Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals

TL;DR

This paper develops a goal-oriented adaptive multilevel stochastic Galerkin finite element method for efficiently approximating nonlinear quantities of interest in parametric elliptic PDEs with many uncertain parameters, ensuring convergence of the error estimates.

Contribution

It introduces a novel adaptive algorithm that combines primal and dual problem approximations with error-guided refinement for nonlinear functionals in high-dimensional stochastic PDEs.

Findings

The algorithm guarantees convergence of the goal functional error to zero.

Numerical experiments confirm the effectiveness and theoretical properties of the proposed method.

The approach efficiently handles nonlinear quantities of interest in complex parametric PDEs.

Abstract

This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of interest represented by continuously G\^ateaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error reduction indicators. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest illustrate and underpin our theoretical findings.