Stokes, Maxwell and Darcy: a single finite element approximation for three model problems
In this work we propose stabilized finite element methods for Stokes’, Maxwell’s and Darcy’s problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantag...
| Autores: | , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2012 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositório: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglês |
| OAI Identifier: | oai:upcommons.upc.edu:2117/15742 |
| Acesso em linha: | https://hdl.handle.net/2117/15742 https://dx.doi.org/10.1016/j.apnum.2011.07.001 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Navier-Stokes equations--Numerical solutions Maxwell equations--Numerical solutions Darcy's law Stabilized finiteelements Compatible approximations Primal and dual problems Singular solutions Nodal interpolations Equacions de Navier-Stokes Maxwell, Equacions de -- Solucions numèriques Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica::Mètodes en elements finits |
| Resumo: | In this work we propose stabilized finite element methods for Stokes’, Maxwell’s and Darcy’s problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantages of our approach. In particular, for Darcy’s problem we are able to design stabilized methods that yield optimal convergence both for the primal and the dual problems. In the case of Maxwell’s problem, the formulation we propose allows one to use continuous finite element interpolations that converge optimally to the continuous solution even if it is non-smooth. Once the formulation is presented for the three model problems independently, we also show how it can be used for a problem that combines all the operators of the independent problems. Stability and convergence is achieved regardless of the fact that any of these operators dominates the others, a feature not possible for the methods of which we are aware. |
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