First order second moment analysis for stochastic interface problems based on low-rank approximation^∗

¹ Helmut Harbrecht, Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland.
helmut.harbrecht@unibas.ch
² Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, P. R. China.
li.jz@sustc.edu.cn

Received: 4 May 2011
Revised: 13 December 2012

Abstract

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method.