Convergence of the mixed finite element method for Maxwell's equations with non-linear conductivity

摘要

In this paper we study a numerical scheme to solve coupled Maxwell's equations with a non-linear conductivity. This model plays an important role in the study of type-II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step we study the stability of the scheme in the quasi-static limit ε→0 and present the corresponding convergence rate. The stability of this singular limit proves existence of the quasistatic model. Finally, we support the theory by several numerical experiments.