On the mixed finite element method with Lagrange multipliers

Babuska I; Gatica, GN

Abstract

In this note we analyze a modified mixed finite element method for second-order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R2. The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babuška-Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart-Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems.

Más información

Título según WOS: On the mixed finite element method with Lagrange multipliers
Título según SCOPUS: On the mixed finite element method with Lagrange multipliers
Título de la Revista: NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
Volumen: 19
Número: 2
Editorial: WILEY-BLACKWELL
Fecha de publicación: 2003
Página de inicio: 192
Página final: 210
Idioma: English
URL: http://doi.wiley.com/10.1002/num.10040
DOI:

10.1002/num.10040

Notas: ISI, SCOPUS