhuh‖Lq1 (T )

+h1−(d−1)/q3T ‖J∇ΓhuhK‖Lq3 (∂T )

)+‖(P−A`

h)∇Γ u`h‖Lq2 (Γ )+Cq4‖µh f `− fh‖Lq4 (Γh).(5.8)

The constant in “.” depends on shape regularity properties of Th and on properties of Γ via the Green’sfunction G, and blows up as q2→ d+, q3→ (d−1)+ or q1,q4→ d

2+

respectively. Cq4 depends on q4,‖ 1

µh‖L∞(Γ ), and ‖P−dH‖L∞(Γ ).

REMARK 5.1 In (5.8) we use Sobolev embeddings in order to define elementwise residuals measuredin Lq norms for q < ∞. This has two advantages. It allows us to admit data f not in L∞, and to measurethe geometric term ‖(P−A`

h)∇Γ u`h‖Lq2 (Γ ) in a weaker norm in the event u /∈W 1∞(Γ ). Our methodology

yields no advantage in the jump residual terms for constant-coefficient operators, but does on the caseof nonconstant diffusion coefficients. In our numerical experiments we simply take qi = ∞ for all i.

REMARK 5.2 Maximum-norm a posteriori estimates for the Laplacian on Euclidean (flat) domains Ω

are contained in Eriksson (1994); Nochetto (1995); Demlow & Georgoulis (2012). They are roughlyspeaking of the form (5.8), but with all geometric terms omitted. A particular focus of those worksis regularity of the domain Ω , as all of them admit nonconvex polygonal or polyhedral domains. Be-cause we prove our results under the assumption that ∂Γ = /0, we avoid technical difficulties associatedwith such low-regularity domains. As discussed in our numerical experiments below, however, it isreasonable to expect that results similar to ours also hold on surface counterparts of polygonal domains.

Proof. We make use of equations (5.1) and (4.22) to rewrite (5.7) as

e(x) =∫

Γ

∇Γ yG(x,y)∇Γ

(u−u`h

)dσ . (5.9)

Let G`(x,y) = G(x,a(y)) for y ∈ Γh and let Gh = IhG`. The residual equation (4.7) yields

e(x) =∫

Γ

∇Γ yG∇Γ

(u(y)−u`h(y)

)dσ =

∫Γh

r(G`−Gh) dσh−∫

Γ

[(P−A`h)∇Γ u`h] ·∇Γ G dσ

− 12 ∑

T∈T

∫∂T

R(G`−Gh) ds+∫

Γh

(µh f `− fh)Gh dσh.(5.10)

For j = 1,2,3,4 let p j,q j,s j, t j > 1 be such that p1, p4 <d

d−2 , p2 <d

d−1 , p3 <d−1d−2 , 1

p j+ 1

q j= 1,

and 1s j+ 1

t j= 1. We first recall (4.22) and apply Holder’s inequality to (5.10). Then we apply (3.14) while

choosing m = 1 or m = 2 according to the criteria explained below. For m = 1 we pick d p jd+p j

= s j <d

d−1 ,


satisfying Lemma 3.2. For m = 2 we pick s j = 1. This yields

|e(x)|. ∑T∈T

‖r‖Lq1 (T )‖G`−Gh‖Lp1 (T )+‖(P−A`

h)∇Γ u`h‖Lq2 (a(T ))‖∇Γ G‖Lp2 (a(T ))

+‖R‖Lq3 (∂T )‖G`−Gh‖Lp3 (∂T )+‖µh f `− fh‖Lq4 (T )‖Gh‖Lp4 (T )

. ∑

T∈T

‖r‖Lq1 (T )h

m−d/s1+d/p1T ∑

Ti∈ωT

|G`|W ms1(Ti)+‖(P−A`

h)∇Γ u`h‖Lq2 (a(T ))‖∇Γ G‖Lp2 (a(T ))

+‖R‖Lq3 (∂T )hm−d/s3+(d−1)/p3T ∑

Ti∈ωT

|G`|W ms3(Ti)+‖µh f `− fh‖Lq4 (T )‖Gh‖Lp4 (T )

.

