Approximative Green’s Functions on Surfaces and Pointwise Error Estimates for the Finite Element Method

Heiko Kröner

doi:10.1515/cmam-2016-0036

Published by De Gruyter November 25, 2016

Approximative Green’s Functions on Surfaces and Pointwise Error Estimates for the Finite Element Method

Heiko Kröner

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2016-0036

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Abstract

In this paper we give a new proof of the L∞-error estimate for the finite element approximation of the Laplace–Beltrami equation with an additional lower order term on a surface. While the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott’s proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green’s function on the surface instead of an exact Green’s function which is obtained by lifting an Euclidean Green’s function locally from the tangent plane to the surface.

Keywords: Finite Elements; Green’s Function; Two-Dimensional Surface; A Priori Error Estimates

MSC 2010: 65N15; 65N30; 65N80

A Appendix

The goal of this appendix is to show (5.7).

We will consider lifts of objects defined on B2⁢r1⁢(z0)⊂S, U⊂Tz0⁢S or a suitable portion of Sh to another of these three surfaces with respect to the representation as a graph over U in (perpendicular) Euclidean coordinates as described in Corollary 2.1. By adding the superscripts S, T or h, we indicate to which surface the object is lifted. For instance, we denote the lift of M⊂B2⁢r1⁢(z0)⊂S to Tz0⁢S by MT. Similar correction terms as in (2.1) and (2.2) appear when we lift integrands of (with a power of r) weighted W1,p-norms. That is, if we estimate such a norm, then the lift produces (at most) a constant as factor on the right-hand side of the estimates.

From (5.9), the triangle inequality and Lemma 5.3 we deduce

(A.1)∥l~-L~h∥L2⁢(S)≤c⁢h.

We recall that l⁢(z)=log⁡|z-z0| is defined in Tz0⁢S≡ℝ2, that r denotes the distance to z0 in Tz0⁢S, and state that l has bounded mean oscillation in the following sense.

Lemma A.1

Let z1∈R2 and 0<ρ<∞. Then there is a constant l0∈R depending on z1 and ρ such that

∫{|z-z1|≤ρ}(l-l0)2≤9⁢π⁢ρ2.

Proof.

This is the assertion of [23, Lemma 2, p. 688]. ∎

We estimate E:=l~-L~h near z0.

Lemma A.2

Let 0<ρ<c1⁢h be given and B={|z-z0|≤ρ} a ball in Tz0⁢S. Then

∫S∩BS|φz0-1⁢(z)-z0|β⁢|D⁢E|p≤c⁢ρβ⁢h2-p for ⁢1≤p<β+2.

Proof.

We have |D⁢l~|≤cr where r=|φz0-1⁢(z)-z0|. We may w.l.o.g. consider r as a function as well on BS and B=BT and get

∫S∩BS|φz0⁢(z)-z0|β⁢|D⁢E|p≤c⁢∫S∩BSrβ⁢(|D⁢l~|p+|D⁢L~h|p)≤c⁢ρβ+2-p+∫Bhrβ⁢|D⁢l~h|p.

Due to Lemma A.1 there is l0∈ℝ so that

(A.2)∥l-l0∥L2⁢(BT)≤c⁢h.

Using an inverse estimate to bound a W1,∞- by a L2-norm, we get

∫Bhrβ⁢|D⁢l~h|p=∫Bhrβ⁢|D⁢(l~h-l0)|p

≤c⁢ρβ⁢h2⁢supBh⁡|D⁢(l~h-l0)|p

≤c⁢ρβ⁢h2⁢h-2⁢p⁢∥l~h-l0∥L2⁢(Bh)p

≤c⁢ρβ⁢h2⁢h-2⁢p⁢(∥L~h-l~∥L2⁢(BS)+∥l-l0∥L2⁢(BT))p

≤c⁢ρβ⁢h2-p

in view of (A.1) and (A.2). ∎

If we choose β=0 in Lemma A.2, we obtain that ∥D⁢E∥L1⁢(S∩BS)=O⁢(h).

