Novel Approach for solving the discrete Stokes problems based on Augmented Lagrangian and Global Techniques: Application to Saddle-Point Linear Systems from Incompressible flow

TL;DR

This paper introduces a new augmented Lagrangian preconditioner combined with global Arnoldi techniques to efficiently solve large saddle-point linear systems from discretized Stokes equations, improving convergence and robustness.

Contribution

The paper presents a novel augmented Lagrangian preconditioner based on global Arnoldi methods tailored for block three-by-three systems from Stokes discretizations, with spectral analysis and numerical validation.

Findings

The new method accelerates convergence of Krylov subspace solvers.

It demonstrates improved robustness and efficiency over existing approaches.

Numerical results show reduced computational time.

Abstract

In this paper, a novel augmented Lagrangian preconditioner based on global Arnoldi for accelerating the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure, these systems typically arise from discretizing the Stokes equations using mixed-finite element methods. In practice, the components of velocity are always approximated using a single finite element space. More precisely, in two dimensions, our new approach based on standard space of scalar finite element basis functions to discretize the velocity space. This componentwise splitting can be shown to induce a natural block three-by-three structure. Spectral analyses is established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring computational time.