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

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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

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## 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*.