Original scientific paper

On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D

Thai Anh Nhan orcid.org/0000-0001-7434-5843 ; Department of Mathematics and Science, Holy Names University, Oakland, USA
Vinh Quang Mai ; Division of Applied Mathematics, Thu Dau Mot University, 6 Tran Van On Street, Thu Dau Mot City, Binh Duong Province, Vietnam

Abstract

For singularly perturbed two-dimensional linear convection-diffusion problems, although optimal error estimates of an upwind finite difference scheme on Bakhvalov-type meshes are widely known, the analysis remains unanswered (Roos and Stynes in Comput. Meth. Appl. Math. 15 (2015), 531--550). In this short communication, by means of a new truncation error and barrier function based analysis, we address this open question for a generalization of Bakhvalov-type meshes in the sense of Boglaev and Kopteva. We prove that the upwind scheme on these mesh modifications is optimal first-order convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm. Furthermore, we derive a sufficient condition on the transition point choices to guarantee that our modified meshes can preserve the favorable properties of the original Bakhvalov mesh.

Keywords

singular perturbation, convection-diffusion, upwind difference scheme, Bakhvalov-type meshes

Hrčak ID:

261505

URI

https://hrcak.srce.hr/261505

Publication date:

26.8.2021.

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