Convergence of Adaptive Stochastic Galerkin FEM

Abstract

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.Date: November 26, 2018. 2010 Mathematics Subject Classification. 35R60, 65C20, 65N12, 65N15, 65N30. Key words and phrases. adaptive methods, a posteriori error analysis, convergence, two-level error estimate, stochastic Galerkin methods, finite element methods, parametric PDEs.Acknowledgements.

T-IFISS: a toolbox for adaptive FEM computation

“…This strategy is referred to as version 1 of the adaptive algorithm implemented in Stochastic T-IFISS. An alternative strategy is referred to as version 2 of the implemented algorithm: here, the refinement type that leads to a larger estimated error reduction is chosen at each iteration; see [13,11]. This strategy exploits the fact that local spatial error indicators (e.g., e X | K 0,K (K ∈ T h ) in the error estimation strategy (EES1)) and individual parametric error indicators e (ν) P 0 (ν ∈ Q P ) provide effective estimates of the error reduction that would be achieved by performing, respectively, a local refinement of the current mesh (e.g., by refining the element K) and a selective enrichment of the parametric component of the current Galerkin approximation (by adding the index ν ∈ Q P to the current index set P).…”

“…This strategy exploits the fact that local spatial error indicators (e.g., e X | K 0,K (K ∈ T h ) in the error estimation strategy (EES1)) and individual parametric error indicators e (ν) P 0 (ν ∈ Q P ) provide effective estimates of the error reduction that would be achieved by performing, respectively, a local refinement of the current mesh (e.g., by refining the element K) and a selective enrichment of the parametric component of the current Galerkin approximation (by adding the index ν ∈ Q P to the current index set P). We refer to [10, Theorem 5.1] and [11,Corollary 3] for the underpinning theoretical results and to [13] and [11,Section 5] for comprehensive numerical studies of the two versions of the adaptive algorithm and different marking strategies.…”

Convergence of adaptive stochastic collocation with finite elements

“…We can finally prove the convergence of the parameter-enrichment algorithm with a technique inspired by [5,Proposition 10].…”

“…For intrusive stochastic Galerkin methods, adaptive algorithms have been investigated in [5,17] and for non-intrusive stochastic collocation methods, an adaptive algorithm was proposed in [22]. The work uses a sparse grid interpolation operator to discretize the parametric domain and proposes an error estimator which consists of an parametric estimator as well as a finite element estimator.…”

“…The work uses a sparse grid interpolation operator to discretize the parametric domain and proposes an error estimator which consists of an parametric estimator as well as a finite element estimator. We extend the algorithm of [22] and include spatial adaptivity by use of a standard h-adaptive algorithm inspired by [5]. Basically, we use Dörfler marking to identify a number of collocation points which require adaptive refinement of the underlying finite element mesh and then use well understood spatial adaptivity to improve the finite element error.…”