Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

TL;DR

This paper studies elliptic PDEs on log-Gaussian shaped domains, establishing regularity and proposing finite element and sparse grid methods for efficient numerical approximation, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a framework for solving elliptic PDEs on log-Gaussian random domains, proving regularity and developing sparse grid discretization techniques.

Findings

Proved existence and uniqueness of solutions on random domains.

Established analytic regularity with respect to random parameters.

Demonstrated efficiency of sparse grid methods through numerical results.

Abstract

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.