IMA Journal of Numerical Analysis 214) Page 1 of 27 doi:1.193/imanum/drn L 2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces Fernando Camacho and Alan Demlow, Department of Mathematics, University of Kentucky, Lexington, KY 456 [Received on 27 June 214] Surface Finite Element Methods SFEM) are widely used to solve surface partial differential equations arising in applications including crystal growth, fluid mechanics and computer graphics. A posteriori error 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 norm errors. In this work we derive a posteriori L 2 and pointwise error estimates for piecewise linear SFEM for the Laplace-Beltrami equation on implicitly defined surfaces. here are two main error sources in SFEM, 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 element equations. Our work includes numerical estimation of the dependence of the error bounds on the geometric properties of the surface. We provide also numerical experiments where the estimators have been 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 surface Γ of class C 3 embedded in R 3 ; extension to higher-dimensional surfaces of codimension one is mostly immediate. A canonical surface finite element method SFEM) was defined in Dziuk 1988). In this method Γ is approximated by a polyhedral surface Γ h having triangular faces and the finite element equations using piecewise linear discrete spaces are then solved over Γ h. his is the method we consider throughout, though extension to higher order FEM and surface approximations could also be considered Demlow 29); Mekchay et al. 211). he application of SFEM in diverse areas including image and surface processing, fluid mechanics, and spectral geometry has become widespread over the past decade; cf. Clarenz et al. 24); Groß et al. 26); Gross & Reusken 27); Olshanskii et al. 29); Reuter et al. 29) among many others. Adaptive finite element methods are also now a standard tool in computational science and engineering because they are computationally more efficient than uniform refinement 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 27). Subsequent works including Ju et al. 29); Wei et al. 21); Mekchay et al. 211); Demlow & Corresponding author. E-mail: fercamacho@uky.edu E-mail: alan.demlow@uky.edu c he author 214. 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 212); Bonito et al. 213) have considered variations on such estimates and related adaptive FEM. A posteriori estimates for SFEM include standard residual a posteriori terms along with geometric components arising from the discretization of Γ. It was shown in Demlow & Dziuk 27) that if the discrete approximation f h to f used to construct the finite element solution is defined properly, then ) 1/2 Γ u u h ) L2 Γ ) C A h l2,l ω )η 1 ) 2 +C P A l h )M Γ u l h L 2 Γ ). 1.2) Here u h is the discrete solution, l denotes a lift of a function from Γ h to Γ, η 1 is a standard energy residual indicator, ω is the patch of elements in the mesh surrounding a triangle, and P is the projection onto the tangent plane to Γ. Also, M is a matrix of unit size. A l h depends on the distance function d of which Γ is the zero level set), the normal ν to Γ, the Weingarten map H whose eigenvalues are the principal curvatures of Γ, and normal ν h to Γ h. A l h satisfies P Al h L ) h 2, where h = diam ). In addition, A h l2,l ω ) 1 as h. hus the first residual term in 1.2) is of order h in an a priori sense) and the geometric error contribution term P A l h )M Γ u l h L 2 Γ ) is of order h 2. Geometric errors caused by approximation of Γ may thus still drive adaptivity on coarse meshes when implemented in adaptive algorithms cf. Demlow & Dziuk 27); Mekchay et al. 211)), but under our assumptions they generally lose importance as the mesh is refined. Also, in 1.2) the constant C does not depend on geometric properties of Γ. In this paper we present efficient a posteriori L 2 and pointwise error estimates for SFEM using piecewise linear finite element spaces. As in 1.2) our estimates include geometric and Galerkin residual) components, but as in L p a priori estimates these contributions are now roughly speaking of the same order as the Galerkin error; cf. Dziuk 1988); Demlow 29). For the L 2 norm we prove that ) 1/2 u u l h L 2 Γ ) C Γ θ 2 ω ) 2 η ) 2 +C [P A l h ) Γ u l h ] L 2 a )) 1.3) +C 1 µ h )u l h L 2 a )). Here η is an L 2 residual indicator which in an a priori sense is of order Oh 2 ). θ 2 is a geometric constant which approaches 1 as h, but it depends on the derivatives H xi of H as well as d, ν, and H and thus incorporates higher-order geometric information than does 1.2). Also, µ h is the ratio of the metrics on Γ h and Γ, and 1 µ h is of order h 2. hus both the first residual term and the second and third geometric terms in 1.3) are of order h 2 in an a priori sense, and in contrast to the energy-norm case geometric contributions may drive refinement on fine as well as on coarse meshes. Finally, C Γ depends on Γ through an H 2 regularity constant which we do not explicitly measure, so in contrast to 1.2) we do 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 domains L 2 a posteriori estimates may be proved by a standard duality argument; cf. Verfürth 1996). Maximum-norm a posteriori bounds on nonconvex Euclidean polygonal domains were first proved in Eriksson 1994); Nochetto 1995), with subsequent extensions and improvements in Dari et al. 2); Nochetto et al. 26); Demlow & Georgoulis 212). echnicalities associated with nonconvex polyhedral domains makes these analyses more challenging; our proofs are simpler because we assume that Γ is closed and smooth. We instead demonstrate via a numerical test that our L estimators are effective on a surface with reentrant corner. Applications where L 2 or L results are of interest include certain
