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A mixed finite element method with Lagrange multipliers for nonlinear exterior transmission problems

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Summary.

We apply a mixed finite element method to numerically solve a class of nonlinear exterior transmission problems in R 2 with inhomogeneous interface conditions. Besides the usual unknowns required for the dual-mixed method, which include the gradient of the temperature in this nonlinear case, our approach makes use of the trace of the outer solution on the transmission boundary as a suitable Lagrange multiplier. In addition, we use a boundary integral operator to reduce the original transmission problem on the unbounded region into a nonlocal one on a bounded domain. In this way, we are lead to a two-fold saddle point operator equation as the resulting variational formulation. We prove that the continuous formulation and the associated Galerkin scheme defined with Raviart-Thomas spaces are well posed, and derive the a-priori estimates and the corresponding rate of convergence. Then, we introduce suitable local problems and deduce first an implicit reliable and quasi-efficient a-posteriori error estimate, and then a fully explicit reliable one. Finally, several numerical results illustrate the effectivity of the explicit estimate for the adaptive computation of the discrete solutions.

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Authors and Affiliations

  1. GI2MA, Departamento de Ingenierí a Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Rommel Bustinza, Galina C. Garcia & Gabriel N. Gatica

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  1. Rommel Bustinza
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  2. Galina C. Garcia
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  3. Gabriel N. Gatica
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Correspondence to Gabriel N. Gatica.

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Mathematics Subject Classification (2000): 65N30, 65N38, 65N22, 65F10

This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.

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Bustinza, R., Garcia, G. & Gatica, G. A mixed finite element method with Lagrange multipliers for nonlinear exterior transmission problems. Numer. Math. 96, 481–523 (2004). https://doi.org/10.1007/s00211-003-0475-8

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  • DOI: https://doi.org/10.1007/s00211-003-0475-8

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