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Compact embeddings of broken Sobolev spaces and applications
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Buffa, Annalisa and Ortner, Christoph (2009) Compact embeddings of broken Sobolev spaces and applications. IMA Journal of Numerical Analysis, Volume 29 (Number 4). pp. 827-855. doi:10.1093/imanum/drn038 ISSN 0272-4979.
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Official URL: http://dx.doi.org/10.1093/imanum/drn038
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, ∞), 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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Galerkin methods, Sobolev spaces, Calculus of variations | ||||||||
Journal or Publication Title: | IMA Journal of Numerical Analysis | ||||||||
Publisher: | Oxford University Press | ||||||||
ISSN: | 0272-4979 | ||||||||
Official Date: | October 2009 | ||||||||
Dates: |
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Volume: | Volume 29 | ||||||||
Number: | Number 4 | ||||||||
Page Range: | pp. 827-855 | ||||||||
DOI: | 10.1093/imanum/drn038 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 20 December 2015 | ||||||||
Date of first compliant Open Access: | 20 December 2015 |
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