Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method for the Biharmonic Problem

  • For the biharmonic problem, we study the convergence of adaptive C0-Interior Penalty Discontinuous Galerkin (C0-IPDG) methods of any polynomial order. We note that C0-IPDG methods for fourth order elliptic boundary value problems have been suggested in [9], whereas a residual-type a posteriori error estimator for a quadratic C0-IPDG method applied to the biharmonic equation has been developed and analyzed in [8]. Following the convergence analysis of adaptive IPDG methods for second order elliptic problems [6], we prove a contraction property for a weighted sum of the C0-IPDG energy norm of the global discretization error and the estimator. The proof of the contraction property is based on the reliability of the estimator, a quasi-orthogonality result, and an estimator reduction property. Numerical results are given that illustrate the performance of the adaptive C0-IPDG approach.

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Author:Thomas Fraunholz, Ronald H. W. HoppeORCiDGND, Malte A. PeterORCiDGND
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Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2012-05)
Publishing Institution:Universität Augsburg
Release Date:2012/03/29
Tag:Numerische Mathematik
C0-Interior Penalty Discontinuous Galerkin method; biharmonic equation; residual type a posteriori error estimator; convergence analysis
GND-Keyword:Diskontinuierliche Galerkin-Methode; Konvergenz; Fehlerabschätzung; A-posteriori-Abschätzung
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehr- und Forschungseinheit Angewandte Analysis
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand