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The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

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Abstract

A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered. The problem is discretized by the upwind finite-difference method. The analysis of this method on Shishkin-type meshes has been well-established, but the discretization mesh in this paper is the original Bakhvalov mesh, introduced in 1969 as the first layer-adapted mesh. We analyze the error of the numerical solution in the maximum norm and prove first-order pointwise accuracy, uniform in the perturbation parameter. This is the first complete analysis of this kind for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh. Our numerical experiments validate the theoretical analysis.

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Authors and Affiliations

  1. Department of Mathematics and Science, Holy Names University, 3500 Mountain Blvd., Oakland, CA, 94619, USA

    Thái Anh Nhan

  2. Department of Mathematical Sciences, Kent State University at Stark, 6000 Frank Ave. NW, North Canton, OH, 44720, USA

    Relja Vulanović

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  1. Thái Anh Nhan
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  2. Relja Vulanović
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Correspondence to Thái Anh Nhan.

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Nhan, T.A., Vulanović, R. The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems. Numer Algor 87, 203–221 (2021). https://doi.org/10.1007/s11075-020-00964-z

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  • DOI: https://doi.org/10.1007/s11075-020-00964-z

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