We study the weak solvability of an interior linear-nonlinear transmission problem arising in steady heat transfer and potential theory. For the variational formulation, we use a Dirichlet-to-Neumann mapping on the interface, which is obtained from the application of the boundary integral method to the linear domain, and we utilize a mixed finite element method in the nonlinear region. Existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete Galerkin method are derived. In particular, a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this Galerkin scheme.
- transmission problems
- Dirichlet-to-Neumann mapping
- boundary integral method
- mixed finite element
- non-conforming Galerkin scheme
Barrenechea, G. R., Gatica, G. N., & Hsiao, G. C. (1999). Weak solvability of interior transmission problems via mixed finite elements and Dirichlet to Neumann mappings. Journal of Computational and Applied Mathematics, 100(2), 145-160. https://doi.org/10.1016/S0377-0427(98)00185-X