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Consider the parametric elliptic problem \begin{equation} - \operatorname{dv}\big(a(y)(x)\nabla u(y)(x)\big) \ = \ f(x) \quad x \in D, \ y \in[-1,1]^\infty, \quad u|_{\partial D} \ = \ 0, \end{equation} where $D \subset{\mathbb R}^m$ is a bounded Lipschitz domain, $[-1,1]^\infty$, $f \in L_2(D)$,and the diffusions $a$ satisfy the uniform ellipticity assumption and areaffinely dependent with respect to $y$. The parametric variable $y$ may bedeterministic or random. In the present paper, a central question to be studiedis as follows. Assume that we have an approximation property that there is asequence of finite element approximations with a certain error convergence ratein energy norm of the space $V:=H^1_0(D)$ for the nonparametric problem $-\operatorname{dv} \big(a(y_0)(x)\nabla u(y_0)(x)\big) = f(x)$ at every point$y_0 \in [-1,1]^\infty$. Then under what assumptions does this sequence inducea sequence of finite element approximations with the same error convergencerate for the parametric elliptic problem in the norm of the Bochner spaces$L_\infty([-1,1]^\infty,V)$ or $L_2([-1,1]^\infty,V)$? We solved this questionby linear collective Taylor, collocation and Galerkin methods, based on Taylorexpansions, Lagrange polynomial interpolations and Legendre polynomialsexpansions, respectively, on the parametric domain $[-1,1]^\infty$. Under verylight conditions, we show that all these approximation methods give the sameerror convergence rate as that by the sequence of finite element approximationsfor the nonparametric elliptic problem. The parametric infinite-variate partcompletely disappears from the convergence rate and influences only theconstant. Hence the curse of dimensionality is broken by linear methods.

Linear collective collocation and Galerkin approximations for parametric and stochastic elliptic PDEs