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Stokes, Maxwell and Darcy: a single finite element approximation for three model problems
Badia, Santiago; Codina, Ramon
Universitat Politècnica de Catalunya. Departament de Resistència dels Materials i Estructures en Enginyeria; Universitat Politècnica de Catalunya. (MC)2 - Grup de Mecànica Computacional en Medis Continus
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.
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica::Mètodes en elements finits
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
Artículo - Borrador

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