(5.11)

Let T0 = T ∈ Th: x ⊂ T. Let also ω ′T = T ′ ∈ T : T ′ ∩ωT 6= /0. Subsequently we split theterms involving |G`|W m

s j(Ti), Ti ∈ ωT in two sets covering Γh. If T ∈ ω ′T0

we choose m = 1 and m = 2 if

T ∈T \ω ′T0. In the first case we pick s1 =

d p1d+p1

so m− ds1+ d

p1= 0, and in the latter case we pick s1 = 1

so m− ds1+ d

p1= 2− d + d

p1and observe that ∑T∈T \ω ′T0

(∑Ti∈ωT |G`|W 2

1 (Ti)

)6 |G`|W 2

1 (Γh\ωT0 ). Then it

follows from (5.5) with rD = hT and p1 <d

d−2 , (5.6) with c1 = hT and the same choice of p1 that

∑T∈ω ′T0

‖r‖Lq1 (T )h1−d+d/p1T ∑

Ti∈ωT

|G`|W 1s1(Ti)

. ‖ 1µh‖L∞(T )‖I−dH‖L∞(T ) max

T∈ω ′T0

‖r‖Lq1 (T )h

2−d/q1T

,

∑T∈T \ω ′T0

‖r‖Lq1 (T )h2−d+d/p1T ∑

Ti∈ωT

|G`|W 21 (Ti)

. θ∞(ωT ) maxT∈T \ω ′T0

‖r‖Lq1 (T )h

2−d/q1T

(1+ | lnh|).

(5.12)

The terms ‖I− dH‖L∞(T ) and θ∞(ωT ) come from the chain rule and Lemma 2.1. Then by a similarargument with s3 = 1 when m = 2 and s3 =

d p3d+p3

when m = 1 we get

∑T∈ω ′T0

‖R‖Lq3 (∂T )h1−d+(d−1)/p3T ∑

Ti∈ωT

|G`|W 1s3(Ti)

.

∥∥∥∥ 1µh

∥∥∥∥L∞(T )

‖I−dH‖L∞(T ) maxT∈ω ′T0

‖R‖Lq(∂T )h

1−(d−1)/q3T

,

∑T∈T \ωT0

‖R‖Lq3 (∂T )h2−d+(d−1)/p3T ∑

Ti∈ωT

|G`|W 21 (Ti)

. θ∞(ωT ) maxT∈T \ω ′T0

‖R‖Lq3 (∂T )h

1−(d−1)/q3T

(1+ | lnh|).

(5.13)


Pick s4 =d p4

d+p4. Then an inverse estimate ‖Gh‖Lp4 . h−d/s4+d/p4

T ‖Gh‖Ls4 (T ) and (3.15) yield

∑T∈T‖µh f `− fh‖Lq4 (T )‖Gh‖Lp4 (T ) 6 ∑

T∈T‖µh f `− fh‖Lq4 (T )h

−d/s4+d/p4T ‖Gh‖Ls4 (T )

6 ∑T∈T‖µh f `− fh‖Lq4 (T ) ∑

K∈ωT

h−d/s4+d/p4T

‖G`‖Ls4 (T )+hT‖∇G`‖Ls4 (T )

.

(5.14)

From Holder’s inequality follows h−d/s4+d/p4T ‖G`‖Ls4 (T ) 6 ‖G`‖Lp

4. 1− d

s4+ d

p4= 0 by our choice of

s4, and since s4 6 p4 it follows that (∑T∈T ‖∇ΓhG`‖p4Ls4 (ωT )

)1/p4 6 (∑T∈T ‖∇ΓhG`‖s4Ls4 (ωT )

)1/s4 . Thus

∑T∈T‖µh f `− fh‖Lq4 (T )‖Gh‖Lp4 (T )

6 ∑T∈T‖µh f `− fh‖Lq4 (T ) ∑

K∈ωT

‖G`‖Lp4 (T )+‖∇ΓhG`‖Ls4 (T )

6CG,q4

(∑

T∈T[(Cp4(ωT )+Cs4(ωT )‖P−dH‖L∞(ωT ))‖µh f `− fh‖Lq4 (T )]

q4

)1/q4

.

