Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

TL;DR

This paper introduces a coupled FEM-BEM approach for efficiently solving elliptic boundary value problems on random domains, especially when only boundary randomness is known, by avoiding the need for full domain mappings.

Contribution

It develops a novel coupling method combining FEM and BEM to handle boundary randomness without computing full domain mappings, verified through regularity analysis and numerical experiments.

Findings

Coupling FEM and BEM is feasible for random boundary problems.

The method achieves efficient multilevel quadrature on random domains.

Numerical results confirm the approach's effectiveness.

Abstract

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify the required regularity of the solution with respect to the random domain mapping for the use of multilevel quadrature, derive the coupling formulation, and show by numerical results that the approach is feasible.