×

zbMATH — the first resource for mathematics

Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. (English) Zbl 1190.65165
Authors’ abstract: We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction.
We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using \(\mathbf H(\text{div},\varOmega)\)-conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers.
Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Becker, R.; Hansbo, P.; Larson, M.G., Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. methods appl. mech. engrg., 192, 5-6, 723-733, (2003) · Zbl 1042.65083
[2] Karakashian, O.A.; Pascal, F., A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. numer. anal., 41, 6, 2374-2399, (2003) · Zbl 1058.65120
[3] Carstensen, C.; Gudi, T.; Jensen, M., A unifying theory of a posteriori error control for discontinuous Galerkin FEM, Numer. math., 112, 3, 363-379, (2009) · Zbl 1169.65106
[4] Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. numer. anal., 45, 4, 1777-1798, (2007) · Zbl 1151.65083
[5] Kim, K.Y., A posteriori error estimators for locally conservative methods of nonlinear elliptic problems, Appl. numer. math., 57, 1065-1080, (2007) · Zbl 1125.65098
[6] S. Cochez-Dhondt, Méthodes d’éléments finis et estimations d’erreur a posteriori, Ph.D. Thesis, Université de Valenciennes et du Hainaut-Cambrésis, 2007
[7] A.F. Stephansen, Méthodes de Galerkine discontinues et analyse d’erreur a posteriori pour les problèmes de diffusion hétérogène, Ph.D. Thesis, Ecole Nationale des Ponts et Chaussées, 2007
[8] Cochez-Dhondt, S.; Nicaise, S., Equilibrated error estimators for discontinuous Galerkin methods, Numer. methods partial differential equations, 24, 5, 1236-1252, (2008) · Zbl 1160.65056
[9] A. Ern, A.F. Stephansen, M. Vohralík, Improved energy norm a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods, HAL Preprint 00193540, version 1 (04-12-2007) Université Paris 6 and Ecole des Ponts, 2007
[10] Lazarov, R.; Repin, S.; Tomar, S.K., Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems, Numer. methods partial differential equations, 25, 4, 952-971, (2009) · Zbl 1167.65451
[11] Prager, W.; Synge, J.L., Approximations in elasticity based on the concept of function space, Quart. appl. math., 5, 241-269, (1947) · Zbl 0029.23505
[12] P. Ladevèze, Comparaison de modèles de milieux continus, Ph.D. Thesis, Université Pierre et Marie Curie (Paris 6), 1975
[13] Ladevèze, P.; Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. numer. anal., 20, 3, 485-509, (1983) · Zbl 0582.65078
[14] Destuynder, P.; Métivet, B., Explicit error bounds in a conforming finite element method, Math. comp., 68, 228, 1379-1396, (1999) · Zbl 0929.65095
[15] Verfürth, R., A posteriori error estimators for convection – diffusion equations, Numer. math., 80, 4, 641-663, (1998) · Zbl 0913.65095
[16] Ern, A.; Stephansen, A.F., A posteriori energy-norm error estimates for advection – diffusion equations approximated by weighted interior penalty methods, J. comput. math., 26, 4, 488-510, (2008) · Zbl 1174.65034
[17] Vohralík, M., A posteriori error estimates for lowest-order mixed finite element discretizations of convection – diffusion – reaction equations, SIAM J. numer. anal., 45, 4, 1570-1599, (2007) · Zbl 1151.65084
[18] Vohralík, M., Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods, Numer. math., 111, 1, 121-158, (2008) · Zbl 1160.65059
[19] Verfürth, R., Robust a posteriori error estimates for stationary convection – diffusion equations, SIAM J. numer. anal., 43, 4, 1766-1782, (2005) · Zbl 1099.65100
[20] Sangalli, G., Analysis of the advection – diffusion operator using fractional order norms, Numer. math., 97, 4, 779-796, (2004) · Zbl 1063.65127
