Dinh D˜ung Galerkin approximation for parametric and stochastic elliptic PDEs - Dinh , Зунг (Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series, 2018)
We study the Galerkin approximation for the parametric elliptic problem \begin{equation} onumber - \operatorname{div} \big(a(y)(x) abla u(y)(x)\big) \ = \ f(x) \quad x \in D, \ y \in \mathbb{I}^{∞}, \quad u|_{\partial D} \ = \ 0, \end{equation}where $D \subset \mathbb{R}^m$ is a bounded Lipschitz domain, $\mathbb{I}^{∞}:=[-1,1]^∞$, $f \in L_2(D)$, and the diffusions $a$ satisfy the uniform ellipticity assumption and are affinely dependent with respect to $y$. Assume that we have an approximation property that there is a sequence of finite element approximations with a certain error convergence rate in energy norm of the space $V:=H^1_0(D)$ for the nonparametric problem $- \operatorname{div}\big(a(y_0)(x) abla u(y_0)(x)\big) = f(x)$ at almost every point $y_0 \in \mathbb{I}^{∞}$ with regard to the uniform probability measure $μ$ on $\mathbb{I}^{∞}$. Based on this assumption we construct a sequence of finite element approximations with the same error convergence rate for the parametric elliptic problem in the norm of the Bochner spaces $L_2(\mathbb{I}^{∞},V,μ)$. This shows that the curse of dimensionality for the parametric elliptic problem is rid by linear methods.