On the convergence of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients

TL;DR

This paper proves a quasi-error reduction for an adaptive Galerkin FEM solving high-dimensional parametric PDEs with lognormal coefficients, advancing understanding of convergence in challenging unbounded coefficient scenarios.

Contribution

It provides the first quasi-error reduction result for adaptive Galerkin FEM applied to PDEs with unbounded lognormal coefficients, extending previous bounded-coefficient analyses.

Findings

Established quasi-error reduction for unbounded coefficients

Guaranteed convergence results for bounded coefficients follow as a corollary

Benchmark example illustrates theoretical findings

Abstract

Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin finite element methods. This work investigates a residual based adaptive algorithm, akin to classical adaptive FEM, used to approximate the solution of the stationary diffusion equation with lognormal coefficients, i.e. with a non-affine parameter dependence of the data. It is known that the refinement procedure is reliable but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical state-of-the-art by providing a quasi-error reduction result for the adaptive solution of the lognormal stationary diffusion problem. The presented analysis generalizes previous results in that guaranteed convergence for uniformly bounded coefficients follows directly as a corollary. Moreover, it highlights the fundamental challenges with unbounded coefficients that cannot be overcome with common techniques. A computational benchmark example illustrates the main theoretical statement.