An hp Finite Element Method for singularly perturbed transmission problems in smooth domains

TL;DR

This paper develops an hp finite element method for a 2D singularly perturbed transmission problem with smooth domains, achieving exponential convergence by leveraging regularity results and tailored mesh and polynomial degree choices.

Contribution

It introduces a robust hp finite element scheme specifically designed for singularly perturbed transmission problems in smooth domains, with proven exponential convergence.

Findings

Method converges exponentially for analytic data

Numerical results confirm theoretical convergence rates

Effective handling of boundary and interface layers

Abstract

We consider a two-dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we design a robust $hp$ finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an e{xponential} rate, provided the mesh and polynomial degree distribution are chosen appropriately. Numerical results illustrating our theoretical findings are also included.