Dinh D˜ung Galerkin approximation for parametric and stochastic elliptic PDEs

Abstract

We study the Galerkin approximation for the parametric elliptic problem

\begin{equation} \nonumber

- \operatorname{div} \big(a(y)(x)\nabla u(y)(x)\big)

\ = \

f(x) \quad x \in D, \ y \in \mathbb{I}^{\infty},

\quad u|_{\partial D} \ = \ 0,

\end{equation}where  $D \subset \mathbb{R}^m$ is a bounded Lipschitz domain, $\mathbb{I}^{\infty}:=[-1,1]^\infty$, $f \in L_2(D)$, 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

$- \operatorname{div}\big(a(y_0)(x)\nabla u(y_0)(x)\big)  = f(x)$

at almost every point $y_0 \in \mathbb{I}^{\infty}$ with regard to the uniform probability measure $\mu$ on $\mathbb{I}^{\infty}$.

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 $L_2(\mathbb{I}^{\infty},V,\mu)$.

This shows that the curse of dimensionality for the parametric elliptic problem is rid by linear methods.