Convergence of adaptive multilevel stochastic Galerkin FEM for parametric PDEs

Abstract

In this talk, we propose and analyze an adaptive multilevel stochastic Galerkin finite element method for a second-order elliptic diffusion problem with random coefficients. The problem is discretized by means of finite generalized polynomial chaos (gpc) expansions in the parameter domain, and standard FEM-discretizations in the spatial domain. Following [1], the adaptive algorithm is driven by a residual-based error estimator, which incorporates both the error due to FEM-discretization and the error due to truncated gpc expansions. Under a local compatibility condition on the mesh sizes of the triangulations associated to an active parameter in the full parameter set, we prove that the proposed algorithm guarantees R-linear convergence of the estimator. To this end, we adopt the approach of [2], and show contraction of a suitable quasi-error quantity. We propose a novel multilevel-refinement algorithm, which simultaneously refines every grid while additionally preserving a local compatibility condition between the meshes in the hierarchy and assigns suitable triangulations to newly activated parameters. Numerical experiments illustrate the theoretical results.