Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations

The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear of system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the Galerkin matrix. We compare the popular...

A preconditioner for the Schur complement matrix

A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann-Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that shar...

A preconditioner for the Schur complement domain decomposition method

A Hierarchical Low-rank Schur Complement Preconditioner for Indefinite Linear Systems

Nonsymmetric and highly indefinite linear systems can be quite difficult to solve via iterative methods. This paper combines ideas from the Multilevel Schur Low-Rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235–259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure ...

Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noise. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random ...