Fully discrete approximation of parametric and stochastic elliptic PDEs

TL;DR

This paper develops a fully discrete approximation method for parametric and stochastic elliptic PDEs, analyzing convergence rates based on spatial and parametric discretizations, and extends results to adaptive spatial discretizations.

Contribution

It introduces a comprehensive analysis of fully discrete approximations for elliptic PDEs with parametric coefficients, including convergence rates and $ ext{ell}^p$ summability in Sobolev norms.

Findings

Convergence rates depend on the total degrees of freedom.

$ ext{ell}^p$ summability results for coefficient sequences.

Extension to adaptive spatial discretizations.

Abstract

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in $\ell^p$ summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for non-Hilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.