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 3 of 27 nonlinear elliptic problems Nochetto et al. 25, 26). L 2 a posteriori bounds for the Euclidean) parabolic Allen-Cahn equation are contained in Bartels & Müller 211), and these employ both elliptic L 2 and elliptic L a posteriori estimates as building blocks. he Allen-Cahn equation may also be considered on surfaces; cf. Dziuk & Elliott 27b). Spectral properties of the Laplace-Beltrami operator have been employed as a shape DNA fingerprint which can be helpful in deciding whether two given surfaces are isometric; cf. Reuter et al. 25, 26); Reuter 21). Maximum-norm error control is natural 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 Γ h and Γ. In Section 3, we prove that Scott-Zhang interpolants yield good approximation properties on Γ h, taking into account the fact that we must consider broken Sobolev W 2 p ) spaces on Γ h instead of globally defined spaces. In Section 4 we prove that an L 2 error estimator is efficient and reliable. In Section 5 we prove similar pointwise results, and in Section 6 we present numerical tests. 2. Geometric preliminaries hroughout we consider 1.1) while assuming Γ u dσ = Γ f dσ = in order to ensure existence and uniqueness of solutions. As above we approximate Γ by a polyhedral surface Γ h having triangular faces and nodes lying on Γ. S h denotes a space of continuous functions that are affine on each face of Γ h. 2.1 Continuous and discrete surfaces Let νx) for all x Γ and ν h x) for almost all x Γ h be the unit outward normal vectors to Γ and Γ h, respectively. Let dx) be the signed distance function from Γ to x where d < inside of Γ and d > in the exterior of Γ and distancex,γ ) = dx). Following Chapter 14 of Gilbarg & rudinger 1998), because Γ is a C 3 surface there exists a tubular region U containing Γ such that the projection ax) : U Γ, ax) = x dx) νx) for x U, 2.1) is unique for all x U c.f. Demlow & Dziuk 27) equation 2.4) for sufficient conditions on the width of U). he unit normal ν is given by νx) = dx). 2.2) Let µ h x) be the Jacobian of the transformation ax) Γh : Γ h Γ, so that µ h dσ h = dσ. 2.3) We let be a shape regular conforming triangular mesh whose elements are the faces of Γ h. hus Γ h =. We also assume that a : Γ h Γ is a bijection, and that ν ν h > everywhere on Γ h. Let h denote the diameter of for any. ypical refinement algorithms in R n preserve shape regularity. his also seems to be the case over surfaces, but we are unaware of a proof c.f. Demlow & Dziuk 27) Section 2.2). We assume that the number of simplices in the patch ω = : / is bounded by a fixed constant for any. his is always true for shape regular meshes over R n, but does not necessarily hold for arbitrary surface meshes. However if the number of elements in each ω is bounded in the initial mesh in an adaptive FEM, standard algorithms maintain the bound for subsequent meshes Demlow & Dziuk 27) section 2.2). Shape regularity implies that there exist fixed constants c 1 and c 2 such that for any the following inequality holds c 1 h i h c 2 h i i ω. 2.4)
4 of 27 F. CAMACHO AND A. DEMLOW In addition, for d L ) + h ν ν h L ) h 2. 2.5) By a b we mean a Cb, where C depends on properties of Γ via PDE regularity constants and on shape regularity of the mesh but not other essential quantities. Let Px) and P h x) be the projection matrices onto the tangent spaces of Γ and Γ h respectively, i.e. Px) = I ν ν, P h x) = I ν h ν h. 2.6) he Weingarten map is given by Hx) = D 2 dx) = νx). Since νx) = 1, ν H = and H ν =. 2.2 Surface derivatives For any function η defined on an open set of R 3 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 function v regarded as a column vector, P postmultiplies the gradient so that Γ v = vp. he Laplace-Beltrami operator is defined by Γ η = Γ Γ η = Γ η ν[ Γ η)] ν. 2.8) In the above equation ν is regarded as a row vector and Γ η) is the matrix corresponding to the total derivative of Γ η. For more details see Demlow & Dziuk 27) and Dziuk & Elliott 27a). 2.3 Finite element approximation he weak form of 1.1) is: find u H 1 Γ ) such that Γ u dσ = and Γ u Γ v dσ = f v dσ v H 1 Γ ). 2.9) Γ Γ Denote by f h x) an approximation of f over Γ h satisfying Γ h f h =, for example, f h x) = µ h f ax)),x Γ h. Recalling that S h is the space of affine Lagrange finite elements on Γ h, our finite element method produces u h S h that solves the problem Γh u h Γh v h dσ h = Γ h f h v h dσ h v h S h. Γ h 2.1) 2.4 Lifts and extensions See Dziuk 1988); Demlow & Dziuk 27) for more details. We extend v defined on Γ to U by v l x) = vax)), x U. 2.11) For v h defined on Γ h we define the lift ṽ h by v h x) = ṽ h ax)), where ax) Γ is as in 2.1). For v h defined on Γ h and x U we extend ṽ h to U by the equation v l h x) = ṽ hax)). he relationship between Γ u l h ax)) and Γ h u h x) is given by Γ u l h ax)) = [I dh)x)] 1 [ I ν h ν ν h ν ] Γh u h x). 2.12)
Following Demlow & Dziuk 27) we define: A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 5 of 27 A h x) = A l h ax)) = 1 µ h x) [Px)][I dh)x)][p hx)][i dh)x)][px)]. 2.13) Equation 2.22 of Demlow & Dziuk 27) yields Γh v h Γh ψ h dσ h = [ Γ v l h ax))] [A l h ][ Γ ψh l ax))] dσ. 2.14) Γ h 2.5 Comparison of Sobolev norms on discrete and continuous surfaces Γ Our main results are proved by duality arguments involving dual functions lying in W 2 p Sobolev spaces. Dziuk 1988) contains a brief comparison of W 2 p Sobolev norms of functions on Γ and their extensions to Γ h. We give a more precise statement about the geometric dependencies of these relationships. LEMMA 2.1 Let and v W 2 p a )) for some 1 p. hen v l W 2 p ) 1 + µ h 1/p L ) P h [I dh] 2 L a )) v W 2 p a )) [ ] P h H L a )) ν ν ν h ) ν h L a )) + max dp hh xi L i=1,2,3 a )) 2.15) ) v W 1 p a )). Before beginning the proof, we mention a couple of notational conventions. First, for vectors a,b,c,d we have a b)c d) = b c)a d. Second, we regard vax)) as a column vector. Proof. By equation 2.11) and the change of variable formula 2.3) we have { } 1/p { v l W 2 p ) = [D α h vl x))] p dσ h = α =2 α =2 a ) 1 } 1/p. [D α h µ vax)))]p dσ 2.16) h 2.7), 2.1), the chain rule and the fact that the projection matrix P h is constant in each triangle yield D 2 h vl x)) = Γh {P h [P dh] vax))} = P h Γh {[P dh] vax))} = P h {[I dh] Γ vax))}p h. Next we expand the right hand side of the previous equation: D 2 h vl x)) =P h { Γ vax))[p dh] H Γ vax)) ν d[h xi Γ vax))] 3 i=1 } dh Γ vax))[p dh] P h. 2.17) Here H xi denotes the derivative of H with respect to x i, and [H xi Γ vax))] 3 i=1 is a matrix whose i-th column is given by H xi Γ vax)). Regrouping terms then yields Γ 2 vax))[i dh] dh Γ 2 vax))[i dh] = [I dh] Γ 2 vax))[i dh]. Hence we write } D 2 h vl x)) = P h {[I dh] Γ 2 vax))[i dh] H Γ vax)) ν d[h xi Γ vax))] 3 i=1 P h. 2.18)
