Quasi-optimal preconditioners for finite element approximations of diffusion dominated convection-diffusion equations on (nearly) equilateral triangle meshes

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 equilateral 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 eigenvalue clustering at unity, the preconditioning matrix sequence and the original matrix sequence are spectrally equivalent, and under the constant coefficients assumption, the eigenvector matrices have a mild conditioning. The obtained results allow to prove the conjugate gradient optimality and the generalized minimal residual quasi-optimality in the case of structured uniform meshes. The interest of such a study relies on the observation that automatic grid generators tend to construct equilateral 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