×

zbMATH — the first resource for mathematics

A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. (English) Zbl 1125.65098
The paper is concerned with the unified analysis of some implicit and explicit error estimators for a general class of locally conservative methods applied to nonlinear elliptic problems. The implicit error estimator is obtained by extending the equilibrated residual method to locally conservative methods an the explicit error estimators are derived by bounding implicit error estimator further from above. The analysis is based in the decomposition of the error into a conforming part and a nonconforming part. The reliability of the methods is established within a unified framework. The results are applied to three locally conservative methods: the P1 nonconforming finite element method, the interior penalty discontinuous Galerkin method and the local discontinuous Galerkin method.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ainsworth, M., Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. numer. anal., 42, 6, 2320-2341, (2005) · Zbl 1085.65102
[2] Ainsworth, M., A synthesis of a posteriori error estimation techniques for conforming, non-conforming and discontinuous Galerkin finite element methods, (), 1-14 · Zbl 1098.65106
[3] Ainsworth, M.; Oden, J.T., A unified approach to a posteriori error estimation using element residual methods, Numer. math., 65, 1, 23-50, (1993) · Zbl 0797.65080
[4] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), John Wiley & Sons New York · Zbl 1008.65076
[5] Bank, R.E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. comp., 44, 170, 283-301, (1985) · Zbl 0569.65079
[6] Bastian, P.; Riviére, B., Superconvergence and \(H(\operatorname{div})\) projection for discontinuous Galerkin methods, Internat. J. numer. methods fluids, 170, 10, 1043-1057, (2003) · Zbl 1030.76026
[7] 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
[8] Braess, D.; Verfürth, R., A posteriori error estimators for the raviart – thomas element, SIAM J. numer. anal., 33, 6, 2431-2444, (1996) · Zbl 0866.65071
[9] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Berlin · Zbl 0788.73002
[10] Bustinza, R.; Gatica, G.N., A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions, SIAM J. sci. comput., 26, 1, 152-177, (2004) · Zbl 1079.65114
[11] Bustinza, R.; Gatica, G.N.; Cockburn, B., An a posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems, J. sci. comput., 22-23, 147-185, (2005) · Zbl 1065.76133
[12] Carstensen, C.; Bartels, S.; Jansche, S., A posteriori error estimates for nonconforming finite element methods, Numer. math., 92, 2, 233-256, (2002) · Zbl 1010.65044
[13] Dari, E.; Duran, R.; Padra, C.; Vampa, V., A posteriori error estimators for nonconforming finite element methods, RAIRO modél. math. anal. numér., 30, 4, 385-400, (1996) · Zbl 0853.65110
[14] Karakashian, O.; 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
[15] Kim, Kwang Y., Mixed finite volume method for nonlinear elliptic problems, Numer. methods partial differential equations, 21, 4, 791-809, (2005) · Zbl 1078.65105
[16] Kwang Y. Kim, A posteriori error analysis for locally conservative mixed methods, Math. Comp., posted on October 4, 2006, PII S0025-5718(06)01903-X, in press
[17] Riviére, R.; Wheeler, M.F., A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Comput. math. appl., 46, 1, 141-163, (2003) · Zbl 1059.65098
[18] Schieweck, F., A posteriori error estimates with post-processing for nonconforming finite elements, M2AN math. model. numer. anal., 36, 3, 489-503, (2002) · Zbl 1041.65083
[19] Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, (1996), Wiley-Teubner Stuttgart · Zbl 0853.65108
[20] Wohlmuth, B., A residual based error estimator for mortar finite element discretizations, Numer. math., 84, 1, 143-171, (1999) · Zbl 0962.65090
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.