6 of 27 F. CAMACHO AND A. DEMLOW Using 2.16), 2.18) and Hölder 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 l { } H 2 ) C h v H 1 a )) + v H 2 a )). he difference is that Lemma 2.1 includes explicit geometric information about the constant. We quickly verify that using equation 2.15) we get Lemma 3 of Dziuk 1988). Because Γ C 3, P h [I dh] L a )) 1 in 2.15). From 2.5) and Γ C 3 it follows that the term multiplying v W 1 p a )) is of order h, reducing to the statement of Dziuk 1988) Lemma 3. 3. Approximation results. Proofs of residual a posteriori error estimates typically employ quasi-interpolants of Clemént or Scott- Zhang type. In Demlow & Dziuk 27) the authors defined and proved approximation properties for such an operator on Γ h, but their operator only yields the first-order approximation properties needed for energy estimates. Such estimates are simpler because u W 1 p Γ ) implies u l W 1 p Γ h ). We must instead consider broken spaces, since u W 2 p Γ ) implies only u l W 2 p ) for. ypical proofs of higher-order approximation properties for Scott-Zhang type interpolants employ a Bramble-Hilbert lemma which in our context would require u l W 2 p ω ) for patches ω, so the standard proof does not apply. he main technical ideas in this section are essentially contained in heorem 3.1 of Veeser 214), though they are applied there in a somewhat different context. 3.1 Interpolant Scott-Zhang) Let and be as defined previously. We consider finite element spaces S h of arbitrary degree n over meshes of arbitrary space dimension, since the proof is no more difficult and more general results are of interest when considering SFEM in higher space dimensions; cf. Demlow 29). Demlow 29) also considers higher-degree surface approximations. Appropriate Scott-Zhang interpolants on such surfaces may also be obtained from the corresponding operator on Γ h by simple composition; we do not give the details. Let N be the set of Lagrange nodes of degree n on. For all nodes z N define: if z is an interior node of, F z := e z if z is an interior point of e z. Here e z is a face of the simplex, 3.1) e z for some arbitrary face containing z, if z is contained in more than one face. Let {ϕ z } z N be the nodal basis for S h, i.e., ϕ zi z j ) = δ i j and degϕ z ) = n. Let {ψ z } z N be the basis dual to {ϕ z } z N, i.e. F z ψ zi ϕ z j = δ i j, where z i,z j are nodes associated with the simplex F z and ψ z : F z R is a polynomial of degree degϕ z ) = n. Following Scott & Zhang 199), we define the interpolant I h v l of v l as I h v l = ϕ z ψ z ζ )v z N F l ζ ) dζ, v W1 1 Γ ). 3.2) z I h is a projection, that is, I h s = s s S h. For any node z N Let dim{ } = d, < h < 1. Since ϕ z 1, it follows that ψ z L q F z ) h dimf z)1 1/q). 3.3) ϕ z W k p ) h k+d/p. 3.4)
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 7 of 27 Finally, for any element and Φ W 1 p ), 1 p <, a standard scaled trace inequality cf. Brenner & Scott 28) heorem 1.6.6) yields 3.2 Approximation properties Φ L p ) h 1/p Φ L p ) + h 1 1/p Φ L p ). 3.5) Equation 4.3) of Scott & Zhang 199) and Lemma 1.13 of Ern & Guermond 24) give approximation properties for the Scott-Zhang interpolator of the form I h Φ Φ W m p ) h l m Φ W l p ω ), m l n + 1, l 1. For our purpose we consider functions Φ that lie in W l pk) for all K ω but that may not be in W l pω ). Assuming Φ W 1 1 ω ) in order to guarantee continuity of traces, our goal is to prove that I h Φ Φ W m p ) h l m K ω Φ W l p K). Lemma 3.1 and heorem 3.1 below were inspired by Veeser 214). LEMMA 3.1 Let z N and let F z be a simplex of dimension d 1 defined as in 3.1) and let be a simplex of dimension d such that z F z. Define ω to be the set of all simplices in that touch. For p 1 let Φ W l p ) for some 1 l n + 1 and for all. Let p S h be the l 1-st degree average aylor polynomial of Φ over the simplex, as defined in Lemma 4.3.8 of Brenner & Scott 28). Pick q such that 1 p + 1 = 1 and assume that satisfies the assumptions of 2.1. hen q ψ z Φ p ) ds F z h d 1)1 1/q)+l 1/p) ω Φ W l p ). 3.6) Proof. If F z is a face of, then the claim follows from the trace inequality 3.5) and the Bramble Hilbert Lemma. Hence assume that F z is a simplex of dimension at most d 2, and let { j } M j=1 be a chain of d-dimensional simplices inside of ω such that j+1 j = F j are d 1)-dimensional simplices, z F j for 1 j M, 1 = and F z is a face of M but F z F M 1. M is uniformly bounded over, by our assumptions. 3.2) yields ψ z Φ p ) ds = I h Φ p )z). F z Let ψ j be the dual to ϕ z on F j as in 3.2) so that F j ψ j ϕ z = 1 and F j ψ j v h = v h z), v h S h. Note that in general ψ z ψ j since F z F j. hen p j+1 p j )z) = F j ψ j p j+1 p j ) dσ, so that using a telescoping sum along with Hölders and triangle inequalities we obtain M 1 ψ z Φ p ) ds = ψ z Φ p M ) ds + F z F z ψ α j p j+1 p j ) ds j=1 F j M 1 ψ z L q F z ) Φ p M L p F z ) + j=1 ψ j L q F j ) ) Φ p j+1 L p F j ) + Φ p j L p F j ). 3.7)
8 of 27 F. CAMACHO AND A. DEMLOW It then follows from 2.4) and 3.3) that ψ z Φ p ) ds F z h d 1)1 1/q) + [ Φ p M L p F z ) M 1 )] Φ p j+1 L p F j ) + Φ p j L p F j ). j=1 hen by the race Inequality 3.5), shape regularity, and the Bramble Hilbert Lemma c.f. Brenner & Scott 28) 4.3.8.) we obtain M 1 )] ψ z Φ p ) ds h d 1)1 1/q)+l 1/p) [ Φ W lp M ) + Φ W l F p j+1 ) + Φ Wp l j ) z thus finishing the proof. h d 1)1 1/q)+l 1/p) ω Φ W l p ), HEOREM 3.1 Let n denote the degree of the finite element space S h, and let 1 l n + 1, 1 p <, and k l. For Φ W 1 p ω ) satisfying Φ W l p ) for all ω, I h Φ Φ W k p ) hl k j=1 3.8) ω Φ W l p ). 3.9) Proof. Let p be as in Lemma 3.1. hen the triangle inequality and 3.4) yield I h Φ Φ W k p ) Φ p W k p ) + I hφ p ) W k p ). Applying the Bramble Hilbert Lemma to the first term in the right hand side and using 3.2), we obtain I h Φ Φ W k p ) hl k Φ W l p ) + ψ z Φ p ) ds ϕ z W k F p ). z Let N denote the set of interior nodes of and let N be the set of boundary nodes of. hen I h Φ Φ W k p ) hl k Φ W l p ) + ψ z Φ p ) ds ϕ z W k z N F p ) z 3.1) + ψ z Φ p ) ds ϕ z W k F p ). z z z N Let q be such that 1 p + q 1 = 1 and let d be the dimension of. Observe that F z = for z N and the number of nodes z is bounded by a fixed constant Cn) depending on n. We use Hölders inequality, 3.3), 3.4), the Bramble Hilbert Lemma and d1 1/q) k + d/p = k to obtain ψ z Φ p ) ds ϕ z W k z N F p ) ψ z L q ) Φ p L p ) ϕ z W k p ) z 3.11) h d1 1/q) k+d/p Φ p L p ) h l k Φ W l p ).
