QMC Galerkin Discretization of Parametric Operator Equations

Abstract

We extend recent results from Kuo et al. (SIAM J Numer Anal 50:3351–3374, 2012) of QMC quadrature and Finite Element discretization for parametric, scalar second order elliptic partial differential equations to general QMC-Galerkin discretizations of parametric operator equations, which depend on possibly countably many parameters. Such problems typically arise in the numerical solution of differential and integral equations with random field inputs. The present setting covers general second order elliptic equations which are possibly indefinite (Helmholtz equation), or which are given in saddle point variational form (such as mixed formulations). They also cover nonsymmetric variational formulations which appear in space-time Galerkin discretizations of parabolic problems or countably parametric nonlinear initial value problems (Hansen and Schwab, Vietnam J. Math 2013, to appear).