We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.
Mots-clés : Noncoercive problem, sign change, metamaterial, a posteriori error estimate, dual norm, distance to energy space, localization, equivalence local–global, minimization, best approximation, equilibrated flux, unified framework, robustness, finite element methods
@article{M2AN_2018__52_5_2037_0,
author = {Ciarlet, Patrick Jr. and Vohral{\'\i}k, Martin},
title = {Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2037--2064},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018034},
mrnumber = {3891753},
zbl = {1417.65187},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018034/}
}
TY - JOUR AU - Ciarlet, Patrick Jr. AU - Vohralík, Martin TI - Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2037 EP - 2064 VL - 52 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018034/ DO - 10.1051/m2an/2018034 LA - en ID - M2AN_2018__52_5_2037_0 ER -
%0 Journal Article %A Ciarlet, Patrick Jr. %A Vohralík, Martin %T Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2037-2064 %V 52 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018034/ %R 10.1051/m2an/2018034 %G en %F M2AN_2018__52_5_2037_0
Ciarlet, Patrick Jr.; Vohralík, Martin. Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2037-2064. doi : 10.1051/m2an/2018034. http://www.numdam.org/articles/10.1051/m2an/2018034/
[1] , Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42 (2005) 2320–2341. | DOI | MR | Zbl
[2] , A framework for obtaining guaranteed error bounds for finite element approximations. J. Comput. Appl. Math. 234 (2010) 2618–2632. | DOI | MR | Zbl
[3] , , , , , , , , and , The FEniCS Project version 1.5. Arch. Numer. Softw. 3 (2015).
[4] and , A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Method. Appl. Mech. Eng. 61 (1987) 1–40. | DOI | MR | Zbl
[5] , and , Local flux reconstructions for standard finite element methods on triangular meshes. SIAM J. Numer. Anal. 54 (2016) 2684–2706. | DOI | MR | Zbl
[6] and , Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579–608. | DOI | MR | Zbl
[7] , Dolfin-tape, DOLFIN tools for a posteriori error estimation, version “paper-norms-nonlin-code-v1.0-rc3” (2016).
[8] , and , Localization of the W-1,q norm for local a posteriori efficiency. HAL Preprint , submitted for publication (2016). | HAL | MR
[9] , and , Mesh requirements for the finite element approximation of problems with sign-changing coefficients. Numer. Math. 138 (2018) 801–838. | DOI | MR | Zbl
[10] , and , T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: M2AN 46 (2012) 1363–1387. | DOI | Numdam | MR | Zbl
[11] , and , Analyse spectrale et singularités d’un problème de transmission non coercif. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 717–720. | DOI | MR | Zbl
[12] , and , Equilibrated residual error estimates are p-robust. Comput. Method. Appl. Mech. Eng. 198 (2009) 1189–1197. | DOI | MR | Zbl
[13] and , Equilibrated residual error estimator for edge elements. Math. Comput. 77 (2008) 651–672. | DOI | MR | Zbl
[14] , Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. | DOI | MR | Zbl
[15] and , Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | DOI | MR | Zbl
[16] , , and , A review of unified a posteriori finite element error control. Numer. Math. Theory Method. Appl. 5 (2012) 509–558. | DOI | MR | Zbl
[17] and , Fully reliable localized error control in the FEM. SIAM J. Sci. Comput. 21 (1999/00) 1465–1484. | DOI | MR | Zbl
[18] and , Computational survey on a posteriori error estimators for nonconforming finite element methods for the Poisson problem. J. Comput. Appl. Math. 249 (2013) 74–94. | DOI | MR | Zbl
[19] and , A posteriori estimation of the linearization error for strongly monotone nonlinear operators. J. Comput. Appl. Math. 205 (2007) 72–87. | DOI | MR | Zbl
[20] and , T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients. Numer. Math. 124 (2013) 1–29. | DOI | MR | Zbl
[21] and , A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials. J. Comput. Appl. Math. 239 (2013) 189–207. | DOI | MR | Zbl
