Optimal Preconditioners for Finite Element Approximations of Convection-Diffusion Equations on structured meshes

TL;DR

This paper analyzes spectral properties of preconditioners for finite element discretizations of convection-diffusion equations on structured meshes, demonstrating their effectiveness and spectral equivalence, leading to optimal Krylov method performance.

Contribution

It provides a spectral analysis of preconditioners for finite element systems on structured meshes, establishing their spectral clustering and equivalence under certain conditions.

Findings

Preconditioned matrices cluster eigenvalues at one.

Spectral equivalence between preconditioned and original matrices.

Numerical tests confirm theoretical results.

Abstract

The paper is devoted to the spectral analysis of effective preconditioners for linear systems obtained via a Finite Element approximation to diffusion-dominated convection-diffusion equations. We consider a model setting in which the structured finite element partition is made by equi-lateral triangles. Under such assumptions, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequences exhibit a strong clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and the eigenvector matrices have a mild conditioning. The obtained results allow to show the optimality of the related preconditioned Krylov methods. %It is important to stress that The interest of such a study relies on the observation that automatic grid generators tend to construct equi-lateral triangles when the mesh is fine enough. Numerical tests, both on the model setting and in the non-structured case, show the effectiveness of the proposal and the correctness of the theoretical findings.