Error estimation and adaptivity for stochastic collocation finite elements Part II: multilevel approximation

TL;DR

This paper extends an adaptive error estimation strategy for stochastic collocation finite element methods to multilevel approaches with tailored meshes, improving efficiency in solving elliptic PDEs with random data.

Contribution

It introduces a multilevel adaptive refinement strategy for stochastic collocation finite elements, extending previous a posteriori error estimation to nonaffine parametric coefficients.

Findings

Multilevel approach improves computational efficiency.

Tailored meshes enhance approximation accuracy.

Codes are publicly available for reproducibility.

Abstract

A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile in 2018 (SIAM J. Numer. Anal, 56, 3121--3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in part I of this work (Bespalov, Silvester and Xu, arXiv:2109.07320). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, are discussed herein. The codes used to generate the numerical results are available online.