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

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

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Error estimates for linear finite elements on Bakhvalov-type meshes

summary:For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh

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

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