Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization
This work addresses uncertainty quantification in PDEs for computational science, but it is incremental as it applies established methods to a specific random domain class.
The paper tackles elliptic PDEs on random log-Gaussian domains by modeling and proving solution existence, uniqueness, and analytic regularity, and numerically approximates them using finite elements and sparse grid methods, with results validated through numerical experiments.
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.