# zbMATH — the first resource for mathematics

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems. (English) Zbl 1163.65077
The authors present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation.
Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation.
Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.

##### 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 35R05 PDEs with low regular coefficients and/or low regular data
Full Text:
##### References:
 [1] Ainsworth, M., A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. numer. anal., 45, 4, 1777-1798, (2007) · Zbl 1151.65083 [2] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), Wiley-Interscience, John Wiley & Sons New York · Zbl 1008.65076 [3] Arnold, D.N., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 4, 742-760, (1982) · Zbl 0482.65060 [4] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 5, 1749-1779, (2002) · Zbl 1008.65080 [5] Babuška, I.; Rheinboldt, W.C., A posteriori error estimates for the finite element method, Int. J. numer. meth. engrg., 12, 1597-1615, (1978) · Zbl 0396.65068 [6] Babuška, I.; Strouboulis, T., The finite element method and its reliability, (2001), Clarendon Press, Oxford University Press New York · Zbl 0997.74069 [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.; Schöberl, J., Equilibrated residual error estimator for edge elements, Math. comp., (2007), (electronic) · Zbl 1135.65041 [9] Brezzi, F.; Cockburn, B.; Marini, L.D.; Süli, E., Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput. methods appl. mech. engrg., 195, 3293-3310, (2006) · Zbl 1125.65102 [10] Burman, E.; Ern, A., Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Math. comp., 76, 259, 1119-1140, (2007) · Zbl 1118.65118 [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] Castillo, P., An a posteriori error estimate for the local discontinuous Galerkin method, J. sci. comput., 22-23, 187-204, (2005) · Zbl 1082.65111 [13] Clément, Ph., Approximations by finite element functions using local regularization, RAIRO anal. numér., 9, 77-84, (1975) · Zbl 0368.65008 [14] Cockburn, B.; Shu, C.W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135 [15] 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 [16] Destuynder, P.; Métivet, B., Explicit error bounds in a conforming finite element method, Math. comp., 68, 228, 1379-1396, (1999) · Zbl 0929.65095 [17] Epshteyn, Y.; Rivière, B., Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J. comput. appl. math., 206, 843-872, (2007) · Zbl 1141.65078 [18] Ern, A.; Proft, J., A posteriori discontinuous Galerkin error estimates for transient convection – diffusion equations, Appl. math. lett., 18, 7, 833-841, (2005) · Zbl 1084.65092 [19] A. Ern, A.F. Stephansen, M. Vohralík, Guaranteed and robust a posteriori error estimation based on flux reconstruction for discontinuous Galerkin methods, SIAM J. Numer. Anal. (2007) (submitted for publication). Open access at http://hal.archives-ouvertes.fr/hal-00193540/ [20] Gockenbach, M.S., Understanding and implementing the finite element method, (2006), SIAM · Zbl 1105.65112 [21] Houston, P.; Perugia, I.; Schötzau, D., An a posteriori error indicator for discontinuous Galerkin discretizations of $$H$$(curl)-elliptic partial differential equations, IMA J. numer. anal., 27, 1, 122-150, (2007) · Zbl 1148.65088 [22] 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, 1, 33-62, (2007) · Zbl 1116.65115 [23] 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), (electronic) · Zbl 1058.65120 [24] Kim, K.Y., A posteriori error analysis for locally conservative mixed methods, Math. comp., 76, 257, 43-66, (2007) · Zbl 1121.65112 [25] 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 [26] 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 [27] R. Lazarov, S. Repin, S. Tomar, Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems, Numer. Methods Partial Differential Equations, in press (doi:10.1002/num.20386) · Zbl 1167.65451 [28] Mikhlin, S., Variational methods in mathematical physics, (1964), Pergamon Oxford · Zbl 0119.19002 [29] Mikhlin, S., Error analysis in numerical processes, (1991), Wiley and Sons Chichester, New York [30] Mozolevski, I.; Bösing, P.R., Sharp expressions for the stabilization parameters in symmetric interior-penalty discontinuous Galerkin finite element approximations of fourth-order elliptic problems, Comput. methods appl. math., 7, 4, 365-375, (2007) · Zbl 1136.65098 [31] Neittaanmäki, P.; Repin, S., Reliable methods for computer simulation, error control and a posteriori estimates, () · Zbl 1076.65093 [32] Repin, S., A posteriori error estimation for nonlinear variational problems by duality theory, Zap. nauchn. sem. S.-peterburg. otdel. mat. inst. Steklov. (POMI), 243, 201-214, (1997), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii. 28, 342 [33] Repin, S., A unified approach to a posteriori error estimation based on duality error majorants, Math. comput. simulation, 50, 1-4, 305-321, (1999), Modelling ’98 (Prague) [34] Repin, S., A posteriori error estimation for variational problems with uniformly convex functionals, Math. comp., 69, 230, 481-500, (2000) · Zbl 0949.65070 [35] Repin, S., Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, (), 143-171, 209 · Zbl 1039.65076 [36] S. Repin, S. Tomar, Helmholtz type decomposition based functional a posteriori error estimates for nonconforming approximation of elliptic problems, RICAM report, 41, IMA J. Numer. Anal. (2007) (submitted for publication) [37] Rivière, B.; Wheeler, M.F., A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Comput. math. appl., 46, 141-164, (2003) · Zbl 1059.65098 [38] Schneider, R.; Xu, Y.; Zhou, A., An analysis of discontinuous Galerkin methods for elliptic problems, Adv. comput. math., 25, 1-3, 259-286, (2006) · Zbl 1099.65116 [39] Schöberl, J., A posteriori error estimates for Maxwell equations, Math. comp., (2007), (electronic) [40] Shahbazi, K., An explicit expression for the penalty parameter of the interior penalty method, J. comput. phys., 205, 401-407, (2005) · Zbl 1072.65149 [41] Sun, S.; Wheeler, M.F., $$L^2(H^1)$$ norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. sci. comput., 22-23, 501-530, (2005) · Zbl 1066.76037 [42] Vejchodsky, T., Guaranteed and locally computable a posteriori error estimate, IMA J. numer. anal., 26, 525-540, (2006) · Zbl 1096.65112 [43] Verfürth, R., A review of a posteriori error estimation and adaptive mesh refinement techniques, (1996), Wiley-Teubner Stuttgart · Zbl 0853.65108 [44] Wheeler, M.F., An elliptic collocation-finite element method with interior penalties, SIAM J. numer. anal., 15, 152-161, (1978) · Zbl 0384.65058 [45] Yang, J.; Chen, Y., A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations, J. comput. math., 24, 3, 425-434, (2006) · Zbl 1142.76034
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.