(5.15)

Here CG,q4 = ‖G‖Lp4 (Γ )+‖∇Γ G‖LS4 (Γ ). Because s4 =d p4

d+p4→ d

d−1−

as p4→ dd−2−

we see that CG,q4

blows up as p4→ dd−2−

(i.e. q4→ d2+

). Thus

∑T∈T‖µh f `− fh‖Lq4 (T )‖Gh‖Lp4 (T ) .CG,q4Cq4‖µh f `− fh‖Lq4 (Γh). (5.16)

Similarly

∑T∈T‖(P−A`

h)∇Γ u`h‖Lq2 (a(T ))‖∇Γ G‖Lp2 (a(T )) .Cq2‖(P−A`h)∇Γ u`h‖Lq2 (Γ ), (5.17)

where Corollary 5.1 gives that Cq2 → ∞ as p2 =→ dd−1− ⇐⇒ q2→ d+. Combining equations (5.11),

(5.12), (5.13), (5.16) and (5.17) to get (5.8) finishes the proof of the Theorem.

5.3 Efficiency

Lemma 4.4 gives that the residual parts of the error indicator are bounded by the L∞ norm of the errorplus some higher order geometric terms when q1 = q3 = ∞.

h2T‖µh f `+4Γhuh‖L∞(T )+hT‖J∇ΓhuhK‖L∞(∂T ) . ‖e‖L∞(ωT )(‖G ‖L∞(ωT )+hT‖DG ‖L∞(ωT ))

+hT‖G ‖L∞(ωT )‖(P−A`h)∇Γ u`h‖L∞(ωT )+h2

T‖µh( f − f )‖L∞(ωT ).(5.18)

Similar estimates follow easily from Holder’s inequality for other allowable choices of q1,q3.

6. Numerical Experiments

In this section we use our a posteriori estimates to implement an Adaptive Surface Finite ElementMethod (ASFEM). We use a maximum marking strategy with threshold 0.25, that is, we mark T for re-finement if η(T )> 0.25maxT ′∈T η(T ′). We tuned our error indicator using empirical constant factors


multiplying the residual components in order to ensure that estimators and errors had similar magni-tudes. For the L2 case the factor chosen was 0.001 and 0.01 for the pointwise case. The error indicatorfor the pointwise estimator is based on (5.8), where we choose all qi = ∞ for i = 1,2,3,4. We usediFEM Chen (2009) as a platform for our numerical experiments.

We first consider the torus obtained by rotating the circle (x−4)2 + z2 = 3.92 about the z-axis. Wetake u = x and show the adaptive results for the L2 estimator. This torus has large curvature inside of its“doughnut hole”, so we expect geometric components of the estimator to be important. In the right chart

101 102 103 104 105 10680

82

84

86

88

90

92

94

96

98

100

Percent of elements refinedwhere the Geometric component of the estimator is dominant

101 102 103 104 105 10610−3

10−2

10−1

100

101

102

103

L2 Estimator.Geometric and residual components of the estimator

C/DOFGeometric componentResidual componentL2 errorL2 Estimator

FIG. 1. Results for ASFEM solving−4Γ u = f with Γ a torus having major radius 4 and minor radius 3.9 and u = x. Left: Percentof elements marked for refinement whose geometric component of the estimator is higher than the residual one. Right: Evolutionof the L2 error, estimator, and the geometric and residual components of the estimator.

in Figure 1 the geometric part of the estimator and the overall estimator practically overlap. The residualpart is about one order of magnitude smaller than the geometric part. Both the geometric and residualcomponents appear to decrease at optimal rate DOF−1. On the left figure we observe that the majorityof elements refined have a dominant geometric component. Thus in this example the refinement ismostly being driven by the geometric component of the estimator. Note however that which componentdominates also depends on the choice of constants multiplying the estimator components.

