A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields

TL;DR

This paper introduces an adaptive stochastic Galerkin finite element method that efficiently solves parametric elliptic PDEs by combining multilevel random field expansions with independent spatial refinements, ensuring uniform error reduction.

Contribution

It presents a novel adaptive method that integrates multilevel random field expansions with independent finite element refinements for improved accuracy in stochastic PDEs.

Findings

Achieves uniform error reduction through multilevel expansions.

Ensures saturation property in the adaptive refinement process.

Effectively handles low-regularity random fields.

Abstract

The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.