We estimate E outside BS, where B is as in Lemma A.2, which means de facto in B2∖BS since we have supp⁡l~⊂B3/2⁢r0⁢(z0). We get

∫B2∖BS|D⁢E|≤(∫B2∖BSr-2)12⁢(∫Sr2⁢|D⁢E|2)12≤c⁢|log⁡h|12⁢(∫Sr2⁢|D⁢E|2)12

and

∫Sr2⁢|D⁢E|2=∫S〈D⁢E,D⁢(r2⁢E)〉-2⁢E⁢r⁢〈D⁢E,D⁢r〉

≤∫S〈D⁢E,D⁢(r2⁢E)〉+2⁢(∫SE2)12⁢(∫Sr2⁢|D⁢E|2)12

which leads by the Peter–Paul inequality to

∫Sr2⁢|D⁢E|2≤2⁢∫S〈D⁢E,D⁢(r2⁢E)〉+4⁢∫SE2≤2⁢∫S〈D⁢E,D⁢(r2⁢E)〉+c⁢h2

in view of (A.1). The next goal is to show

(A.3)∫S〈D⁢E,D⁢(r2⁢E)〉≤14⁢∫Sr2⁢|D⁢E|2+c⁢h2⁢|log⁡h|

which implies (5.7). Define

T1={τ∈Th:dist⁡(z~0,τ)≥h},Ω1=⋃τ∈T1τ⊂Sh.

Note that, for small h,

{z∈S:distS⁢(z,z0)≥3⁢h}⊂Ω1S

and l~∈C∞⁢(Ω1S). Let l~I be a function in Vh which equals l~ at all nodes in Ω1. Let l¯I denote the lift of l~I to S.

Lemma A.3

There hold

∫Ω1S(l¯I-l~)2≤c⁢h2,∫Ω1Sr-2⁢|D⁢(r2⁢(l¯I-l~))|2≤c⁢h2⁢|log⁡h|.

Proof.

Let τ∈T1. Then

∥l~-l¯I∥Ws,∞⁢(τS)≤c⁢h2-s⁢∥l~∥W2,∞⁢(τS)≤c⁢h2-s⁢(minτ⁡r)-2,s=0,1.

Since minτ⁡r≥h and maxτ⁡r-minτ⁡r≤h, we have

(A.4)maxτ⁡rminτ⁡r≤2

and hence, for β≥0,

∫τSrβ⁢(l~-l¯I)2+h2⁢∫τSrβ⁢|D⁢(l~-l¯h)|2≤c⁢∫τSrβ⁢h4⁢(minτ⁡r)-4≤c⁢∫τSrβ-4⁢h4.

Summing over all τ∈T1 implies the lemma since

∫τSrβ-4⁢h4≤{c⁢hβ+2if ⁢β<2,c⁢|log⁡h|⁢h4if ⁢β=2

and

r-2⁢|D⁢(r2⁢(l~-l¯I))|2≤8⁢(l~-l¯I)2+2⁢r2⁢|D⁢(l~-l¯I)|2.∎

We conclude

∥l~I-l~h∥L∞⁢(Ω1)≤c⁢h-1⁢∥l~I-l~h∥L2⁢(Ω1)

≤c⁢h-1⁢(∥l¯I-l~∥L2⁢(Ω1S)+∥L~h-l~∥L2⁢(Ω1S))

(A.5)≤c

in view of Lemma A.3, (5.8) and (A.1).

Lemma A.4

Let φ∈Vh and v=(r2⁢φ)I∈Vh the linear interpolation of r2⁢φ in Ω1. Then

∫Ω1r-2⁢|D⁢(r2⁢φ-v)|2≤c⁢∫Ω1φ2.

Proof.

For τ∈T1 we have

(A.6)∥r2⁢φ-v∥W1,∞⁢(τ)≤c⁢h⁢∥r2⁢φ∥W2,∞⁢(τ)≤c⁢h⁢∑j=12∥r2∥Wj,∞⁢(τ)⁢∥φ∥W2-j,∞⁢(τ)

because D2⁢(φ|τ)=0. In view of (A.4) and r≥h on Ω1 there hold

∥r2∥Wj,∞⁢(τ)≤c⁢infτ⁡r2-j≤c⁢infτ⁡r⁢h1-j

and, in view of an inverse estimate,

∥φ∥W2-j,∞⁢(τ)≤c⁢hj-3⁢∥φ∥L2⁢(τ).

Applying these estimates in (A.6) gives

∥r2⁢φ-v∥W1,∞⁢(τ)≤c⁢h-1⁢infτ⁡r⁢∥φ∥L2⁢(τ)

which leads to

∫τr-2⁢|D⁢(r2⁢φ-v)|2≤c⁢∫τφ2

by estimating the integrand in the L∞-norm. Summing over τ∈T1 gives the claim. ∎

Lemma A.5

Estimate (A.3) holds.

