QMC integration for lognormal-parametric, elliptic PDEs: local supports imply product weights

We analyze convergence rates of quasi-Monte Carlo (QMC) quadratures for countablyparametric solutions of linear, elliptic partial differential equations (PDE) in divergence form with log-Gaussian diffusion coefficient, based on the error bounds in [James A. Nichols and Frances Y. Kuo: Fast CBC construction of randomly shifted lattice rules achieving O(N−1+δ) convergence for unbounded integrands over R in weighted spaces with POD weights. J. Complexity, 30(4):444-468, 2014]. We prove, for representations of the Gaussian random field PDE input with locally supported basis functions, and for continuous, piecewise polynomial Finite Element discretizations in the physical domain error bounds in weighted spaces with product weights that exploit localization of supports. The convergence rate O(N−1+δ) (independent of the parameter space dimension s) is achieved under weak summability conditions on the expansion coefficients.