Convergence Rates of Best Nterm Galerkin Approximations for a Class of Elliptic sPDEs
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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 01 (D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a socalled "generalized polynomial chaos" (gpc) expansion of u. Convergence estimates of approximations of u by best Nterm 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∞ ⊆ V of finite element spaces in D of the coefficients in the Nterm 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) ∩ H 01 (D) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N dofs 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 Nterm approximation rate r and the approximation order of the space discretization in D. © 2010 SFoCM.
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Cohen, A., DeVore, R., & Schwab, C.
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