Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs I: Analytic regularity and gpc-approximation

For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an apriori error analysis for sparse tensor, space-time discretizations. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameterspace by Galerkin projection onto finitely supported polynomial systems in the parameterspace. Uniform stability with respect to the support of the resulting coupled parabolic systems is established. Analyticity of the solution with respect to the countably many parameters is established, and a regularity result of the parametric solution is proved for both, compatible as well as incompatible initial data and source terms. The present results will be used in [6] to obtain convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.