On the Convergence of Stochastic Finite Elements

We investigate the rate of convergence of stochastic basis elements to the solution of a stochastic operator equation. As in deterministic finite elements, the solution may be approximately represented as the linear combination of basis elements. In the stochastic case, however, the solution belongs to a Hilbert space of functions defined on a cross product domain endowed with the product of a deterministic and probabilistic measure. We show that if the dimension of the stochastic space is n, and the desired accuracy is of order {var_epsilon}, the number of stochastic elements required to achieve this level of precision, in the Galerkin method, is on the order of | ln {var_epsilon} |{sup n}