Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method

  • We study the convergence of an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method for a 2D model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods, the convergence analysis does not require multiple interior nodes for refined elements of the triangulation and thus leads to a more efficient adaptive scheme. In fact, it will be shown that bisection of elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to theWe study the convergence of an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method for a 2D model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods, the convergence analysis does not require multiple interior nodes for refined elements of the triangulation and thus leads to a more efficient adaptive scheme. In fact, it will be shown that bisection of elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. Results of numerical experiments are given to illustrate the performance of the adaptive method.show moreshow less

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Metadaten
Author:Ronald H. W. HoppeGND, Guido Kanschat, Tim Warburton
URN:urn:nbn:de:bvb:384-opus4-4728
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/594
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2007-36)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:University of Houston, Texas A&M University, Rice University
Release Date:2007/10/19
Tag:adaptive finite elements; convergence analysis; interior penalty discontinuous Galerkin method
GND-Keyword:Finite-Elemente-Methode; Konvergenz; Diskontinuierliche Galerkin-Methode
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik