Ern, Alexandre; Nicaise, Serge; Vohralík, Martin An accurate \(\mathbf H\)(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. (English) Zbl 1129.65085 C. R., Math., Acad. Sci. Paris 345, No. 12, 709-712 (2007). Summary: We introduce a new \(\mathbf H\)(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. The reconstructed flux is computed elementwise and its divergence equals the \(L^{2}\)-orthogonal projection of the source term onto the discrete space. Moreover, the energy-norm of the error in the flux is bounded by the discrete energy-norm of the error in the primal variable, independently of diffusion heterogeneities. Cited in 29 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N15 Error bounds for boundary value problems involving PDEs Keywords:error bounds; \(\mathbf H\)(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems PDF BibTeX XML Cite \textit{A. Ern} et al., C. R., Math., Acad. Sci. Paris 345, No. 12, 709--712 (2007; Zbl 1129.65085) Full Text: DOI References: [1] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), John Wiley and Sons New York, NY · Zbl 1008.65076 [2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 5, 1749-1779, (2002) · Zbl 1008.65080 [3] 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 [4] 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 [5] S. Cochez-Dhondt, S. Nicaise, Equilibrated error estimators for discontinuous Galerkin methods, Technical report, Université de Valenciennes, 2007, NMPDE, submitted for publication · Zbl 1160.65056 [6] Ern, A.; Guermond, J.-L., Discontinuous Galerkin methods for Friedrichs’ systems. II. second-order elliptic pdes, SIAM J. numer. anal., 44, 6, 2363-2388, (2006), (electronic) · Zbl 1133.65098 [7] A. Ern, A.F. Stephansen, A posteriori energy-norm error estimates for advection – diffusion equations approximated by weighted interior penalty methods, Technical Report 364, Ecole nationale des ponts et chaussées, 2007 · Zbl 1174.65034 [8] A. Ern, A.F. Stephansen, M. Vohralík, Improved energy norm a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods, Technical report, Ecole nationale des ponts et chaussées and Université Pierre et Marie Curie, 2007 [9] A. Ern, A.F. Stephansen, P. Zunino, A discontinuous Galerkin method with weighted averages for advection – diffusion equations with locally small and anisotropic diffusivity, Technical Report 332, Ecole nationale des ponts et chaussées, 2007, IMAJNA, submitted for publication · Zbl 1165.65074 [10] Neitaanmaäki, P.; Repin, S., Reliable methods for computer simulation: error control and a posteriori error estimates, (2004), Elsevier Amsterdam, The Netherlands 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.