A note on the domain mapping method with rough diffusion coefficients
For researchers working on stochastic PDEs with rough coefficients, this provides a new analytical framework, though the contribution is incremental as it extends existing perturbation methods.
The paper addresses elliptic diffusion problems on random domains with non-smooth diffusion coefficients by proposing a perturbation method that decomposes the coefficient into an analytic part and a rough random perturbation, yielding approximation results in terms of perturbation amplitude. Numerical examples validate the theoretical findings.
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.