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A dual-dual mixed formulation for nonlinear exterior transmission problems

Authors: Gabriel N. Gatica and Salim Meddahi
Journal: Math. Comp. 70 (2001), 1461-1480
MSC (2000): Primary 65N30, 65N38, 65J15, 35J65, 35J05
Published electronically: May 23, 2000
MathSciNet review: 1836913
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We combine a dual-mixed finite element method with a Dirichlet-to-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

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Additional Information

Gabriel N. Gatica
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Salim Meddahi
Affiliation: Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, España

Keywords: Mixed finite elements, boundary elements, coupling
Received by editor(s): April 13, 1999
Received by editor(s) in revised form: October 13, 1999
Published electronically: May 23, 2000
Additional Notes: This research was partially supported by Fondecyt-Chile through research project 1980122, and by FONDAP-Conicyt through Program A on Numerical Analysis.
Dedicated: Dedicated to Professor Dr. George C. Hsiao on the occasion of his 65th birthday
Article copyright: © Copyright 2000 American Mathematical Society