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 equation onumber - div (a(y)(x) abla u(y)(x)) = f(x) x D, y I^∞, u|_D = 0, equationwhere D R^m is a bounded Lipschitz domain, I^∞:=[-1,1]^∞, f 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 - div(a(y_0)(x) abla u(y_0)(x)) = f(x) at almost every point y_0 I^∞ with regard to the uniform probability measure μ on 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(I^∞,V,μ). This shows that the curse of dimensionality for the parametric elliptic problem is rid by linear methods.