Saddle point least squares preconditioning of mixed methods

Abstract

We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided. For the proposed discrete spaces and solvers, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf − sup and sup − sup constants of the pair of discrete spaces.2000 Mathematics Subject Classification. 74S05, 74B05, 65N22, 65N55.

Efficient discretization and preconditioning of the singularly perturbed Reaction Diffusion problem

“…Preconditioning the SPLS discretization. We summarize a general preconditioning framework to approximate the solution of (3.1) that is presented in [5,7]. We plan to combine this framework with the new concept of optimal test norm.…”

“…According to [7], the UPCG iterations p j converge to the solution p h of (3.14), and the rate of convergence for p j − p h Sh depends on the condition number of Sh that satisfies (3.15) κ( Sh ) ≤ κ(S h ) • κ(P h A h ).…”

“…To improve the rate of convergence (in the energy norm) for the case ε < ε 0 , we proceed by using two main tools. The first is a mixed formulation that is suitable to Saddle Point Least Squares (SPLS) discretization [7,8,9,11]. This allows for a higher order approximation of the flux ∇u using the same discretization spaces as in the standard discretization of (1.1).…”

“…Many aspects of the SPLS formulation are also common to the DPG approach [13,17,21,22,24,27]. The SPLS method as a general method for solving mixed variational formulations is presented in [7,8,9,11]. In this paper, we also combine SPLS formulation with the concept of optimal test norm [16,19,20,21,24,25,27] and a general preconditioning technique, introduced in [7], in order to improve the stability and approximability of the final discretization process.…”

A Non-conforming Saddle Point Least Squares Approach for an Elliptic Interface Problem

“…Due to the iterative process we choose to solve the discrete SPLS formulation, assembly of the stiffness matrices for the trial spaces is avoided. The SPLS method can be also be combined with multilevel preconditioning techniques in order to address particular challenges of the PDE to be solved due to discontinuous coefficients or multidimensional domains [6]. In contrast with the SPLS work presented in [10,11], where both the test and trial spaces were chosen to be conforming finite element spaces, this paper considers trial spaces which are non-conforming finite element spaces.…”