Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM

TL;DR

This paper introduces a two-level a posteriori error estimator for adaptive multilevel stochastic Galerkin finite element methods, providing reliable error bounds and guiding spatial and parametric refinement for parametric PDEs.

Contribution

It presents a novel two-level a posteriori estimator that guarantees lower bounds and guides adaptive refinement in stochastic Galerkin FEM for parametric PDEs.

Findings

The estimator always provides a lower bound for the error.

Three adaptive algorithms are empirically compared.

Implementation aspects for multilevel stochastic Galerkin approximations are discussed.

Abstract

The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.