Analysis and Implementation of Stochastic Collocation Method for Parabolic Partial Differential Equations with Random Input Data

Introduction

We consider the following linear parabolic stochastic PDE (1). The coefficient and the forcing term are represented not only by KL expansion but also by a nonlinear function of a random vector;

Both bounded and unbounded random variables are considered;

Errors are analyzed for both a semi-discrete scheme and a fully-discrete scheme for the parabolic PDE (1).

The error of the numerical solution splits into ε = (u − uh) + (uh − uh,p). We mainly analyze the second term, i.e. the interpolation error in the probability space and obtain the first term by classic finite element analysis. Moreover, as long as uh is analytic with respect to the random parameters, the interpolation error will decay exponentially by using typical approximation theories. There- fore, in what follows the analyticity of uh is the key point of our analysis. The procedure of the error analysis is shown in the flowchart below...