A locally modified second-order finite element method for interface problems and its implementation in 2 dimensions

Abstract

The locally modified finite element method, which is introduced in Frei and Richter [SIAM J. Numer. Anal.52(2014) 2315–2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method in two space dimensions to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimala priorierror estimates in theL2-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.