Saddle Point Least Squares Preconditioning of Mixed Methods

TL;DR

This paper introduces a new discretization and preconditioning approach for mixed variational problems, leveraging saddle point theory to improve iterative solution efficiency and compatibility of discrete spaces.

Contribution

It develops a simple, effective preconditioning method for mixed formulations that avoids global assembly and provides sharp error and convergence estimates.

Findings

Efficient iterative solvers for mixed formulations are proposed.

Discrete spaces with compatible properties are identified.

Sharp bounds for discretization and convergence rates are established.

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.