A Convergent Adaptive Finite Element Stochastic Galerkin Method Based on Multilevel Expansions of Random Fields

SIAM Journal on Numerical Analysis, vol. 63, pp. 1776–1807

Abstract

Abstract. The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.

Authors 4

1. Markus Bachmayr Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52062 Aachen, Germany

2. Martin Eigel

   Weierstrass Institute for Applied Analysis and Stochastics

   Affiliation as printed

   Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany

3. Henrik Eisenmann Aachen Institut für Geometrie und Praktische Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52062 Aachen, Germany

4. Igor Voulis

   University of Göttingen

   Affiliation as printed

   Institute for Numerical and Applied Mathematics, University of Göttingen, 37083 Göttingen, Germany