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 9 of 27 Lemma 3.1) and the fact that the number of nodes of any is bounded by Cn) imply that ψ z Φ p ) ds ϕ z W k F p ) h d 1)1 1/q)+l 1/p) z Φ W l p ) ϕ z W k p ). ω z N 3.4) applied to ϕ z W k p ), d 1)1 1/q) + l 1/p) k + d/p = l k, and shape regularity yield z N ψ z Φ p ) ds ϕ z W K F p ) z hl k ω Φ W l p ). 3.12) We substitute 3.11) and 3.12) into 3.1) to obtain 3.9), finishing the proof. he following are scaled versions of standard Sobolev embedding theorems; cf. Adams & Fournier 23) heorem 4.12. We only consider ranges of indices used in our proofs below. LEMMA 3.2 Assume that 1 p 1 < d 2 d 1, 1 p 2 < d p j d+p j s < d 1 d, s p j for j = {1,2}. hen d d 2 and either m = 2 and s = 1 or m = 1 and Φ L p 1 ) Φ L p 2 ) m j= m j= h j d/s+d 1)/p 1 Φ j W s ), for Φ W s j ), h j d/s+d/p 2 Φ j W s ), for Φ W s j ). 3.13a) 3.13b) In the following sections we apply our approximation results in the following form. COROLLARY 3.1 Assume that either p 1 = p 2 = s = 2 and m = 1 or m = 2, or that p 1, p 2, s, and m are related as in Lemma 3.2 above. hen for, I h Φ Φ L p 1 ) h m d/s+d 1)/p 1 I h Φ Φ L p 2 ) h m d/s+d/p 2 i ω Φ W m i ω Φ W m s i ). s i ), 3.14a) 3.14b) In addition, for 1 p ) I h Φ L p ) Φ L p i ) + h Φ L p i ). 3.15) i ω Proof. We easily verify 3.14) by combining heorem 3.1 and 3.13). 3.15) follows from the triangle inequality and heorem 3.1. 3.3 A generalized Bramble-Hilbert Lemma In Scott & Zhang 199) a Bramble-Hilbert Lemma is applied over element patches in order to prove approximation properties for the Scott-Zhang interpolant. Employing the same notation as above, let
1 of 27 F. CAMACHO AND A. DEMLOW j n, l n + 1, and u W l pω ) with 1 p <. hen inf u v j v S W h p ω ) inf j u p j p P n W hl p ω ) u W l p ω ). 3.16) Lemma 3.1 and heorem 3.1 may be rewritten as a Bramble-Hilbert lemma for broken Sobolev spaces. Let j n, k = max{ j,1}, 1 l n + 1, u W 1 p ω ), and u W l p ) for each ω. hen inf v S h u v p Wp j ω ) )1/p ω inf h k j p P n u p p Wp k ) )1/p h l j u p Wp l ) )1/p. 3.17) ω he two differences between 3.16) and 3.17) are that the former uses standard and the latter broken Sobolev spaces, and that 3.17) requires k,l 1. heorem 3.2 of Veeser 214) establishes that continuous and discontinuous finite element spaces yield equivalent approximation in the H 1 seminorm not only asymptotically but on any mesh satisfying reasonable assumptions; this is essentially the first inequality in 3.17) with p = 2 and j = k = 1. We thus again emphasize that we apply techniques in Veeser 214) in a different context but with only modest generalization of the basic ideas. 4. L 2 a posteriori estimate In this section we derive an L 2 a posteriori error estimator. We first state a standard regularity result. LEMMA 4.1 Regularity c.f. Demlow 29) Lemma 2.1). Let f L 2 Γ ) satisfy f dσ =, and assume that Γ is a C 2 surface. hen the problem Lu,v) = f,v) v H 1 Γ ) has a unique weak solution satisfying u dσ = and Γ u H 2 Γ ) C f L 2 Γ ). 4.1) Γ We next define the error e := u u l h. From this point on v h will be used to denote the interpolant of v l, i.e. v h I h v l. Our main result is stated in the following theorem. HEOREM 4.1 Assume that Γ is a C 3 surface. Let ux) be the solution to 1.1) and define C p K) := 1 1/p µ, C p ω K ) = 1 1/p h L K) µ h L ω K ) θ p K) := C p K) P h [I dh] 2 L ak)) + P hh L ak)) ν ν ν h ) ν h L a )) ) + max dp hh xi L i=1,2,3 ak)), θ p ω K ) p = θ p K ) p, 1 p <, K ω K θ ω K ) = max K ω K θ K ) γ 2 K) = C 2 K)1 + h P dh L K)), γ 2 ω K ) 2 = K ω K γ 2 K ) 2. hen the following bound holds: u u l h L 2 Γ ) [ θ 2 ω ) 2{ h 4 µ h f l + Γh u h 2 L 2 ) + h3 Γh u h 2 L 2 ) + [P A l h ) Γ u l h ] 2 L 2 a )) + ul h µ hu l h 2 L 2 a )) + γ 2ω ) 2 µ h f l f h 2 L 2 ) } 4.2) ] 1/2. 4.3)
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 11 of 27 he constant hidden in depends on the regularity constant in 4.1) but not other essential quantities. he main difference between our estimators and those arising in the case of flat Euclidean domains is that we include the geometric terms C p ), θ p ), θ ω K ) and γ 2 ). hese terms arise naturally when we move between the discrete and the continuous surface as discussed in Lemma 2.1. As discussed in the introduction, Θ 2 ) 1 as h, while the remaining additive) geometric terms disappear as h. Note also that the term Θ 2 includes derivatives of H and thus requires C 3 regularity of Γ. All other portions of our estimator require only C 2 regularity of Γ. he corresponding energy norm estimates in Demlow & Dziuk 27) require C 2 regularity of Γ. he higher regularity required here is due to the fact that we map higher-order derivatives between the discrete and continuous surfaces as part of our duality