FEM-BEM coupling in Fractional Diffusion

TL;DR

This paper introduces a novel FEM-BEM coupling scheme for fractional PDEs on unbounded domains, utilizing Caffarelli-Silvestre extension, boundary integral reformulation, and diagonalization to ensure well-posedness and provide numerical validation.

Contribution

It develops a fully discrete FEM-BEM coupling method for fractional PDEs on ull space, combining boundary integral equations with finite elements and diagonalization techniques.

Findings

The scheme is well-posed and stable.

A-priori error estimates are established.

Numerical examples demonstrate effectiveness.

Abstract

We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space $\mathbb{R}^d$ . Based on a reformulation using the well-known Caffarelli-Silvestre extension, we study a modified variational formulation to obtain well-posedness of the discrete problem. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.