A New Strategy Based on Augmented Lagrangian for a Class of Block Three-by-Three Linear Systems: An Application for Stokes Problems

In this paper, we propose a novel augmented Lagrangian preconditioner based on the global Arnoldi method for accelerating the convergence of Krylov subspace solvers applied to linear systems with a block three-by-three structure. Such systems typically arise from mixed finite element discretizations of the Stokes equations. In practice, the velocity components are approximated using a common finite element space. More specifically, in two dimensions, our approach relies on a standard scalar finite element basis to discretize the velocity space. This componentwise splitting naturally induces the desired block three-by-three structure. We establish spectral analyses for the exact versions of the proposed preconditioners. Numerical experiments further demonstrate that the new approach is more robust and efficient for solving discrete Stokes problems. In particular, efficiency is assessed in terms of reduced computational time, confirming the practical advantages of our method.