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A posteriori error control for discontinuous Galerkin methods for parabolic problems

journal contribution
posted on 2023-06-07, 23:39 authored by Emmanuil H Georgoulis, Omar LakkisOmar Lakkis, Juha M Virtanen
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.

History

Publication status

  • Published

Journal

SIAM Journal on Numerical Analysis

ISSN

1095-7170

Publisher

Society for Industrial and Applied Mathematics

Issue

2

Volume

49

Page range

427-458

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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