A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case

TL;DR

This paper develops a posteriori error estimation and adaptive algorithms for stochastic Galerkin finite element methods applied to parametric elliptic PDEs with non-affine coefficients, enhancing accuracy and efficiency.

Contribution

It extends error estimation and adaptivity techniques to non-affine parametric coefficients in stochastic Galerkin FEM, beyond the affine case.

Findings

Reliable and efficient energy error estimates derived.

Numerical tests confirm improved adaptivity for non-affine coefficients.

Adaptive algorithm effectively reduces errors in model problems.

Abstract

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient.