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

Abstract

In this article we investigate the analysis of a finite element method for solving H(curl; Ω)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; Ω)-conforming finite elements on a family of tetrahedral meshes which 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 triangulation. Optimal error estimates in the H(curl; Ω)-norm are obtained for the first time. The analysis is based on a δ-strip argument, a new extension theorem for H1(curl)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution. © 2012 Springer-Verlag.