Further developments on boundary-field equation methods for nonlinear transmission problems

Abstract

This paper is concerned with new insights on the application of the boundary-field equation approach, which refers basically to the combined use of finite element and boundary integral equation methods, to numerically solve nonlinear exterior transmission problems in 2D and 3D. As a model, we consider a nonlinear second-order elliptic equation in divergence form holding in an annular domain coupled with the Laplace equation in the corresponding unbounded exterior region, together with transmission conditions on the interface and a suitable radiation condition at infinity. We first extend the classical Johnson & Nédélec coupling procedure, which makes use of only one boundary integral equation, to our nonlinear model without assuming any restrictive smoothness requirement on the boundary but only Lipschitz-continuity. Next, we extend the applicability of a recently introduced modification of the Costabel & Han coupling method, which employs the two boundary integral equations arising from the Green representation formula, to our nonlinear model as well. This new boundary-field equation method is based on the introduction of both Cauchy data on the boundary as independent unknowns. Primal and dual-mixed variational formulations are analyzed for each extension described above, and suitable hypotheses on the nonlinear coefficient of the elliptic equation allow to establish well-posedness of the corresponding continuous and discrete schemes by using monotone operator and nonlinear Babuška-Brezzi theories. Finally, a priori error estimates are established.