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

We apply a mixed finite element method to numerically solve a class of nonlinear exterior transmission problems in R2 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.