argument. It has been observed in the case of energy norm a posteriori estimates that it is possible to assume only Lipschitz regularity of Γ cf. Mekchay et al. 211); Bonito et al. 213)), but at the expense of a larger geometric error. In particular, the geometric and PDE energy errors are of the same order if a Lipschitz map is used to relate the continuous and discrete surfaces. he higher rate of convergence of the geometric error when a C 2 surface is assumed is due to the fact that the orthogonal projection a used in the current work has special orthogonality properties. ranslating these observations to the case of L 2 error estimates, it would be possible to prove a posteriori bounds under the assumption that Γ is only C 1,1. However, the resulting geometric errors would converge at a lower rate than the residual PDE) portion of the error measured in L 2, and would thus dominate the estimator. We thus assume a C 3 surface. We use the following lemma to prove heorem 4.1. LEMMA 4.2 Let u and u l h be the continuous and discrete solutions of equation 1.1). respectively. Let v solve Γ v = u µ h u l h in Γ, Γ v dσ =. hen the following bound holds: u u l h 2 L 2 Γ ) I + II + III + IV + µ hu l h ul h 2 L 2 Γ ), 4.4) where I = µ h f l + Γh u h )v l v h ) dσ h, Γ h III = 1 2 II = Γh u h v l v h ) ds, IV = Γ [P A l h ) Γ u l h ] Γ v dσ, Γ h µ h f l f h )v h dσ h, and v l x), x Γ h is the lift of v to Γ h defined in 2.11). Proof. Since Γ u = and Γ h u h =, it follows that Γ u µ hu l h ) dσ =. Now we compute the L 2 norm of the error u u l h 2 L 2 Γ ) = u ul h,u µ hu l h + µ hu l h ul h ) = u u l h, Γ v) + u u l h, µ hu l h ul h ) = A + u ul h, µ hu l h ul h ), 4.5) where A = u u l h, Γ v). By integration by parts we get since Γ = /: A = u u l h, v) = Γ Γ u u l h ) Γ v dσ. 4.6)
12 of 27 F. CAMACHO AND A. DEMLOW he residual identity given in equation 3.5) of Demlow & Dziuk 27) gives us A = Γ u u l h ) Γ v dσ = µ h f l + Γh u h )v l v h ) dσ h [P A l h ) Γ u l h ] Γ v dσ Γ Γ h Γ 1 2 Γh u h v l v h ) ds + µ h f l f h )v h dσ h Γ h = I + II + III + IV. Combining equations 4.5) and 4.7) we easily get for any ε > 4.7) u u l h 2 L 2 Γ ) I + II + III + IV + u ul h L 2 Γ ) µ hu l h ul h L 2 Γ ) I + II + III + IV + ε 2 u ul h 2 L 2 Γ ) + 1 2ε µ hu l h ul h 2 L 2 Γ ). 4.8) aking ε = 2 1 and rescaling concludes the proof of the Lemma. 4.1 A posteriori upper bound Proof of heorem 4.1) We now prove bounds for elements I through IV of 4.7). Bound for I. Hölder s inequality yields µ h f l + Γh u h )v l v h ) dσ h µ h f l + Γh u h L Γ 2 ) vl v h L 2 ). 4.9) h Recall that we defined v h = I h v l. hen by 3.14b) with p = s = m = 2 we get v l v h L 2 ) h2 v l H 2 K). 4.1) K ω Next we apply Lemma 2.1, 4.2), and observe that H 2 ) bounds the H1 and H 2 semi-norms to get K ω v l H 2 K) θ 2ω ) v H 2 aω )). Finite overlap of the patches ω then yields ) 1/2 µ h f l + Γh u h )v l v h ) dσ h h 4 µ h f l + Γh u h 2 L Γ 2 ) θ 2ω ) 2 v H 2 Γ ). 4.11) h Bound for II. he Cauchy-Schwarz inequality and v H 1 Γ ) v H 2 Γ ) yields [P A l h ) Γ u l h ] Γ v dσ [P A l h ) Γ u l h ] L Γ 2 Γ ) Γ v L 2 Γ ) { } 1/2 4.12) = [P A l h ) Γ u l h ] 2 L 2 a )) v H2Γ ). Bound for III. Using Hölder s inequality, 3.14a) with p = s = m = 2, and compute as in 4.12) yields Γh u h v l v h ) ds Γh u h L 2 ) vl v h L 2 ) τ Γh u h L 2 ) h3/2 v l H 2 K) K ω 4.13) ) 1/2 Γh u h L 2 ) h3 θ 2 ω ) 2 v H 2 Γ ).
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 13 of 27 Bound for IV. It follows from Hölder s inequality and 3.15) that µ h f l f h )v h dσ h µ h f l f h ) L Γ 2 ) v h L 2 ), h µ h f l f h L 2 ) v l L 2 K) + h Γ v l L 2 K) ). K ω 4.14) hen we use 2.1), 2.3), 3.15) and Hölder s inequality to deduce that Bound for 4.7). Let Γ h µ h f l f h )v h dσ h + h I dh) v L 2 ak)) )] [ µh f l f h L 2 ) K ω C 2 K) v L 2 ak)) γ 2 ω ) µ h f l f h L 2 ) v H 2 ak)) 4.15) ) 1/2 γ 2 ω ) 2 µ h f l f h 2 L 2 ) v H2Γ )). η 2 = h 4 µ h f l + Γh u h 2 L 2 ) θ 2ω ) 2 + [P A l h ) Γ u l h ] 2 L 2 Γ ) + Γh u h 2 L 2 ) h3 θ 2 ω ) 2 + γ 2 ω ) 2 µ h f l f h 2 L 2 ). 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 µ h u l h L 2 Γ ). 4.17) We combine 4.17), 4.7), 4.4) and then use Cauchy and the triangle inequality to get u u l h 2 L 2 Γ ) η u µ hu l h L 2 Γ ) + ul h µ hu l h 2 L 2 Γ ) 1 4ε η2 + ε u µ h u l h 2 L 2 Γ ) + ul h µ hu l h 2 L 2 Γ ) 1 4ε η2 + 2ε u u l h 2 L 2 Γ ) + 2ε ul h µ hu l h 2 L 2 Γ ) + ul h µ hu l h 2 L 2 Γ ). Rearranging terms and taking ε sufficiently small write finalizes the proof of heorem 4.1. We define the error indicator ˆη ) in each triangle as follows: { } ˆη ) := h 2 µ h f l + Γh u h L 2 ) + h3/2 Γ h u h L 2 ) θ 2 ω ) + [P A l h ) Γ u l h ] L 2 a )) 4.18) + u l h µ hu l h L 2 a )) + γ 2ω ) µ h f l f h L 2 ). hus we write u u l h L 2 Γ ) ˆη 2 ) ) 1/2.