[21] Sangalli, G., A uniform analysis of nonsymmetric and coercive linear operators, SIAM J. math. anal., 36, 6, 2033-2048, (2005), (electronic) · Zbl 1114.35060
[22] Sangalli, G., Robust a-posteriori estimator for advection – diffusion-reaction problems, Math. comp., 77, 261, 41-70, (2008), (electronic) · Zbl 1130.65083
[23] Schötzau, D.; Zhu, L., A robust a-posteriori error estimator for discontinuous Galerkin methods for convection – diffusion equations, Appl. numer. math., 59, 9, 2236-2255, (2009) · Zbl 1169.65108
[24] Burman, E.; Zunino, P., A domain decomposition method based on weighted interior penalties for advection – diffusion – reaction problems, SIAM J. numer. anal., 44, 4, 1612-1638, (2006), (electronic) · Zbl 1125.65113
[25] Ern, A.; Stephansen, A.F.; Zunino, P., A discontinuous Galerkin method with weighted averages for advection – diffusion equations with locally small and anisotropic diffusivity, IMA J. numer. anal., 29, 2, 235-256, (2009) · Zbl 1165.65074
[26] Di Pietro, D.A.; Ern, A.; Guermond, J.-L., Discontinuous Galerkin methods for anisotropic semidefinite diffusion with advection, SIAM J. numer. anal., 46, 2, 805-831, (2008) · Zbl 1165.49032
[27] Bastian, P.; Rivière, B., Superconvergence and \(H(\operatorname{div})\) projection for discontinuous Galerkin methods, Internat. J. numer. methods fluids, 42, 10, 1043-1057, (2003) · Zbl 1030.76026
[28] Ern, A.; Nicaise, S.; Vohralík, M., An accurate \(\mathbf{H}(\operatorname{div})\) flux reconstruction for discontinuous Galerkin approximations of elliptic problems, C. R. math. acad. sci. Paris, 345, 709-712, (2007) · Zbl 1129.65085
[29] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, () · Zbl 1009.65067
[30] Payne, L.E.; Weinberger, H.F., An optimal Poincaré inequality for convex domains, Arch. ration. mech. anal., 5, 286-292 (1960), (1960) · Zbl 0099.08402
[31] Bebendorf, M., A note on the Poincaré inequality for convex domains, Z. anal. anwend., 22, 4, 751-756, (2003) · Zbl 1057.26011
[32] Carstensen, C.; Funken, S.A., Constants in clément-interpolation error and residual based a posteriori error estimates in finite element methods, East – west J. numer. math., 8, 3, 153-175, (2000) · Zbl 0973.65091
[33] Vohralík, M., On the discrete poincaré-friedrichs inequalities for nonconforming approximations of the Sobolev space \(H^1\), Numer. funct. anal. optim., 26, 7, 925-952, (2005) · Zbl 1089.65124
[34] Cheddadi, I.; Fučík, R.; Prieto, M.I.; Vohralík, M., Guaranteed and robust a posteriori error estimates for singularly perturbed reaction – diffusion problems, M2AN math. model. numer. anal., 43, 5, 867-888, (2009) · Zbl 1190.65164
[35] Verfürth, R., A note on constant-free a posteriori error estimates, SIAM J. numer. anal., 47, 4, 3180-3194, (2009) · Zbl 1197.65177
[36] Ainsworth, M., Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. numer. anal., 42, 6, 2320-2341, (2005) · Zbl 1085.65102
[37] Bernardi, C.; Verfürth, R., Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. math., 85, 4, 579-608, (2000) · Zbl 0962.65096
[38] Burman, E.; Ern, A., Continuous interior penalty \(h p\)-finite element methods for advection and advection – diffusion equations, Math. comp., 76, 259, 1119-1140, (2007) · Zbl 1118.65118
[39] M. Ainsworth, R. Rankin, Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes, Research Report, University of Strathclyde, 2008 · Zbl 1208.65155
[40] Ern, A.; Vohralík, M., Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids, C. R. math. acad. sci. Paris, 347, 441-444, (2009) · Zbl 1161.65085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.