Proof.

For vh∈Vh we have

a⁢(E,Vh)=O⁢(h2)⁢∥L~h∥H1⁢(S)⁢∥Vh∥H1⁢(S)

and estimate

∫S〈D⁢E,D⁢(r2⁢E)〉=∫S〈D⁢E,D⁢(r2⁢E-Vh)〉-∫SE⁢Vh+O⁢(h2)⁢∥L~h∥H1⁢(S)⁢∥Vh∥H1⁢(S)

(A.7)≤∫Ω1〈D⁢E,D⁢(r2⁢E-Vh)〉+c⁢(h2+h⁢|Vh|W1,∞⁢(S∖Ω1))+∫SVh2+O⁢(h2)⁢∥L~h∥H1⁢(S)⁢∥Vh∥H1⁢(S),

where the inequality follows from Lemma A.2 and (A.1). If vh interpolates r2⁢(l¯I-L~h) in Ω1, then

∫Ω1S〈D⁢E,D⁢(r2⁢E-Vh)〉≤116⁢∫Sr2⁢|D⁢E|2+4⁢∫Ω1Sr-2⁢|D⁢(r2⁢E-Vh)|2

≤116⁢∫Sr2⁢|D⁢E|2+8⁢∫Ω1Sr-2⁢|D⁢(r2⁢(l~-l¯I))|2+8⁢∫Ω1Sr-2⁢|D⁢(r2⁢(l¯I-L~h)-Vh)|2

≤116⁢∫Sr2⁢|D⁢E|2+c⁢h2⁢|log⁡h|+c⁢∫Ω1S(l¯I-L~h)2

≤116⁢∫Sr2⁢|D⁢E|2+c⁢h2⁢|log⁡h|+c⁢(∫Ω1S(l¯I-l~)2+∫Ω1S(l~-L~h)2)

(A.8)≤116⁢∫Sr2⁢|D⁢E|2+c⁢h2⁢|log⁡h|,

where the third inequality follows from Lemmas A.3 and A.4, and the fifth by Lemma A.3 and (A.1). We use (A.8) to estimate the first summand on the right-hand side of (A.7) and obtain

∫S〈D⁢E,D⁢(r2⁢E)〉≤c⁢h2⁢|log⁡h|+116⁢∫Sr2⁢|D⁢E|2+c⁢h⁢∥Vh∥W1,∞⁢(S∖Ω1S)+∫SVh2+O⁢(h2)⁢∥L~h∥H1⁢(S)⁢∥Vh∥H1⁢(S).

We estimate vh with standard interpolation estimates:

h⁢∥vh∥W1,∞⁢(Sh∖Ω1)+h-1⁢∥vh∥L2⁢(Sh∖Ω1)≤c⁢supS∖Ω1S⁡|r2⁢(l¯I-L~h)|=sup∂⁡Ω1S⁡|r2⁢(l¯I-L~h)|≤c⁢h2,

where we assume w.l.o.g. that vh is zero at all nodes in the interior of Sh∖Ω1 and for the last inequality estimate (A.5). Furthermore, we have

∥vh∥L2⁢(Ω1)≤c⁢∥r2⁢(l¯I-L~h)∥L2⁢(Ω1S)≤c⁢∥l¯I-l~∥L2⁢(Ω1S)+∥l~-L~h∥L2⁢(Ω1S)≤c⁢h

in view of Lemma A.3, (5.8), (A.1) and

∥Vh∥H1⁢(S)≤c⁢h-1⁢∥Vh∥L2⁢(S)≤c⁢h

and

∥L~h∥H1⁢(S)≤c⁢h-1⁢∥L~h∥L2⁢(S)≤c.∎

References

[1] Bertalmio M., Cheng L.-T., Osher S. and Sapiro G., Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2001), no. 2, 759–780. 10.1006/jcph.2001.6937Search in Google Scholar

[2] Brenner S. C. and Scott L. R., The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math., Springer, Berlin, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[3] Dedner A., Madhavan P. and Stinner B., Analysis of the discontinuous Galerkin method for elliptic problems on surfaces, IMA J. Numer. Anal. 33 (2013), no. 3, 952–973. 10.1093/imanum/drs033Search in Google Scholar

