School of Applied Mathematics and Informatics, J.J. Strossmayer University of Osijek/Osijek Mathematical Society
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.
Nhan, T. A., & Mai, V. Q. (2021). On Bakhvalov-type meshes for a linear convection-diffusion problem in 2D. Mathematical Communication 26 (2021), 121–130.