We next take Γ as above but u = exp(

162.6975−x2

). The residual component of the estimator is more

important than when u = x above (left chart in Figure 2), which is expected because u has an exponentialpeak on the outer radius of the torus were the curvatures and thus geometric error effects are small. Inthe right chart of Figure 2 we observed unexpected oscillations in the geometric component of the L2estimator and to some extent also the error even on fine meshes. This initially seems counterintuitivesince refinement usually yields nearly monotonically decreasing estimators. After a careful analysis weobserved that although the initial mesh is nearly transverse to Γ , some of the intermediate meshes arenot, as illustrated in Figure 3. We identify this phenomena as the cause of the oscillations. In particular,the quality of the approximation of ν by νh may be worse on a finer mesh, affecting all the quantitieswhose calculation depends on it. These include the Jacobian µh = ν ·νh(1−d(x)κ1(x))(1−d(x)κ2(x))Demlow & Dziuk (2007), Ah defined in (2.13) and Ph. When we performed uniform refinement ofthe mesh, the oscillation and non-transverse intermediate meshes were not observed. Even for adaptiverefinement asymptotic convergence rates are not affected by these geometric artifacts, and a quasi-monotone decrease of the geometric error may still be expected Bonito et al. (2013).


101 102 103 104 105 1060

10

20

30

40

50

60

70

80

90

100

Percent of elements refinedwhere the Geometric component of the estimator is dominant

101 102 103 104 105 10610−3

10−2

10−1

100

101

102

103

L2 Estimator.Geometric and residual components of the estimator

C/DOFGeometric componentResidual componentL2 errorL2 Estimator

FIG. 2. Results for ASFEM on a torus with major and minor radii 4 and 3.9 and u = exp(

162.6975−x2

). In the left plot we graph

the percent of elements refined whose geometric component of the estimator is higher than the residual one. In the right plot weshow the evolution of the L2 error, residual and its components.

FIG. 3. Left: The initial transverse triangulation. Right: An intermediate triangulation. The right mesh contains darkly-shadednon-transverse elements that cause “kinks” in Γh. These were marked for refinement by ASFEM due to large geometric estimators.

We also use this surface along with the exponential peak example to test the effect of the multiplica-tive geometric constant θ2 appearing in Theorem 4.1. In our code we set CP(K) to 1. Note that |1−‖Ph[I− dH]‖L∞(a(K))| = O(h2

T ), ‖~ν − (~ν · ~nuh)~νh‖L∞(a(T )) = O(hT ), and maxi=1,2,3 ‖dPhHxi‖L∞(a(K)) =

O(h2T ). Thus we expect |θ2(T )− 1| = O(hT ). In Figure 4 we plot maxT∈T |θ(T )− 1| for uniform

mesh refinement, and indeed observe O(h) behavior for sufficiently refined meshes. However, in thepreasymptotic range the higher-order geometric term maxi=1,2,3 ‖dPhHxi‖L∞(a(K)) is dominant and wethus observe O(h2) behavior there. In the second plot in Figure 4 we compare adaptive computationscarried out with θ2 estimated accurately, and with θ(T ) = 1 for all T . We observe little difference inthe ability of the adaptive codes to effectively reduce the error. In addition, the residual component ofthe estimator is only slightly larger when we compute θ2 accurately as compared with when we simplyset θ2 = 1. This is in part due to the fact that the areas where θ2(T ) is expected to be large (the insideof the torus) are largely disjoint from the areas where the residual components of the error might natu-rally be expected to be large (around the exponential peak on the outside of the torus). However, it alsoindicates that computation of geometric information in θ may not enhance the overall accuracy of thecode in many situations. We also carried out a similar comparison with the test solution u = x. There


was slightly more difference between the overall size of the residual components of the estimator whenθ2 was computed accurately as compared with when we set θ2(T ) = 1, but as in the pictured examplethe adaptive code worked equally well with both versions of θ .

101 102 103 104 105 10610−3

10−2

10−1

100

101

102

max

T!

T|!

(T)

!1|

DOF

maxT !T |!(T ) ! 1| vs Degrees of Freedom

maxT !T

|! (T ) ! 1|

C/DOF

C/DOF 1/2

101 102 103 104 105 10610−3

10−2

10−1

100

101

102

103

DOF

L2 Estimator.Comparison when varying theta

C/DOFResidual componentL2 errorResidual component, theta=1L2 error, theta=1

FIG. 4. Left: Asymptotic decrease of Θ to 1. Right: Comparison of adaptive computation of exponential peak solution withgeometric term θ estimated accurately and with θ = 1.

