Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs

Abstract

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D_Rd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(!)=(yi(!)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x 2 D and and the in general countably many parameters y. We establish new regularity theorems decribing the smoothness properties of the solution u as a map from y 2 U = (-1, 1) to V = H1 0 (D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos”(gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V -valued polynomials in the variable y 2 U are established. These estimates are of the form N-r, where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte-Carlo simulations with N "samples" (i.e. deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family (V) 1 1=0_V of finite element spaced in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficients. New regularity theorems decribing the smoothness properties of the solution u as a map from y 2 U = (-1,1)1 to a smoothness space W_V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H2(D)/H1 0 (D) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N-s d.o.f in terms of the total number of degrees of freedom Nd.o.f can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.