Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids
Autor(a) principal: | |
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Data de Publicação: | 2006 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/8984 https://doi.org/10.1385/CBB:44:3:539 |
Resumo: | In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s? [1,2] order O(hs)-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H1+s(O). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h3/2-e) with e > 0 if u? H3(O). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s? , the given error estimates are strictly local. |
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Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform GridsIn this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s? [1,2] order O(hs)-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H1+s(O). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h3/2-e) with e > 0 if u? H3(O). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s? , the given error estimates are strictly local.http://www.informaworld.com/10.1080/016305606007964852006info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/8984http://hdl.handle.net/10316/8984https://doi.org/10.1385/CBB:44:3:539engNumerical Functional Analysis and Optimization - Taylor & Francis. 27:5 (2006) 539-564Ferreira, J. A.Grigorieff, R. D.info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2020-11-06T17:00:14Zoai:estudogeral.uc.pt:10316/8984Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:00:43.168417Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
title |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
spellingShingle |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids Ferreira, J. A. |
title_short |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
title_full |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
title_fullStr |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
title_full_unstemmed |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
title_sort |
Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids |
author |
Ferreira, J. A. |
author_facet |
Ferreira, J. A. Grigorieff, R. D. |
author_role |
author |
author2 |
Grigorieff, R. D. |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Ferreira, J. A. Grigorieff, R. D. |
description |
In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s? [1,2] order O(hs)-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H1+s(O). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h3/2-e) with e > 0 if u? H3(O). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s? , the given error estimates are strictly local. |
publishDate |
2006 |
dc.date.none.fl_str_mv |
2006 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/8984 http://hdl.handle.net/10316/8984 https://doi.org/10.1385/CBB:44:3:539 |
url |
http://hdl.handle.net/10316/8984 https://doi.org/10.1385/CBB:44:3:539 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Numerical Functional Analysis and Optimization - Taylor & Francis. 27:5 (2006) 539-564 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799133897582182400 |