A posteriori error control for discontinuous Galerkin methods for parabolic problems

Georgoulis, Emmanuil H, Lakkis, Omar and Virtanen, Juha M (2011) A posteriori error control for discontinuous Galerkin methods for parabolic problems. SIAM Journal on Numerical Analysis, 49 (2). pp. 427-458. ISSN 1095-7170

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We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To use this method in practice, we apply it to the interior penalty discontinuous Galerkin method, for which new a posteriori error bounds are derived. For the analysis of the time-dependent problems we use the elliptic reconstruction technique, and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it. We illustrate the theory with a series of numerical experiments aimed at (1) exploring practically the reliability and efficiency of the derived a posteriori estimates, and (2) testing them in an adaptive algorithm implementation.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Omar Lakkis
Date Deposited: 06 Feb 2012 19:39
Last Modified: 07 Nov 2012 16:16
URI: http://sro.sussex.ac.uk/id/eprint/21656
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