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A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. (English) Zbl 1169.65108
The authors propose and analyze a robust a-posteriori error estimator for discontinuous Galerkin discretizations of stationary convection-diffusion problems. They utilize the upwind discretization of the transport terms and the classical interior penalty method for the diffusive terms. The estimator yields global upper and lower bounds of the error measured in terms of the energy norm and a semi-norm associated with the convective term in the equation. The ratio of the upper and lower bounds is independent of the magnitude of the Péclet number of the problem, and hence the estimator is fully robust for convection-dominated problems. The error measure used in the paper includes a non-local norm. Numerical examples indicate that this error contribution is smaller than the energy error and of high order once the mesh is sufficiently refined.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
Software:
deal.ii
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[1] Ainsworth, M.; Oden, J., A-posteriori error estimation in finite element analysis, Wiley-interscience series in pure and applied mathematics, (2000), Wiley New York · Zbl 1008.65076
[2] Alaoui, L.E.; Ern, A.; Burman, E., A-priori and a-posteriori analysis of non-conforming finite elements with face penalty for advection – diffusion equations, IMA J. numer. anal., 27, 151-171, (2007) · Zbl 1112.65109
[3] Arnold, D., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1980) · Zbl 0482.65060
[4] Arnold, D.; Brezzi, F.; Cockburn, B.; Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 1749-1779, (2002) · Zbl 1008.65080
[5] Bangerth, W.; Hartmann, R.; Kanschat, G., differential equations analysis library, (2005), Technical Reference, fifth ed., URL:
[6] W. Bangerth, R. Hartmann, G. Kanschat, deal.II — a general purpose object oriented finite element library, ACM Trans. Math. Software 33, 24:1-24:27 · Zbl 1365.65248
[7] Becker, R.; Hansbo, P.; Larson, M., Energy norm a-posteriori error estimation for discontinuous Galerkin methods, Comput. methods appl. mech. engrg., 192, 723-733, (2003) · Zbl 1042.65083
[8] Becker, R.; Hansbo, P.; Stenberg, R., A finite element method for domain decomposition with non-matching grids, Modél. math. anal. numér., 37, 209-225, (2003) · Zbl 1047.65099
[9] Bustinza, R.; Gatica, G.; Cockburn, B., An a-posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems, J. sci. comput., 22, 147-185, (2005) · Zbl 1065.76133
[10] Cockburn, B., Discontinuous Galerkin methods for convection-dominated problems, (), 69-224 · Zbl 0937.76049
[11] ()
[12] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[13] Girault, V.; Raviart, P.-A., Finite element methods for navier – stokes equations: theory and algorithms, Springer series in computational mathematics, vol. 5, (1986), Springer
[14] Harriman, K.; Houston, P.; Senior, B.; Süli, E., hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form, () · Zbl 1037.65117
[15] P. Houston, D. Schötzau, T. Wihler, hp-adaptive discontinuous Galerkin finite element methods for the Stokes problem, in: P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, D. Knörzer (Eds.), Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, vol. II, University of Jyväskylä, Jyväskylä, 2004, URL: http://www.mit.jyu.fi/eccomas2004/proceedings/proceed.html · Zbl 1179.76053
[16] Houston, P.; Schötzau, D.; Wihler, T., Energy norm a-posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem, J. sci. comput., 22, 357-380, (2005) · Zbl 1065.76139
[17] Houston, P.; Schötzau, D.; Wihler, T., An hp-adaptive discontinuous Galerkin FEM for nearly incompressible linear elasticity, Comput. methods appl. mech. engrg., 195, 3224-3246, (2006) · Zbl 1118.74049
[18] Houston, P.; Schötzau, D.; Wihler, T.P., Energy norm a-posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. models methods appl. sci., 17, 33-62, (2007) · Zbl 1116.65115
[19] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. comp., 46, 1-26, (1986) · Zbl 0618.65105
[20] Kanschat, G.; Rannacher, R., Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems, J. numer. math., 10, 249-274, (2002) · Zbl 1022.65123
[21] Karakashian, O.A.; Pascal, F., A-posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. numer. anal., 41, 2374-2399, (2003) · Zbl 1058.65120
[22] Kay, D.; Silvester, D., The reliability of local error estimators for convection – diffusion equations, IMA J. numer. anal., 21, 107-122, (2001) · Zbl 0980.65116
[23] Lesaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport equation, (), 89-145
[24] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Math. abh. sem. univ. Hamburg, 36, 9-15, (1971) · Zbl 0229.65079
[25] W. Reed, T. Hill, Triangular mesh methods for the neutron transport equation, Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973
[26] Rivière, B.; Wheeler, M., A-posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Comput. math. appl., 46, 141-163, (2003) · Zbl 1059.65098
[27] Sangalli, G., A uniform analysis of nonsymmetric and coercive linear operators, SIAM J. math. anal., 36, 2033-2048, (2005) · Zbl 1114.35060
[28] Sangalli, G., Robust a-posteriori estimator for advection – diffusion – reaction problems, Math. comp., 77, 41-70, (2008) · Zbl 1130.65083
[29] Verfürth, R., A review of A-posteriori error estimation and adaptive mesh-refinement techniques, (1996), Teubner Stuttgart · Zbl 0853.65108
[30] Verfürth, R., A-posteriori error estimators for convection – diffusion equations, Numer. math., 80, 641-643, (1998) · Zbl 0913.65095
[31] Verfürth, R., Robust a-posteriori error estimates for stationary convection – diffusion equations, SIAM J. numer. anal., 43, 1766-1782, (2005) · Zbl 1099.65100
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