The Mortar Finite Element Method for 3D Maxwell Equations: First Results

Abstract

In the framework of domain decomposition methods, we extend the main ideas of the mortar element method to the numerical solution of Maxwell's equations (in wave form) by H( curl)-conforming finite elements. The method we propose turns out to be a new nonconforming, nonoverlapping domain decomposition method where nonmatching grids are allowed at the interfaces between subdomains. A model problem is considered, the convergence of the discrete approximation is analyzed, and an error estimate is provided. The method is proven to be slightly suboptimal with a loss of a factor $\scriptstyle\sqrt{|{\rm ln} h|}$ with respect to the degree of polynomials. In order to achieve this convergence result we nevertheless need extra-regularity assumptions on the solution of the continuous problem.