14 of 27 F. CAMACHO AND A. DEMLOW 4.2 Efficiency Next we verify that the residual part of the estimator 4.18) is bounded above by the true error plus data oscillation and geometric terms. We use standard techniques developed in Verfürth 1994). We first clarify the role of geometric terms in mappings between Γ h and Γ needed in the course of our arguments. LEMMA 4.3 Assume that and γ is an edge of. Let P be the piecewise constants on. For x Γ h let x = ax), v h ax)) = v h x), and = a ). Let φ, φ γ be the squares of the interior bubble function associated with and the edge bubble function associated with γ respectively and define [ G := [I dh] 1 I ν ] h ν. 4.19) ν h ν hen for v h P and 1 p Γ φ ṽ h ) L p ) h 2 G L ) + h 1 DG L )) v L p ), 4.2a) ) Γ φ γ ṽ h ) L p ) h 2+1/p G L ) + h 1+1/p DG L ) v L p γ). 4.2b) Proof. Let w := φ v. 2.8) yields Γ w = Γ w ν[d Γ w] ν. We apply Demlow & Dziuk 27) equation 2.19) to this equation. Let e i denote the column unit vector with entry-i equal to one and zero everywhere else, let [ ) i ], 1 i 3 denote a matrix where the i-th column is given by ) i and get Γ w = G Γh w ) ν [ D G Γh w )] ν = 3 i=1 e i G xi Γh w) ) ν [ G xi Γh w) ] 3 i=1 ν + 3 i=1 e i xi G ) Γh w ) ν [ xi G ) Γh w ] 3 i=1 ν. aking L p norm of both sides and using the triangle and Hölder s inequalities yields thus 3 Γ w L p ) e i G L i=1 ) xi Γh w L p ) + e ) i xi G L ) Γh w L p ) + νg L ) D Γh w) ν L p ) + DG L Γh w L p ), Γ w L p ) G L ) φ v W 2 p ) + DG L ) φ v W 1 p ). Because v is constant in each we can use an inverse estimate to deduce 4.2a). he same argument, after observing that we apply a scaling argument to go from to γ, yields 4.2b). We now prove our main efficiency result. Note that this bound will also be used in the following section to establish efficiency of maximum-norm error estimators. LEMMA 4.4 Let f be a piecewise constant approximation to f. Let 1 q, f L q Γ ). For x Γ h let x = ax), v h ax)) = v h x), choose µ h f l = f h, and let G be defined by equation 4.19). hen for any elementwise constant function f, h 2 µ h f l + Γh u h L q ) + h 1+1/q Γh u h L q ) e L q ω ) G L ω ) + h DG L ω )) + h G L ω ) P A l h ) Γ u l h L q ω ) + h 2 µ h f l f L q ω ). 4.21)
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 15 of 27 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 l h ) Γ u l h L 2 Γ ) in our efficiency analysis. A major reason for this exclusion is that efficiency estimates for the residual component of 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. 213). Secondly, if 4.21) were used to derive a global efficiency bound by adding the contributions over h, then 4.21) should first be multiplied through by θ 2 ω ) for the sake of consistency with 4.3). We also comment briefly on the data oscillation term h 2 µ h f l f Lq ω ). Because our residual terms are defined on Γ, this term is defined with respect to the natural mapping µ h f l of f to Γ h rather than with respect to f. In addition, it is multiplied by h 2, which is the natural scaling for L q norms. Proof. We introduce a more compact notation for the residuals: and define r := µ h f l + Γh u h, R := Γh u h. 4.22) G 1 := h 2 G L ) + h 1 DG L ). 4.23) Let r and R denote piecewise constant approximations of r and R respectively. We choose p such that 1 p + 1 q = 1, use the residual equation 4.7) and let v = φ r, v h =. Since φ and Γh φ vanish on we obtain Γ e Γ φ r) dσ = φ r r dσ h P A l h ) Γ u l h Γ φ r) dσ. hus after adding and subtracting the appropriate terms and rearranging we get φ r 2 dσ h = Γ e + P A l h ) Γ u l h ) Γ φ r) dσ + φ r r r) dσ h. 4.24) Integration by parts together with Lemma 4.3 and 4.23) gives Γ e Γ e L q ) φ r) dσ = e Γ φ r) dσ e L q ) Γ φ r L p ) h 2 G L ) + h 1 DG L )) r L p ) = G 1 e L q ) r L p ). 4.25) In a similar way we apply Hölder s inequality, use Demlow & Dziuk 27) equation 2.19 together with the definition of G, Ainsworth & Oden 2) Lemma 2.1 and heorem 2.2 to obtain P A l h ) Γ u l h Γ φ r) dσ G L ) P A l h ) Γ u l h L q ) Γ h φ r) L p ) h 1 G L ) P A l h ) Γ u l h L q ) r L p ). We combine 4.24), 4.25) and 4.26) to get φ r 2 dσ h r L p ) 4.26) { } G 1 e L q ) + h 1 G L ) P A l h ) Γ u l h L q ) + r r L q ). 4.27) It follows from Ainsworth & Oden 2) Lemma 2.1 that r 2 L 2 ) φ r 2 dσ h. Let d be the dimension of the simplex. he equivalence of norms in a finite dimensional space and a scaling argument
16 of 27 F. CAMACHO AND A. DEMLOW gives the bound h d/p+d/2 r L p ) r L 2 ). hus we have hd d1/p+1/q) r L p ) r L q ) r 2 L 2 ). Observe that d d1/p + 1/q) = and finally apply the triangle inequality to r L q ) to obtain r L q ) G 1 e L q ) + h 1 G L ) P A l h ) Γ u l h L q ) + r r L q ). 4.28) Let γ be an edge of, v := φ γ R, v h =, and µ h f l = f h. Let ω γ := { γ /} and observe that φ γ vanishes outside of ω γ. hen it follows from 4.7) that R 2 ds φ γ R 2 ds = Γ e + P A l h ) Γ u l h ) Γ φ l R) γ dσ φ γ r R dσ h γ γ ω γ ω γ + φ γ R R R) dσ. γ Following the steps used to derive 4.28) we use Hölder s inequality, Lemma 4.3 of Ainsworth & Oden 2), Lemma 2.1, 4.23), 4.28), the triangle inequality, and inverse estimates to deduce that h 1/p R L q γ) G 1 e L qω γ + h 1 G L ω γ ) P A l h ) Γ u l h L q ω γ ) + r r L q ω γ ) + R R L q γ). he result follows after multiplying this inequality by h 2, equation 4.28) by h2, substituting G 1 for the right 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 212)), we represent the error by writing a weak form of the problem using the Green s function as the auxiliary function. In this section we thus establish several properties of the Green s function for Γ. We first cite Demlow 29) Lemma 2.2. LEMMA 5.1 here exists a function Gx,y) unique up to a constant) such that for all functions ux) C 2 Γ ), ux) 1 ux) dσ = Gx,y) Γ uy)) dσ = Γ,y Gx,y) Γ,y uy) dσ. 5.1) Γ Γ Γ Γ Let αx,y) be the surface distance between x,y Γ, and let d denote the dimension of Γ. Further assume that αx,y) < 1. hen c.f. Demlow 29) Lemma 2.2, Aubin 1982) heorem 4.17) { ) ln C Gx,y) αx,y) for d = 2, 5.2) αx,y) 2 d d > 2. Let γ + β > where γ,β are multiindices. hen D γ Γ,y Dβ Γ,x Gx,y) αx,y)2 d γ+β. 5.3)
