Sparse Tensor Galerkin Discretization of Parametric and Random Parabolic PDEs---Analytic Regularity and Generalized Polynomial Chaos Approximation

Abstract

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 a priori error analysis for $N$-term generalized polynomial chaos approximations in a scale of Bochner spaces. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space by Galerkin projection onto finitely supported polynomial systems in the parameter space. 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 imply convergence rates and stability of sparse, adaptive space-time tensor product Galerkin discretizations of these infinite dimensional, parametric problems in the parameter space recently proposed in [C. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291--467; C. J. Gittelson, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations, Ph.D. thesis, ETH Zürich, 2011].