[4] Demlow A., Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 2, 805–827. 10.1137/070708135Search in Google Scholar

[5] Dziuk G., Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155. 10.1007/BFb0082865Search in Google Scholar

[6] Dziuk G. and Elliott C. M., Finite elements on evolving surfaces, IMA J. Numer. Anal. 25 (2007), 385–407. 10.1093/imanum/drl023Search in Google Scholar

[7] Dziuk G. and Elliott C. M., Finite element methods for surface PDEs, Acta Numer. 22 (2013), 289–396. 10.1017/S0962492913000056Search in Google Scholar

[8] Dziuk G. and Elliott C. M., L2 estimates for the evolving surface finite element method, SIAM J. Numer. Anal. 50 (2013), 2677–2694. 10.1137/110828642Search in Google Scholar

[9] Dziuk G., Elliott C. M. and Heine C.-J., A h-narrow band finite-element method for elliptic equations on implicit surfaces, IMA J. Numer. Anal. 30 (2013), no. 2, 351–376. 10.1093/imanum/drn049Search in Google Scholar

[10] Dziuk G., Lubich C. and Mansour D., Runge–Kutta time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 32 (2012), 394–416. 10.1093/imanum/drr017Search in Google Scholar

[11] Haverkamp R., Eine Aussage zur L∞-Stabilität und zur genauen Konvergenzordnung der H01-Projektionen, Numer. Math. 44 (1984), 393–405. 10.1007/BF01405570Search in Google Scholar

[12] Kovács B. and Power C., Maximum norm stability and error estimates for the evolving surface finite element method, preprint 2015, https://arxiv.org/abs/1510.00605. 10.1002/num.22212Search in Google Scholar

[13] Lubich C., Mansour D. and Venkataraman C., Backward difference time discretization of parabolic differential equations on evolving surfaces, IMA J. Numer. Anal. 33 (2013), no. 4, 1365–1385. 10.1093/imanum/drs044Search in Google Scholar

[14] Natterer F., Über die punkweise Konvergenz finiter Elemente, Numer. Math. 25 (1975/76), 67–77. 10.1007/BF01419529Search in Google Scholar

[15] Nitsche J., L∞ convergence of finite element approximation, Mathematical Aspects of Finite Element Methods (Rome 1975), Lecture Notes in Math. 606, Springer, Berlin (1977), 261–274. 10.1007/BFb0064468Search in Google Scholar

[16] Nitsche J. A. and Schatz A. H., Interior estimates for Ritz–Galerkin methods, Math. Comp. 28 (1974), 937–958. 10.1090/S0025-5718-1974-0373325-9Search in Google Scholar

[17] Olshanskii M. A. and Reusken A., Error analysis of a space-time finite element method for solving PDEs on evolving surfaces, SIAM J. Numer. Anal. 52 (2014), no. 4, 2092–2120. 10.1137/130936877Search in Google Scholar

[18] Olshanskii M. A., Reusken A. and Grande J., A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3339–3358. 10.1137/080717602Search in Google Scholar

[19] Olshanskii M. A., Reusken A. and Grande J., A finite element method for surface PDEs: Matrix properties, Numer. Math. 114 (2010), no. 3, 491–520. 10.1007/s00211-009-0260-4Search in Google Scholar

[20] Olshanskii M. A., Reusken A. and Xu X., An Eulerian space-time finite element method for diffusion problems on evolving surfaces, SIAM J. Numer. Anal. 52 (2014), no. 3, 1354–1377. 10.1137/130918149Search in Google Scholar

[21] Schatz A. H., Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global Estimates, Math. Comp. 67 (1998), 877–899. 10.1090/S0025-5718-98-00959-4Search in Google Scholar

[22] Schatz A. H. and Wahlbin L. B., Interior maximum-norm estimates for finite element methods. Part II, Math. Comp. 64 (1995), 907–928. 10.2307/2153476Search in Google Scholar

[23] Scott R., Optimal L∞-estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. 10.1090/S0025-5718-1976-0436617-2Search in Google Scholar

Received: 2015-11-26

Revised: 2016-10-16

Accepted: 2016-10-20

Published Online: 2016-11-25

Published in Print: 2017-1-1

Approximative Green’s Functions on Surfaces and Pointwise Error Estimates for the Finite Element Method

Abstract

A Appendix

Proof.

Proof.

Proof.

Proof.

Proof.

References

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