Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems

Abstract

In this article we analyze a finite element method for solving H(div;Ω)-elliptic interface problems in general three-dimensional Lipschitz domains with smooth material interfaces. The continuous problems are discretized by means of lowest order H(div;Ω)-conforming finite elements of the first family (Raviart–Thomas or Nédélec face elements) on a family of unstructured oriented tetrahedral meshes. These resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H(div;Ω)-norms are obtained for the first time. The analysis is based on a so-called δ-strip argument, a new extension theorem for H¹(div)-functions across smooth interfaces, a novel non-standard interfaceaware interpolation operator, and a perturbation argument for degrees of freedom in H(div;Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.