A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes

Abstract

We analyze a-posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countablyparametric, elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this end we establish a contraction property satisfied by its iterates. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.

A posteriori error estimation for elliptic partial differential equations with small uncertainties

“…, which is due to the uncertainty in the diffusion coefficient. Proceeding as in (17) and using the fact that D (a j ∇u 0 + a 0 ∇U j ) · ∇v = 0 for all v ∈ V , the following equalities hold for any v ∈ V and for a.e. y ∈…”

Adaptive algorithm for stochastic Galerkin method

“…They are based on the idea that the error estimates with respect to spatial and stochastic approximation spaces can be separated in some sense [4], [9], [13], [12]. Eigel et al in [12], [13] describe and prove residual based a posteriori error estimates derived from the adequate approaches for deterministic problems. A marking strategy for both physical and stochastic degrees of freedom is based on the Dorfler property [12].…”

“…Eigel et al in [12], [13] describe and prove residual based a posteriori error estimates derived from the adequate approaches for deterministic problems. A marking strategy for both physical and stochastic degrees of freedom is based on the Dorfler property [12]. For dealing with the stochastic part of the error, the equivalence between the energy norm of the underlying problem and the energy norm of some related deterministic problem is used.…”

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

“…In the work [34], the authors examined the use of stochastic collocation for the numerical solution of optimal control problems subject to SPDEs, discussing generalized polynomial chaos thoroughly and presenting computational examples to show the performance of their method. Also, after finishing this paper the authors became aware of the work [35] including recent developments on the adaptive stochastic Galerkin FEM approches, which gives us some future research ideas. In the work [35], the authors developed adaptive refinement algorithms for SGFEM for countably-parametric, elliptic boundary value problems and proved the convergence of their adaptive algorithm.…”

“…Also, after finishing this paper the authors became aware of the work [35] including recent developments on the adaptive stochastic Galerkin FEM approches, which gives us some future research ideas. In the work [35], the authors developed adaptive refinement algorithms for SGFEM for countably-parametric, elliptic boundary value problems and proved the convergence of their adaptive algorithm.…”