Prime sample scheme for almost sure convergence of a Galerkin approximation (Q1179376)

This paper is concerned with Galerkin approximation of the mean of the solution of a random partial differential equation of the form \(\partial u/\partial t=\partial^ 2u/\partial x^ 2\) in \(D\times(0,T]\times\Omega\); \(u(x,0,\omega)=u_ 0(x,\omega)\), \(u(x,t,\omega)=0\) in \(\partial D\times(0,T]\times\Omega\), where \(D=(0,1)\), \(0<T<\infty\), and \((\Omega,\Sigma,P)\) is a complete probability space. A method of choosing grid mesh sizes as reciprocals of prime numbers is presented which results in a Galerkin approximation with less computational complexity. A table is given which summarizes a comparison for a simple example of numerical approximations obtained from this method with those obtained by using a standard Galerkin approximation.