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Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. (English)
Anal. Appl., Singap. 9, No. 1, 11-47 (2011).
This paper deals with simultaneously solving a family of elliptic equations on a bounded Lipschitz domain $D\subset {\bbfR}\sp{d}$ of the form $-\nabla\cdot(a\nabla u)=f$ in $D$, $u\vert\sb{ tial D}=0$, where the diffusion coefficients $a(x,y)$ are functions of $x\in D$ and of parameters $y=(y\sb 1,y\sb2,\dots)$ which may be finite or infinite in number. The right-hand side $f$ is a fixed function on $D$ and the gradient operator $\nabla$ is taken with respect to $x$. It is shown that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then used to prove that under weak assumptions on the diffusion coefficients, the entire family of solutions to given equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space $V=H\sb0\sp1(D)$ of weak solutions of the elliptic problem with a controlled number of terms $N$. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in $D$ is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number $N\sb{dof}$ of degrees of freedom is the minimum of the convergence rates afforded by the best $N$-term sequence approximations in the parameter space and the rate of finite element approximations in $D$ for a single instance of the parametric problem.
Reviewer: Aleksandr D. Borisenko (Kyïv)