Robust coupling of DPG and BEM for a singularly perturbed transmission problem

TL;DR

This paper develops a robust numerical coupling scheme combining DPG and BEM for a singularly perturbed transmission problem, ensuring stability and efficiency through local computation of test functions and validated by numerical experiments.

Contribution

It adapts existing DPG-BEM coupling methods to handle singular perturbations with proven robustness and local test function computation.

Findings

The scheme is robust in balanced norms for singularly perturbed problems.

Optimal test functions are computed locally, enhancing computational efficiency.

Numerical experiments confirm the method's effectiveness in two dimensions.

Abstract

We consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the Laplacian in the exterior, connected through standard transmission conditions. We establish a DPG scheme coupled with Galerkin boundary elements for its discretization, and prove its robustness for the field variables in so-called balanced norms. Our coupling scheme is the one from [F\"uhrer, Heuer, Karkulik: On the coupling of DPG and BEM, arXiv:1508.00630], adapted to the singularly perturbed case by using the scheme from [Heuer, Karkulik: A robust DPG method for singularly perturbed reaction diffusion problems, arXiv:1509.07560]. Essential feature of our method is that optimal test functions have to be computed only locally. We report on various numerical experiments in two dimensions.