Foundations of Computational Mathematics

, Volume 10, Issue 6, pp 615–646

Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

Authors

    • Laboratoire Jacques-Louis Lions, UMR 7598UPMC Univ. Paris 06
    • Laboratoire Jacques-Louis Lions, UMR 7598CNRS
  • Ronald DeVore
    • Department of MathematicsTexas A& M University
  • Christoph Schwab
    • Seminar for Applied MathematicsETH Zürich
Article

DOI: 10.1007/s10208-010-9072-2

Cite this article as:
Cohen, A., DeVore, R. & Schwab, C. Found Comput Math (2010) 10: 615. doi:10.1007/s10208-010-9072-2

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(ω)=(yi(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable xD 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 yU=(−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 yU are established. These estimates are of the form Nr, 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 yU=(−1,1) to a smoothness space WV 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 Ndof 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.

Keywords

Stochastic and parametric elliptic equationsWiener polynomial chaosApproximation ratesNonlinear approximationSparsity

Mathematics Subject Classification (2000)

41A65N65C30

Copyright information

© SFoCM 2010