Error Analysis of a Stochastic Collocation Method for Parabolic Partial Differential Equations with Random Input Data

Abstract

A stochastic collocation method for solving linear parabolic partial differential equations with random coefficients, forcing terms, and initial conditions is analyzed. The input data are assumed to depend on a finite number of random variables. Unlike previous analyses, a wider range of situations are considered, including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element, Crank--Nicolson fully discrete space-time discretizations. Ingredients in the convergence analysis include the proof of the analyticity, with respect to the probabilistic parameters, of the semidiscrete and fully discrete approximate solutions. A numerical example is provided to illustrate the analyses.