Stochastic Galerkin locking free mixed finite element methods for parameter-dependent linear elasticity equations

Abstract: It is the aim of this talk to give an overview of some recent work [1], [2] on the use of stochastic Galerkin mixed finite element methods(SG-MFEMs) for parameter-dependent linear elasticity equations. Starting from a novel three-field PDE model in which the Young's modulus is represented as an affine function of a finite/countable set of parameters, we discuss SG-MFEM approximation and introduce a novel a posteriori error estimation scheme. We examine the error in the natural weighted norm with respect to which the weak formulation is stable.  Exploiting the connection between this norm and the underlying PDE operator also leads to an efficient preconditioning strategy for the associated discrete problems. Unlike standard residual-based error estimation schemes, the proposed strategy requires the solution of auxiliary problems on carefully constructed detail spaces on both the spatial and parameter domains. We show that the proposed error estimator is reliable and efficient. The constants in the bounds are independent of the Poisson ratio as well as the SG-MFEM discretisation parameters, meaning that the estimator is robust in the incompressible limit. Finally, we also discuss proxies for the error reduction associated with potential enrichments of the SG-MFEM spaces and suggest how to use these to develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance.