Robust Preconditioning for Stochastic Galerkin Formulations of Parameter-Dependent Nearly Incompressible Elasticity Equations

Abstract

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 Young's 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.Key words. uncertainty quantification, linear elasticity, mixed approximation, stochastic Galerkin finite element method, a posteriori error estimation, adaptivity.

Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds

“…Similar idea was, in a simpler form, used already in [28,29]. In the current paper, it is applied in a more general setting, and we believe that the derived technique may lead to an improvement of some other recently introduced estimates, such as [17,23]. The derived technique is also applicable to systems in the form of multi-term matrix equation (see [25, eq.…”

“…[34, p.120], the eigenvalues of (2s − 2) for uniform distribution; see [13,17,23,24,35]. The main difference between (2.12) and (2.13) is that the former is considered point-wise, while the latter uses the norms of a k over D. The condition (2.12) allows us to obtain not only more accurate two-sided guaranteed bounds to the spectra, but these bounds also apply to parameter distribution and functions a k for which no estimate could be obtained using the standard approach; see subsection 4.4.…”

“…For special classes of parameters, it was shown, see, e.g., [24,32], that certain block-diagonal matrices are spectrally equivalent to A independently of the degree of polynomials and the number of random parameters, and thus they can be used for preconditioning. Having a good preconditioning method or, in other words, a good and feasible approximation of A −1 , we may also efficiently estimate a posteriori the energy norm of the error during iterative solution processes [1,5,6,9,17]. This estimate can be used in adaptive algorithms [5,7,8].…”

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

“…For instance, one might consider the Young modulus E (e.g. in [42]) or the bulk modulus K and the Poisson ratio ν (e.g. in [56,18]), as the independent driving coefficients.…”

Convergence of Adaptive Stochastic Galerkin FEM

“…Typically, they are developed by extending the a posteriori error estimation techniques commonly used for deterministic problems to parametric settings. For example, dual-based a posteriori error estimates are employed in [MLM07]; implicit error estimators (in the spirit of [AO00]) are used in [WK09] for the sGFEM based on multi-element generalized polynomial chaos expansions; explicit residual-based a posteriori error estimators provide spatial and stochastic error indicators for adaptive refinement in [Git13,EGSZ14,EGSZ15]; local equilibration error estimators are utilized in [EM16]; and hierarchical error estimators and the associated estimates of error reduction drive adaptive algorithms proposed in [BS16,BR18a,BPRR18,CPB18,KBPS18].…”