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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 17, Issue 3

# A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L∞-Coefficients

Monika Weymuth
/ Stefan Sauter
/ Sergey Repin
• Corresponding author
• Russian Academy of Sciences, Saint Petersburg Department of V. A. Steklov Institute of Mathematics, Fontanka 27, 191 011 Saint Petersburg, Russia; and University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland
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Published Online: 2017-06-21 | DOI: https://doi.org/10.1515/cmam-2017-0015

## Abstract

We consider elliptic problems with complicated, discontinuous diffusion tensor ${A}_{0}$. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say ${A}_{\epsilon }$, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error was derived under the assumption that the difference ${A}_{0}-{A}_{\epsilon }$ becomes small with respect to the ${L}^{\mathrm{\infty }}$-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of ${A}_{0}$ have to be matched exactly by the coefficient ${A}_{\epsilon }$. Therefore the efficient application of that theory to problems with complicated or curved interfaces is limited. In this paper, we will present a refined theory, where the difference ${A}_{0}-{A}_{\epsilon }$ is measured in the ${L}^{q}$-norm for some appropriate $q\in \right]2,\mathrm{\infty }\left[$ and, hence, the geometric resolution condition is significantly relaxed.

MSC 2010: 65N30; 65N15; 35J25

## References

• [1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. 140, Elsevier, Amsterdam, 2003. Google Scholar

• [2]

I. Babuška, U. Banerjee and J. E. Osborn, Generalized finite element methods – Main ideas, results and perspective, Int. J. Comput. Methods 1 (2004), no. 1, 67–103.

• [3]

I. Babuška, G. Caloz and J. E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), no. 4, 945–981.

• [4]

I. Babuška and J. M. Melenk, The partition of unity method, Int. J. Numer. Meths. Engng. 40 (1997), no. 4, 727–758.

• [5]

I. Babuška and J. E. Osborn, Can a finite element method perform arbitrarily badly?, Math. Comp. 69 (2000), no. 230, 443–462. Google Scholar

• [6]

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators. Part I: Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), no. 6, 2294–2312.

• [7]

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators. Part II: General unstructured grids, SIAM J. Numer. Anal. 41 (2003), no. 6, 2313–2332.

• [8]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. Google Scholar

• [9]

A. Bonito, R. A. Devore and R. H. Nochetto, Adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal. 51 (2013), no. 6, 3106–3134.

• [10]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999. Google Scholar

• [11]

W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys. 2 (2007), no. 3, 367–450. Google Scholar

• [12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. Google Scholar

• [13]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. Google Scholar

• [14]

M. Križek and P. Naittaanmäki, Superconvergence phenomenon in the finite element method arising from averaging of gradients, Numer. Math. 45 (1984), 105–116.

• [15]

J. M. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1–4, 289–314.

• [16]

N. G. Meyers, An ${L}^{p}$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 17 (1963), no. 3, 189–206. Google Scholar

• [17]

T. Preusser, M. Rumpf, S. Sauter and L. O. Schwen, 3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients, SIAM J. Sci. Comput. 33 (2011), no. 5, 2115–2143.

• [18]

S. I. Repin, A Posteriori Error Estimates for Partial Differential Equations, Walter de Gruyter, Berlin, 2008. Google Scholar

• [19]

S. I. Repin, T. S. Samrowski and S. A. Sauter, Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces, ESAIM Math. Model. Numer. Anal. 46 (2012), 1389–1405.

• [20]

C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains: A New Approach to Weak, Strong and $\left(2+k\right)$-Solutions in Sobolev-Type Spaces, Pitman Research Notes in Math. 360, Addison Wesley Longman, Harlow, 1996. Google Scholar

• [21]

T. Strouboulis, I. Babuška and K. Copps, The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg. 181 (2000), no. 1–3, 43–69.

• [22]

T. Strouboulis, K. Copps and I. Babuška, The generalized finite element method: An example of its implementation and illustration of its performance, Int. J. Numer. Meths. Engng. 47 (2000), no. 8, 1401–1417.

• [23]

T. Strouboulis, K. Copps and I. Babuška, The generalized finite element method, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 32–33, 4081–4193.

• [24]

M. Weymuth, Adaptive local basis for elliptic problems with ${L}^{\mathrm{\infty }}$-coefficients, Ph.D. thesis, University of Zurich, 2016. Google Scholar

• [25]

Z. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM J. Sci. Comput. 26 (2005), 1192–1213.

## About the article

Received: 2017-04-06

Revised: 2017-05-18

Accepted: 2017-05-19

Published Online: 2017-06-21

Published in Print: 2017-07-01

Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 3, Pages 515–531, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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© 2017 Walter de Gruyter GmbH, Berlin/Boston.

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