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⊂ℝ^d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L ²(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y _i (ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parameters y.

We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)^∞ to $V=H^{1}_{0}(D)$. 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∈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 $\{V_{l}\}_{l=0}^{\infty}\subset V$ of finite element spaces 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 coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−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 $H^{2}(D)\cap H^{1}_{0}(D)$ in the case where D is a smooth or convex domain.

Our analysis shows that in realistic settings a convergence rate $N_{\mathrm{dof}}^{-s}$ in terms of the total number of degrees of freedom N _dof 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.