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 17 of 27 LEMMA 5.2 Let d 2 denote the dimension of the surface Γ. hen Gx,y) W 1 p Γ ), where p < d d 1. Proof. By 5.3) we have G p W 1 p Γ ) = Γ G p dσ Γ αx,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. here is a local isomorphism χ that maps U to a disk D contained in a plane of dimension d embedded in R d+1. We let µ denote the Jacobian of the transformation χ : U D. hen G p W 1 p U) µ L D) D r 1 d)p dσ D, where r := χx) χy), y U. By a linear scaling we can choose χ such that r D = r U, where r D and r U represent the radii of D and U respectively. hen we use polar coordinates to get: G p W 1 p U) µ L D) S d rd he last integral is finite whenever p1 d) + d 1 > 1 i.e. p < r p1 d) r d 1 dr ds d. 5.4) d 1 d. his completes the proof. COROLLARY 5.1 Let p satisfy p < d 1 d. hen there is a constant, Cp,d), depending on p and d such that G W 1 p U) Cp,d) and Cp,d) + as p d. d 1) Proof. From 5.4) we get where r D > is a fixed constant and clearly Cp,d) := G W 1 p U) 1 r1 d+d/p [d pd 1)] 1/p D, 5.5) 1 d pd 1) ) 1/p + as p d d 1). LEMMA 5.3 Let x be the singularity of the Green s function and let U be a neighborhood of x such that there is a constant c 1 for which the disk of radius c 1 centered at x is contained in the interior of U i.e B c1 x ) Ů). hen G W 2 1 Γ \U) 1 + ln 1 c 1 ). 5.6) Proof. Let c 2 denote the diameter of Γ. By equation 5.3), G W 2 1 Γ \U) ) c 2 c 1 r 1 dr 1 + ln 1c1. 5.2 Estimator We now state and prove the main result of this section. Define: e := ux) u l h x) + 1 u l h x) dσ. 5.7) Γ Γ
18 of 27 F. CAMACHO AND A. DEMLOW HEOREM 5.1 Let ux) be the solution to the Laplace-Beltrami equation 1.1), let h = min {h }, { } d 2, q 1 > d 2, q 2 > d, q 3 > d 1, and q 4 > d 2. Let θ ) be as in 4.2) and define ˆθ ) := θ ) + µ 1 h L ) I dh L ) and similarly for ˆθ ω ). hen for x Γ, there holds { ex) max ˆθ ω )1 + lnh ) h 2 d/q 1 µ h f l + Γh u h L q 1 ) + h 1 d 1)/q 3 Γh u h L q 3 ) )} + P A l h ) Γ u l h L q 2 Γ ) + C q4 µ h f l f h L q 4 Γh ). 5.8) he constant in depends on shape regularity properties of h and on properties of Γ via the Green s function G, and blows up as q 2 d +, q 3 d 1) + or q 1,q 4 d 2 + respectively. C q4 depends on q 4, 1 µ h L Γ ), and P dh L Γ ). REMARK 5.1 In 5.8) we use Sobolev embeddings in order to define elementwise residuals measured in L q norms for q <. his has two advantages. It allows us to admit data f not in L, and to measure the geometric term P A l h ) Γ u l h L q 2 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 case of nonconstant diffusion coefficients. In our numerical experiments we simply take q i = 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 212). hey are roughly speaking of the form 5.8), but with all geometric terms omitted. A particular focus of those works is regularity of the domain Ω, as all of them admit nonconvex polygonal or polyhedral domains. Because we prove our results under the assumption that Γ = /, we avoid technical difficulties associated with such low-regularity domains. As discussed in our numerical experiments below, however, it is reasonable 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 ) ex) = Γ y Gx,y) Γ u u l h dσ. 5.9) Γ Let G l x,y) = Gx,ay)) for y Γ h and let G h = I h G l. he residual equation 4.7) yields ) ex) = Γ y G Γ uy) u l h y) dσ = rg l G h ) dσ h [P A l h ) Γ u l h ] Γ G dσ Γ Γ h Γ 1 2 RG l G h ) ds + µ h f l f h )G h dσ h. Γ h 5.1) For j = {1,2,3,4} let p j,q j,s j,t j 1 be such that p 1, p 4 < d 2, p 2 < d 1 d, p 3 < d 1 d 2, 1 p j + q 1 j = 1, and s 1 j + t 1 j = 1. We first recall 4.22) and apply Hölder s inequality to 5.1). hen we apply 3.14) while choosing m = 1 or m = 2 according to the criteria explained below. For m = 1 we pick d p j d+p j = s j < d d d 1,
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 19 of 27 satisfying Lemma 3.2. For m = 2 we pick s j = 1. his yields { ex) r L q 1 ) G l G h L p 1 ) + P A l h ) Γ u l h L q 2 a )) Γ G L p 2 a )) } + R L q 3 ) G l G h L p 3 ) + µ h f l f h L q 4 ) G h L p 4 ) { r L q 1 ) h m d/s 1+d/p 1 G l W m s1 i ) + P A l h ) Γ u l h L q 2 a )) Γ G L p 2 a )) i ω + R L q 3 ) h m d/s 3+d 1)/p 3 G l W m s3 i ) + µ h f l f h L q 4 ) G h L p 4 ) }. i ω 5.11) Let = { h : x }. Let also ω = { : ω /}. Subsequently we split the terms involving G l W m s j i ), i ω in two sets covering Γ h. If ω we choose m = 1 and m = 2 if \ω. In the first case we pick s 1 = d p 1 d+p 1 so m s d 1 + p d 1 =, and in the latter case we pick s 1 = 1 ) so m s d 1 + p d 1 = 2 d + p d 1 and observe that \ω i ω G l W 2 1 i ) G l W 2 1 Γ h \ω ). hen it follows from 5.5) with r D = h and p 1 < d 2 d, 5.6) with c 1 = h and the same choice of p 1 that r L q 1 ) h 1 d+d/p1 G l W 1 ω i ω s1 i ) 1 { µ L ) I dh L ) max h ω r L q 1 ) h 2 d+d/p1 G l W 2 \ω 1 i ) i ω { ˆθ ω ) max r L \ω q 1 ) h 2 d/q 1 r L q 1 ) h 2 d/q 1 } 1 + lnh ). }, 5.12) he terms I dh L ) and ˆθ ω ) come from the chain rule and Lemma 2.1. hen by a similar argument with s 3 = 1 when m = 2 and s 3 = d p 3 d+p 3 when m = 1 we get R L q 3 ) h 1 d+d 1)/p3 G l W 1 ω i ω 1 I dh L ) max L ) µ h s3 i ) { R L ω q )h 1 d 1)/q 3 R L q 3 ) h 2 d+d 1)/p 3 G l W 2 1 i ) \ω i ω { } ˆθ ω ) max R L \ω q 3 ) h 1 d 1)/q 3 1 + lnh ). }, 5.13)
2 of 27 F. CAMACHO AND A. DEMLOW Pick s 4 = d p 4 d+p 4. hen an inverse estimate G h L p 4 h d/s 4+d/p 4 G h L s 4 ) and 3.15) yield µ h f l f h L q 4 ) G h L p 4 ) µ h f l f h L q 4 ) h d/s 4+d/p 4 G h L s 4 ) µ h f l f h L q 4 ) h d/s 4+d/p 4 K ω { G l L s4 ) + h G l L s4 ) }. 5.14) From Hölder s inequality follows h d/s 4+d/p 4 G l L s 4 ) G l p. 1 d L 4 s 4 + p d 4 = by our choice of s 4, and since s 4 p 4 it follows that Γh G l p 4 L s 4 ω ) )1/p 4 Γh G l s 4 L s 4 ω ) )1/s 4. hus µ h f l f h L q 4 ) G h L p 4 ) µ h f l f h L q 4 ) K ω { G l L p4 ) + Γh G l L s4 ) } ) 1/q4 C G,q4 [C p4 ω ) +C s4 ω ) P dh L ω )) µ h f l f h L q 4 ) ] q4. Here C G,q4 = G Lp4 Γ ) + Γ G LS4 Γ ). Because s 4 = d p 4 blows up as p 4 Similarly d 2 d i.e. q4 d + 2 ). hus d+p 4 5.15) d 1 d as p4 d 2 d we see that CG,q4 µ h f l f h L q 4 ) G h L p 4 ) C G,q4 C q4 µ h f l f h L q 4 Γh ). 5.16) P A l h ) Γ u l h L q 2 a )) Γ G L p 2 a )) C q2 P A l h ) Γ u l h L q 2 Γ ), 5.17) where Corollary 5.1 gives that C q2 as p 2 = d 1 d q2 d +. Combining equations 5.11), 5.12), 5.13), 5.16) and 5.17) to get 5.8) finishes the proof of the heorem. 5.3 Efficiency Lemma 4.4 gives that the residual parts of the error indicator are bounded by the L norm of the error plus some higher order geometric terms when q 1 = q 3 =. h 2 µ h f l + Γh u h L ) + h Γh u h L ) e L ω ) G L ω ) + h DG L ω )) + h G L ω ) P A l h ) Γ u l h L ω ) + h 2 µ h f f ) L ω ). 5.18) Similar estimates follow easily from Hölder s inequality for other allowable choices of q 1,q 3. 6. Numerical Experiments In this section we use our a posteriori estimates to implement an Adaptive Surface Finite Element Method ASFEM). We use a maximum marking strategy with threshold.25, that is, we mark for refinement if η ).25max η ). We tuned our error indicator using empirical constant factors
