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IMA Journal of Numerical Analysis Advance Access originally published online on July 4, 2008
IMA Journal of Numerical Analysis 2009 29(4):827-855; doi:10.1093/imanum/drn038
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Compact embeddings of broken Sobolev spaces and applications

Annalisa Buffa

Istituto di Matematica Applicata e Tecnologie Informatiche del Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy

Christoph Ortner{dagger}

Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK

{dagger} Email: christoph.ortner{at}comlab.ox.ac.uk

Received on 19 April 2007. Revised on 18 May 2008.


  Abstract

In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, {infty}), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincaré inequalities.

Key Words: discontinuous Galerkin method; broken Sobolev spaces; embedding theorems; compactness; {Gamma}-convergence


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