Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

Abstract

A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard & Nobile in 2018 (SIAM J. Numer. Anal., 56, 3121-3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available online.

Error estimation and adaptivity for stochastic collocation finite elements Part II: multilevel approximation

“…The numerical results presented in [5] demonstrate the effectivity and robustness of our error estimation strategy as well as the utility of the error indicators guiding the adaptive refinement process. The results in [5] also showed that optimality of convergence is difficult to achieve using a simple single-level approach where a single finite element space is associated with all active collocation points. The main aim of this contribution is to see if optimal convergence rates can be recovered by computing results using a multilevel implementation of the algorithm outlined in [5].…”

“…The combination of adaptive sparse grid methods with a hierarchy of spatial approximations is a relatively new development, see for example, [12,13]. In our precursor paper [5] (part I), we extended the adaptive framework developed by Guignard & Nobile [10] and presented a critical comparison of alternative strategies in the context of solving a model problem that combines strong anisotropy in the parametric dependence with singular behavior in the physical space. The numerical results presented in [5] demonstrate the effectivity and robustness of our error estimation strategy as well as the utility of the error indicators guiding the adaptive refinement process.…”

“…In our precursor paper [5] (part I), we extended the adaptive framework developed by Guignard & Nobile [10] and presented a critical comparison of alternative strategies in the context of solving a model problem that combines strong anisotropy in the parametric dependence with singular behavior in the physical space. The numerical results presented in [5] demonstrate the effectivity and robustness of our error estimation strategy as well as the utility of the error indicators guiding the adaptive refinement process. The results in [5] also showed that optimality of convergence is difficult to achieve using a simple single-level approach where a single finite element space is associated with all active collocation points.…”

“…The results in [5] also showed that optimality of convergence is difficult to achieve using a simple single-level approach where a single finite element space is associated with all active collocation points. The main aim of this contribution is to see if optimal convergence rates can be recovered by computing results using a multilevel implementation of the algorithm outlined in [5].…”

“…The model problems that are of interest are stated in section 2. The only difference from the problem statement in [5] is that we also cover the case where the right-hand side function has a parametric dependence. The adaptive solution algorithm from [5] is extended to cover the case of a non-deterministic right-hand side function in section 3.…”