Numerical solution of spectral stochastic finite element systems

This paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element (SSFEM) formulation. Two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems. A brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure. The differences between these matrices from their deterministic finite element counterparts are illustrated. An iterative solution scheme is proposed, which utilizes their specific properties for efficient memory management and enhanced convergence behavior. Results from numerical tests are presented. Comparisons with standard algorithms illustrate the efficiency of the proposed algorithm. The second solution procedure presented in this paper is based on hierarchical basis concepts. Results from numerical tests are again provided, and the limitations of this approach are assessed. The performance of both proposed algorithms indicates that the linear algebraic systems from the underlying SSFEM formulation can be solved with considerably less effort in memory and computation time than their size suggests. Furthermore, the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic FEM.