Almost sure convergence of Galerkin approximations for a heat equation with a random initial condition (Q1825021)

This paper deals with the convergence of random Galerkin approximations of the solution of the random partial differential equation \(\partial u/\partial t=\partial^ 2u/\partial x^ 2\) in \(D\times (0,T]\times \Omega\) subject to \(u(x,0,\omega)=u_ 0(x,\omega)\) and \(u(x,t,\omega)=0\) in \(\partial D\times (0,T]\times \Omega,\) where \(u_ 0\) is a known random function. Theorems are proved which establish the convergence in probability and the rate of convergence of the Galerkin sample means to the mean of the solution under the \(L^ 2\) norm in x for all t in (0,T]. Then theorems are proved which establish the same sort of convergence results at the nodes of t when differentiation with respect to t is replaced by a backward difference approximation.