Robust a posteriori error estimation for stochastic Galerkin formulations of parameter-dependent linear elasticity equations

TL;DR

This paper develops a reliable and efficient a posteriori error estimation method for stochastic Galerkin finite element approximations of parameter-dependent linear elasticity equations, enabling adaptive refinement and accurate error control.

Contribution

It introduces a novel a posteriori error estimator with bounds independent of Poisson ratio and discretization parameters, and develops an adaptive algorithm for stochastic Galerkin methods.

Findings

The error estimator provides reliable bounds for the SG-MFEM error.

The adaptive algorithm effectively reduces error below prescribed tolerance.

Numerical experiments validate the theoretical error bounds and adaptivity.

Abstract

The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Young's modulus is an affine function of a countable set of parameters. We analyse the weak formulation, its stability with respect to a weighted norm and discuss approximation using stochastic Galerkin mixed finite element methods (SG-MFEMs). We introduce a novel a posteriori error estimation scheme and establish upper and lower bounds for the SG-MFEM error. The constants in the bounds are independent of the Poisson ratio as well as the SG-MFEM discretisation parameters. In addition, we discuss proxies for the error reduction associated with certain enrichments of the SG-MFEM spaces and we use these to develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. We prove that both the a posteriori error estimate and the error reduction proxies are reliable and efficient in the incompressible limit case. Numerical results are presented to validate the theory. All experiments were performed using open source (IFISS) software that is available online.