[22] and , Robust a posteriori error control for transmission problems with sign changing coefficients using localization of dual norms. HAL Preprint (2015). | HAL | Numdam
[23] , The Finite Element Method for Elliptic Problems. Vol. 4 of Studies in Mathematics and its Applications. North-Holland, Amsterdam (1978). | MR | Zbl
[24] , and , Convergence rates of AFEM with H-1 data. Found. Comput. Math. 12 (2012) 671–718. | DOI | MR | Zbl
[25] and , Explicit error bounds for a nonconforming finite element method. SIAM J. Numer. Anal. 35 (1998) 2099–2115. | DOI | MR | Zbl
[26] and , Explicit error bounds in a conforming finite element method. Math. Comput. 68 (1999) 1379–1396. | DOI | MR | Zbl
[27] and , Mathematical aspects of discontinuous Galerkin methods. Vol. 69 of Mathématiques & Applications. (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012). | MR | Zbl
[28] , and , hp-adaptation driven by polynomial-degree-robust a posteriori error estimates for elliptic problems. SIAM J. Sci. Comput. 38 (2016) A3220–A3246. | DOI | MR | Zbl
[29] , and , Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Method. Appl. Mech. Eng. 200 (2011) 2782–2795. | DOI | MR | Zbl
[30] and , Theory and Practice Of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | DOI | MR | Zbl
[31] and , Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35 (2013) A1761–A1791. | DOI | MR | Zbl
[32] and , Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53 (2015) 1058–1081. | DOI | MR | Zbl
[33] and , Stable broken H1 and H polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. HAL Preprint (2018). | HAL | MR | Zbl
[34] , The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method. Int. J. Numer. Method. Eng. 20 (1984) 1491–1506. | DOI | MR | Zbl
[35] and , Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. ESAIM: M2AN 50 (2016) 1333–1369. | DOI | Numdam | MR | Zbl
[36] and , Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485–509. | DOI | MR | Zbl
[37] and , A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42 (2004) 1394–1414 | DOI | MR | Zbl
[38] , and , Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comput. 72 (2003) 1067–1097. | DOI | MR | Zbl
[39] and , A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235 (2011) 4272–4282. | DOI | MR | Zbl
[40] , and , An a posteriori error estimator for the Lamé equation based on equilibrated fluxes. IMA J. Numer. Anal. 28 (2008) 331–353. | DOI | MR | Zbl
[41] and , An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl
[42] and , Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5 (1947) 241–269. | DOI | MR | Zbl
[43] , A posteriori estimates for partial differential equations. Vol. 4 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2008). | DOI | MR | Zbl
[44] and , Mixed and hybrid methods. In Vol. II of Handbook of Numerical Analysis. North-Holland, Amsterdam (1991) 523–639. | MR | Zbl
[45] , Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16 (2016) 723–750. | DOI | MR | Zbl
[46] and , Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal. 47 (2009) 2387–2405. | DOI | MR | Zbl
[47] and , Poincaré constants for finite element stars. IMA J. Numer. Anal. 32 (2012) 30–47. | DOI | MR | Zbl
[48] , A posteriori error estimates for non-linear parabolic equations. Tech. report, Ruhr-Universität Bochum (2004).
[49] , Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766–1782. | DOI | MR | Zbl
[50] , A posteriori error estimation techniques for finite element methods, in Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). | MR | Zbl
[51] , On the discrete Poincaré–Friedrichs inequalities for nonconforming approximations of the Sobolev space H1. Numer. Funct. Anal. Optim. 26 (2005) 925–952. | DOI | MR | Zbl
[52] , Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. J. Sci. Comput. 46 (2011) 397–438. | DOI | MR | Zbl
[53] , and , Surface modes of negative-parameter interfaces and the importance of rounding sharp corners. Metamaterials 2 (2008) 113–121. | DOI
Cité par Sources :
