A new upwind difference analysis of an exponentially graded Bakhvalov-type mesh for singularly perturbed elliptic convection–diffusion problems

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We are concerned with numerical methods for singularly perturbed elliptic convection–diffusion differential equations in two dimensions. Due to the presence of exponential boundary layers in these problems, special meshes – which are dense in layer regions – are designed to robustly resolve the layers. Recently, the so-called exponentially graded mesh has been extensively studied by Xenophontos et al. (see, e.g., (Xenophontos et al., 2016)) for which the authors classify the mesh as a Shishkin-type mesh. In this paper, we propose a modification of the exponentially graded mesh as a Bakhvalov-type mesh. The modified mesh is constructed by incorporating Bakhvalov’s idea that allows the mesh not only to be gradually graded in the layer regions, but also to preserve the favorable properties of the original Bakhvalov mesh. Our numerical experiments show an improvement in the accuracy of the computed solution when an upwind difference scheme is discretized on our modified mesh. More importantly, the main purpose of the paper is to propose a novel finite-difference error analysis, which is based on the truncation error and barrier function approach, for a class of Bakhvalov-type meshes; that is although such analysis for two-dimensional singularly perturbed convection–diffusion problems on Shishkin-type meshes was derived two decades ago, but no similar results are known for Bakhvalov-type meshes (Roos and Stynes, 2015).