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Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces

Published online by Cambridge University Press:  11 April 2012

Sergey I. Repin ,
Tatiana S. Samrowski  and
Stéfan A. Sauter
Sergey I. Repin
Affiliation:
V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia. repin@pdmi.ras.ru
Tatiana S. Samrowski
Affiliation:
Institut für Mathematik, Universität Zürich, 8057 Zürich, Switzerland; tatiana.samrowski@math.uzh.ch; stas@math.uzh.ch
Stéfan A. Sauter
Affiliation:
Institut für Mathematik, Universität Zürich, 8057 Zürich, Switzerland; tatiana.samrowski@math.uzh.ch; stas@math.uzh.ch
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Abstract

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of functional type. An efficient numerical strategy is based upon balancing the modeling and discretization errors, which provides an economical way of finding an approximate solution with an a priori given accuracy. Numerical tests demonstrate the reliability and efficiency of this combined modeling-discretization method.

Keywords

A posteriori error estimatecomplicated diffusion coefficientdefeaturing of modelscombined modeling discretization adaptive strategy
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1389 - 1405
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Ainsworth, M., A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains. Numer. Math. 80 (1998) 325–362. Google Scholar
M. Ainsworth and A. Arnold, A reliable a posteriori error estimator for adaptive hierarchic modeling, in Adv. Adap. Comp. Meth. Mech., edited by P. Ladevéze and J.T. Oden (1998) 101–114.
M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000).
Babuška, I. and Rheinboldt, W.C., A posteriori error estimates for the finite element method. Int. J. Numer. Math. Eng. 12 (1978) 1597–1615. Google Scholar
Babuška, I. and Rheinboldt, W.C., Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. Google Scholar
Babuška, I. and Rodríguez, R., The problem of the selection of an a posteriori error indicator based on smoothing techniques. Int. J. Numer. Methods Eng. 36 (1993) 539–567. Google Scholar
Babuška, I. and Schwab, C., A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal. 33 (1996) 221–246. Google Scholar
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Amsterdam, North-Holland (1978).
D. Braess, Finite elements : Theory, Fast Solvers and Application in Solid Mechanics. Cambridge University Press (2007).
Braess, D., An a posteriori error estimate and a comparison theorem for the nonconforming P1 element. Calcolo 46 (2009) 149–155. Google Scholar
Braess, D. and Schöberl, J., Equilibrated residual error estimator for edge elements. Math. Comp. 77 (2008) 651–672. Google Scholar
Carstensen, C. and Sauter, S., A posteriori error analysis for elliptic PDEs on domains with complicated structures. Numer. Math. 96 (2004) 691–721. Google Scholar
M. Chipot, Elliptic Equations : An Introductory Course. Birkhäuser Verlag AG (2009).
Clément, Ph., Approximations by finite element functions using local regularization. RAIRO Anal. Numer. 9 (1975) 77–84. Google Scholar
Dörfler, W., Rumpf, M., An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation. Math. Comp. 67 1361–1382 (1998). Google Scholar
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994).
Jiranek, P., Strakos, Z. and Vohralik, M., A posteriori error estimates including algebraic error : computable upper bounds and stopping criteria for iterative solvers, SIAM J. Sci. Comput. 32 (2010) 1567–1590. Google Scholar
P. Neittaanmäki and S.I. Repin, Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Elsevier, New York (2004).
Repin, S., A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauch. Semin. (POMI) 243 (1997) 201–214. Google Scholar
Repin, S., A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp. 69 (2000) 481–500. Google Scholar
S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, Proc. of the St. Petersburg Mathematical Society IX, Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI 209 (2003) 143–171.
S. Repin, A Posteriori Error Estimates For Partial Differential Equations. Walter de Gruyter, Berlin (2008).
Repin, S. and Sauter, S., Functional a posteriori estimates for the reaction-diffusion problem. C. R. Math. Acad. Sci. Paris 343 (2006) 349–354. Google Scholar
Repin, S. and Sauter, S., Computable estimates of the modeling error related to Kirchhoff-Love plate model. Anal. Appl. 8 (2010) 1–20. Google Scholar
Repin, S. and Valdman, J., Functional a posteriori error estimates for problems with nonlinear boundary conditions. J. Numer. Math. 16 (2008) 51–81. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing 70 (2003) 205–233. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., Duality based a posteriori error estimator for the Dirichlet problem. Proc. Appl. Math. Mech. 2 (2003) 513–514. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., A posteriori estimation of dimension reduction errors for elliptic problems in thin domains. SIAM J. Numer. Anal. 42 (2004) 1435–1451. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., A posteriori control of dimension reduction errors on long domains. Proc. Appl. Math. Mech. 4 (2004) 714–715. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. J. Comput. Appl. Math. 164-165 (2004) 601–612. Google Scholar
Repin, S., Sauter, S. and Smolianski, A., Two-sided a posteriori error estimates for mixed formulations of elliptic problems. SIAM J. Numer. Anal. 45 (2007) 928–945. Google Scholar
Schwab, C., A posteriori modeling error estimation for hierarchic plate model. Numer. Math. 74 (1996) 221–259. Google Scholar
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Amsterdam (1996).
J. Valdman, Minimization of functional majorant in a posteriori error analysis based on H(div) multigrid-preconditioned CG method. Adv. Numer. Math., Advances Numer. Anal. (2009) 164519.
Vogelius, M. and Babuška, I., On a dimensional reduction method I. The optimal selection of basis functions. Math. Comput. 37 (1981) 31–46. Google Scholar