Convergence of adaptive stochastic collocation with finite elements

TL;DR

This paper proves the convergence of an adaptive algorithm combining stochastic collocation and finite element methods for solving elliptic PDEs with random parameters, enhancing efficiency in uncertainty quantification.

Contribution

It introduces a novel adaptive algorithm that simultaneously refines the parameter space and spatial meshes, ensuring convergence for stochastic PDE discretizations.

Findings

Proves convergence of the adaptive algorithm.

Demonstrates improved accuracy through adaptive enrichment.

Provides theoretical guarantees for combined stochastic and spatial refinement.

Abstract

We consider an elliptic partial differential equation with a random diffusion parameter discretized by a stochastic collocation method in the parameter domain and a finite element method in the spatial domain. We prove convergence of an adaptive algorithm which adaptively enriches the parameter space as well as refines the finite element meshes.