Sparse tensor Galerkin discretization of parametric and random parabolic PDEs - analytic regularity and generalized polynomial chaos approximation.
Hoang, Viet Ha.
Date of Issue2013
School of Physical and Mathematical Sciences
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 Z¨urich, 2011].
SIAM journal on mathematical analysis
© 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Mathematical Analysis and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/100793682]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.