Adaptive data refinement in the spectral stochastic finite element method

One version of the stochastic finite element method involves representing the solution with respect to a basis in the space of random variables and evaluating the co-ordinates of the solution with respect to this basis by relying on Hilbert space projections. The approach results in an explicit dependence of the solution on certain statistics of the data. The error in evaluating these statistics, which is directly related to the amount of available data, can be propagated into errors in computing probabilistic measures of the solution. This provides the possibility of controlling the approximation error, due to limitations in the data, in probabilistic statements regarding the performance of the system under consideration. In addition to this error associated with data resolution, is added the more traditional error, associated with mesh resolution. This latter also contributes to polluting the estimated probabilities associated with the problem. The present paper will develop the above concepts and indicate how they can be coupled in order to yield a more meaningful and useful measure of approximation error in a given problem. Copyright (C) 2002 John Wiley Sons, Ltd.