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.
Mots-clés : a posteriori error estimate, complicated diffusion coefficient, defeaturing of models, combined modeling discretization adaptive strategy
@article{M2AN_2012__46_6_1389_0,
author = {Repin, Sergey I. and Samrowski, Tatiana S. and Sauter, St\'efan A.},
title = {Combined \protect\emph{a posteriori }modeling-discretization error estimate for elliptic problems with complicated interfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1389--1405},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
doi = {10.1051/m2an/2012007},
zbl = {1281.65144},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012007/}
}
TY - JOUR AU - Repin, Sergey I. AU - Samrowski, Tatiana S. AU - Sauter, Stéfan A. TI - Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1389 EP - 1405 VL - 46 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012007/ DO - 10.1051/m2an/2012007 LA - en ID - M2AN_2012__46_6_1389_0 ER -
%0 Journal Article %A Repin, Sergey I. %A Samrowski, Tatiana S. %A Sauter, Stéfan A. %T Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1389-1405 %V 46 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012007/ %R 10.1051/m2an/2012007 %G en %F M2AN_2012__46_6_1389_0
Repin, Sergey I.; Samrowski, Tatiana S.; Sauter, Stéfan A. Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1389-1405. doi : 10.1051/m2an/2012007. http://www.numdam.org/articles/10.1051/m2an/2012007/
[1] , A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains. Numer. Math. 80 (1998) 325-362. | MR | Zbl
[2] and , 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.
[3] and , A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000). | MR | Zbl
[4] and , A posteriori error estimates for the finite element method. Int. J. Numer. Math. Eng. 12 (1978) 1597-1615. | Zbl
[5] and , Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | MR | Zbl
[6] and , The problem of the selection of an a posteriori error indicator based on smoothing techniques. Int. J. Numer. Methods Eng. 36 (1993) 539-567. | Zbl
[7] and , A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal. 33 (1996) 221-246. | MR | Zbl
[8] , and , Asymptotic Analysis for Periodic Structures. Amsterdam, North-Holland (1978). | MR | Zbl
[9] , Finite elements : Theory, Fast Solvers and Application in Solid Mechanics. Cambridge University Press (2007). | Zbl
[10] , An a posteriori error estimate and a comparison theorem for the nonconforming P1 element. Calcolo 46 (2009) 149-155. | MR | Zbl
[11] and , Equilibrated residual error estimator for edge elements. Math. Comp. 77 (2008) 651-672. | MR | Zbl
[12] and , A posteriori error analysis for elliptic PDEs on domains with complicated structures. Numer. Math. 96 (2004) 691-721. | MR | Zbl
[13] , Elliptic Equations : An Introductory Course. Birkhäuser Verlag AG (2009). | MR | Zbl
[14] , Approximations by finite element functions using local regularization. RAIRO Anal. Numer. 9 (1975) 77-84. | Numdam | Zbl
[15] , , An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation. Math. Comp. 67 1361-1382 (1998). | MR | Zbl
[16] , and , Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). | MR | Zbl
[17] , and , A posteriori error estimates including algebraic error : computable upper bounds and stopping criteria for iterative solvers, SIAM J. Sci. Comput. 32 (2010) 1567-1590. | MR | Zbl
[18] and , Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Elsevier, New York (2004). | MR | Zbl
[19] , A posteriori error estimation for nonlinear variational problems by duality theory. Zapiski Nauch. Semin. (POMI) 243 (1997) 201-214. | MR | Zbl
[20] , A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp. 69 (2000) 481-500. | MR | Zbl
[21] , 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. | MR | Zbl
[22] , A Posteriori Error Estimates For Partial Differential Equations. Walter de Gruyter, Berlin (2008). | Zbl
[23] and , Functional a posteriori estimates for the reaction-diffusion problem. C. R. Math. Acad. Sci. Paris 343 (2006) 349-354. | MR | Zbl
[24] and , Computable estimates of the modeling error related to Kirchhoff-Love plate model. Anal. Appl. 8 (2010) 1-20. | MR | Zbl
[25] and , Functional a posteriori error estimates for problems with nonlinear boundary conditions. J. Numer. Math. 16 (2008) 51-81. | MR | Zbl
[26] , and , A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing 70 (2003) 205-233. | MR | Zbl
[27] , and , Duality based a posteriori error estimator for the Dirichlet problem. Proc. Appl. Math. Mech. 2 (2003) 513-514. | Zbl
[28] , and , A posteriori estimation of dimension reduction errors for elliptic problems in thin domains. SIAM J. Numer. Anal. 42 (2004) 1435-1451. | MR | Zbl
[29] , and , A posteriori control of dimension reduction errors on long domains. Proc. Appl. Math. Mech. 4 (2004) 714-715.
[30] , and , A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. J. Comput. Appl. Math. 164-165 (2004) 601-612. | MR | Zbl
[31] , and , Two-sided a posteriori error estimates for mixed formulations of elliptic problems. SIAM J. Numer. Anal. 45 (2007) 928-945. | MR | Zbl
[32] , A posteriori modeling error estimation for hierarchic plate model. Numer. Math. 74 (1996) 221-259. | MR | Zbl
[33] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Amsterdam (1996). | Zbl
[34] , 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. | MR | Zbl
[35] and , On a dimensional reduction method I. The optimal selection of basis functions. Math. Comput. 37 (1981) 31-46. | MR | Zbl
Cité par Sources :
