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Analysis of a class of penalty methods for computing singular minimizers
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Carstensen, Carsten and Ortner, Christoph (2010) Analysis of a class of penalty methods for computing singular minimizers. Computational Methods in Applied Mathematics, Volume 10 (Number 2). pp. 137-163. ISSN 1609-4840.
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Abstract
Amongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computa- tional calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter.
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): | Finite element method, Calculus of variations | ||||
Journal or Publication Title: | Computational Methods in Applied Mathematics | ||||
Publisher: | Vydavetstvo Belaruskaya Navuka | ||||
ISSN: | 1609-4840 | ||||
Official Date: | 2010 | ||||
Dates: |
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Volume: | Volume 10 | ||||
Number: | Number 2 | ||||
Page Range: | pp. 137-163 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Date of first compliant deposit: | 28 July 2016 | ||||
Date of first compliant Open Access: | 28 July 2016 | ||||
Funder: | Deutsche Forschungsgemeinschaft (DFG), National Research Foundation of Korea (NRF), Hausdorff Institute of Mathematics (Bonn, Germany), Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | R31-2008-000-10049-0 (NRF), 797 (DFG) |
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