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

Thai Anh Nhan, Vinh Quang Mai


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.


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

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ISSN: 1331-0623 (Print), 1848-8013 (Online)