A POSERIORI ESIMAES IN L 2 AND L FOR SFEM 21 of 27 multiplying the residual components in order to ensure that estimators and errors had similar magnitudes. For the L 2 case the factor chosen was.1 and.1 for the pointwise case. he error indicator for the pointwise estimator is based on 5.8), where we choose all q i = for i = {1,2,3,4}. We used ifem Chen 29) as a platform for our numerical experiments. We first consider the torus obtained by rotating the circle x 4) 2 + z 2 = 3.9 2 about the z-axis. We take u = x and show the adaptive results for the L 2 estimator. his torus has large curvature inside of its doughnut hole, so we expect geometric components of the estimator to be important. In the right chart 1 98 96 94 92 Percent of elements refined where the Geometric component of the estimator is dominant 1 3 1 2 1 1 L2 Estimator. Geometric and residual components of the estimator C/DOF Geometric component Residual component L2 error L2 Estimator 9 1 88 86 1 1 84 82 1 2 8 1 3 1 1 1 2 1 3 1 4 1 5 1 6 1 1 1 2 1 3 1 4 1 5 1 6 FIG. 1. Results for ASFEM solving Γ u = f with Γ a torus having major radius 4 and minor radius 3.9 and u = x. Left: Percent of elements marked for refinement whose geometric component of the estimator is higher than the residual one. Right: Evolution of the L 2 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. he residual part is about one order of magnitude smaller than the geometric part. Both the geometric and residual components appear to decrease at optimal rate DOF 1. On the left figure we observe that the majority of elements refined have a dominant geometric component. hus in this example the refinement is mostly being driven by the geometric component of the estimator. Note however that which component dominates also depends on the choice of constants multiplying ) the estimator components. 1 We next take Γ as above but u = exp. he residual component of the estimator is more 62.6975 x 2 important than when u = x above left chart in Figure 2), which is expected because u has an exponential peak on the outer radius of the torus were the curvatures and thus geometric error effects are small. In the right chart of Figure 2 we observed unexpected oscillations in the geometric component of the L 2 estimator and to some extent also the error even on fine meshes. his initially seems counterintuitive since refinement usually yields nearly monotonically decreasing estimators. After a careful analysis we observed that although the initial mesh is nearly transverse to Γ, some of the intermediate meshes are not, 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 ν h may be worse on a finer mesh, affecting all the quantities whose calculation depends on it. hese include the Jacobian µ h = ν ν h 1 dx)κ 1 x))1 dx)κ 2 x)) Demlow & Dziuk 27), A h defined in 2.13) and P h. When we performed uniform refinement of the mesh, the oscillation and non-transverse intermediate meshes were not observed. Even for adaptive refinement asymptotic convergence rates are not affected by these geometric artifacts, and a quasimonotone decrease of the geometric error may still be expected Bonito et al. 213).
22 of 27 F. CAMACHO AND A. DEMLOW 1 9 8 7 6 Percent of elements refined where the Geometric component of the estimator is dominant 1 3 1 2 1 1 L2 Estimator. Geometric and residual components of the estimator C/DOF Geometric component Residual component L2 error L2 Estimator 5 1 4 3 1 1 2 1 1 2 1 1 1 2 1 3 1 4 1 5 1 6 1 3 1 1 1 2 1 3 1 4 1 5 1 6 1 FIG. 2. Results for ASFEM on a torus with major and minor radii 4 and 3.9 and u = exp. 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 we show the evolution of the L 2 error, residual and its components. 62.6975 x 2 ) FIG. 3. Left: he initial transverse triangulation. Right: An intermediate triangulation. he right mesh contains darkly-shaded non-transverse elements that cause kinks in Γ h. hese 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 multiplicative geometric constant θ 2 appearing in heorem 4.1. In our code we set C P K) to 1. Note that 1 P h [I dh] L ak)) = Oh 2 ), ν ν nu h) ν h L a )) = Oh ), and max i=1,2,3 dp h H xi L ak)) = Oh 2 ). hus we expect θ 2 ) 1 = Oh ). In Figure 4 we plot max θ ) 1 for uniform mesh refinement, and indeed observe Oh) behavior for sufficiently refined meshes. However, in the preasymptotic range the higher-order geometric term max i=1,2,3 dp h H xi L ak)) is dominant and we thus observe Oh 2 ) behavior there. In the second plot in Figure 4 we compare adaptive computations carried out with θ 2 estimated accurately, and with θ ) = 1 for all. We observe little difference in the ability of the adaptive codes to effectively reduce the error. In addition, the residual component of the estimator is only slightly larger when we compute θ 2 accurately as compared with when we simply set θ 2 = 1. his is in part due to the fact that the areas where θ 2 ) is expected to be large the inside of the torus) are largely disjoint from the areas where the residual components of the error might naturally be expected to be large around the exponential peak on the outside of the torus). However, it also indicates that computation of geometric information in θ may not enhance the overall accuracy of the code in many situations. We also carried out a similar comparison with the test solution u = x. here