For the second example we use the torus obtained by rotating the circle (x−4)2+ z2 = 1 and chooseu = exp

(1

25.2875−x2

). The solution has an exponential peak around the points (±1,0,0). We use AS-

FEM based on the L2 and pointwise error estimators. All components of the estimator converge withoptimal rate DOF−1 (the error plots are standard and thus not pictured). In Figure 5 we present meshesobtained by our L2 and pointwise ASFEMs showing more refinement near the points (±1,0,0). This isexpected since the solution has exponential peaks there and the geometric quantities H,Hxi are relativelysmall on Γ . The pointwise estimator gives a higher density of refinement near (±1,0,0) than the L2estimator, as is expected since the maximum norm is stronger.

−5

0

5

−505−1

0

1

L2 EstimatorDOF = 3596

−5

0

5

−505−1

0

1

Pointwise EstimatorDOF = 3180

FIG. 5. Intermediate meshes obtained by adaptive refinement based on L2 (left) and pointwise (right) estimators.

For the final example we apply our estimator to a spherical wedge Γ := (ρ,φ ,θ) : ρ = 1,06 φ 6π,0 6 θ 6 5π

3 (Figure 6). We chose u = sin(λθ)sin(φ)λ . Our theory does not apply to this examplesince Γ is not closed. Γ has a re-entrant corner and is thus a surface counterpart of a nonconvex

24 of 27 REFERENCES

polygonal (Euclidean) domain. Our proof for the L2 a posteriori estimator relies on H2 regularity,which does not hold on non-convex polygonal domains or for Γ . Thus we expect the L2 estimator to beunreliable as on nonconvex polyhedron; cf. Liao & Nochetto (2003); Wihler (2007). This is confirmedin the left plot of Figure 6, which shows that the L2 error decreases at a slower rate than our estimator.We point out that the L2 error estimator is dominated by the geometric component for the range degreesof freedom considered. However, it is clear that the jump term ‖J∇ΓhuhK‖L2(∂T )h

3/2T θ2(ωT ) is decreasing

at a slower rate than the overall estimator and thus is expected to dominate the estimator asymptotically.Comparing the jump estimator to the L2 error in Figure 6 corroborates that the L2 estimator is notreliable. On the other hand we expect the pointwise estimator to be reliable as on nonconvex polyhedra

100 101 102 103 104 105 10610−8

10−6

10−4

10−2

100

102

L2 Estimator and residual components.Uniform refinement

C/DOF

!µhf ! +"!huh!L2(T )h2T !2("T )

!jump(#!huh)!L2(#T )h3/2T !2"T

L2 errorL2 Estimator

−0.50

0.51

−0.8−0.6−0.4−0.2

00.2

0.40.6

0.8

−1

−0.5

0

0.5

1

DOF = 1868

FIG. 6. In the left plot we show the L2 error decrease versus the residual part of the L2 estimator. On the right plot we show amesh obtained by adaptive refinement based on our pointwise estimator

Nochetto (1995); Demlow & Georgoulis (2012) and the corresponding ASFEM to yield optimal meshrefinement. This is confirmed in Figure 7, which shows that our estimator is reliable under both uniformand adaptive refinement and that the pointwise ASFEM achieves optimal convergence. Note that similartest problems can be constructed on the closed unit sphere; cf. Demlow & Olshanskii (2012). These fitwithin our theory and are examples of cases where adaptivity is needed to recover optimal convergencerates even on smooth surfaces.

Funding

The work of both authors was partially supported by the U.S. National Science Foundation under grantDMS-1016094.

References

ADAMS, R. A. & FOURNIER, J. J. F. (2003) Sobolev spaces. Pure and Applied Mathematics (Amster-dam), vol. 140, second edn. Elsevier/Academic Press, Amsterdam, pp. xiv+305.

REFERENCES 25 of 27

100 101 102 103 104 105 10610−5

10−4

10−3

10−2

10−1

100

101

Pointwise EstimatorAdaptive Refinement

C/DOFMax errorPointwise Estimator

100 101 102 103 104 105 10610−5

10−4

10−3

10−2

10−1

100

101

Pointwise EstimatorUniform Refinement

C/DOFMax errorPointwise Estimator

FIG. 7. We show the error and estimator plots for the pointwise estimator using adaptive refinement and uniform refinement

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