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

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 and 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.

Truncation preconditioners for stochastic Galerkin finite element discretizations

Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.

Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM

The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.

Convergence of adaptive stochastic Galerkin FEM

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.

Robust a posteriori error estimation for stochastic Galerkin formulations of parameter-dependent linear elasticity equations

The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Youngs modulus is an affine function of a countable set of parameters. We analyse the weak formulation, its stability with respect to a weighted norm and discuss approximation using stochastic Galerkin mixed finite element methods (SG-MFEMs). We introduce a novel a posteriori error estimation scheme and establish upper and lower bounds for the SG-MFEM error. The constants in the bounds are independent of the Poisson ratio as well as the SG-MFEM discretisation parameters. In addition, we discuss proxies for the error reduction associated with certain enrichments of the SG-MFEM spaces and we use these to develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. We prove that both the a posteriori error estimate and the error reduction proxies are reliable and efficient in the incompressible limit case. Numerical results are presented to validate the theory. All experiments were performed using open source (IFISS) software that is available online.

Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation

Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be activated in the approximation space in order to estimate a quantity of interest to a prescribed error tolerance are needed. For elliptic PDEs with parameter-dependent coefficients, stochastic Galerkin finite element methods (SGFEMs) have been well studied. Under certain assumptions, it can be shown that there exists a sequence of SGFEM approximation spaces for which the energy norm of the error decays to zero at a rate that is independent of the number of input parameters. However, it is not clear how to adaptively construct these spaces in a practical and computationally efficient way. We present a new adaptive SGFEM algorithm that tackles elliptic PDEs with parameter-dependent coefficients quickly and efficiently. We consider approximation spaces with a multilevel structure---where each solution mode is associated with a finite element space on a potentially different mesh---and use an implicit a posteriori error estimation strategy to steer the adaptive enrichment of the space. At each step, the components of the error estimator are used to assess the potential benefits of a variety of enrichment strategies, including whether or not to activate more parameters. No marking or tuning parameters are required. Numerical experiments for a selection of test problems demonstrate that the new method performs optimally in that it generates a sequence of approximations for which the estimated energy error decays to zero at the same rate as the error for the underlying finite element method applied to the associated parameter-free problem.

Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs

We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).