A Posteriori Error Estimation for Stochastic Static Problems

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Abstract-To solve stochastic static field problems, a discretization by the Finite Element Method can be used. A system of equations is obtained with the unknowns (scalar potential at nodes for example) being random variables. To solve this stochastic system, the random variables can be approximated in a finite dimension functional space -a truncated polynomial chaos expansion. The error between the exact solution and the approximated one depends not only on the spatial mesh but also on the discretization along the stochastic dimension. In this paper, we propose an a posteriori estimation of the error due to the discretization along the stochastic dimension.

Reduction of a Finite-Element Parametric Model Using Adaptive POD Methods—Application to Uncertainty Quantification

“…Several error estimators derived from the residual have been proposed in the literature. In the following, an estimator is derived from [10], which is not necessarily efficient for large FE problems but provides error bounds. We suppose that the stiffness matrix S(p) is symmetric positive definite, which is the case with standard FE gaged potential formulations of static field problems.…”

Residual-based a posteriori error estimation for stochastic magnetostatic problems

“…A richer basis in the stochastic dimension is applied using polynomials of higher order than the ones used for the solution. An error estimator evaluated from the stochastic residual and the mean value of the stiffness matrix has also been proposed recently in [24]. These papers [23,24] focus on the stochastic error and the spatial error is assumed to be negligible.…”