A note on the domain mapping method with rough diffusion coefficients

TL;DR

This paper investigates elliptic diffusion problems on random domains with non-smooth coefficients, proposing a perturbation method to improve approximation accuracy and validating results through numerical examples.

Contribution

It introduces a novel perturbation approach for elliptic problems with rough diffusion coefficients, enhancing approximation methods for quantities of interest.

Findings

Perturbation method effectively handles non-smooth coefficients.

Approximation errors are quantified in terms of perturbation amplitude.

Numerical results confirm theoretical predictions.

Abstract

In this article, we consider elliptic diffusion problems on random domains with non-smooth diffusion coefficients. We start by illustrating the problems that arise from a non-smooth diffusion coefficient by recapitulating the corresponding regularity analysis. Then, we propose an alternative approach to address this problem by means of a perturbation method. Based on the assumption that the diffusion coefficient can be decomposed in a possibly deterministic, analytic part and a rough random perturbation, we derive approximation results in terms of the perturbations amplitude for the approximation of quantities of interest of the solution. Numerical examples are given in order to validate and quantify the theoretical results.