Parameter-uniform convergence analysis on a Bakhvalov-type mesh with a smooth mesh-generating function using the preconditioning approach
- Authors
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- Keywords:
- singular perturbation, convection-diffusion, boundary-value problem, Bakhvalov-type mesh, finite differences, uniform convergence, preconditioning
- Abstract
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The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on a Bakhvalov-type mesh generated by a smooth mesh-generating function is analyzed. The preconditioning technique is used to obtain the first-order pointwise convergence uniform in the perturbation parameter.
- References
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[1] N. S. Bakhvalov, The optimization of methods of solving boundary value problems with a boundary layer, USSR Computational Mathematics and Mathematical Physics 9 (1969), 139–166.
[2] E. Bohl, Finite Modelle gewöhnlicher Randwertaufgaben, Teubner, Stuttgart, 1981.
[3] T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Lecture Notes in Mathematics, vol. 1985, Springer, Berlin and Heidelberg, 2010.
[4] T. Linß, H.-G. Roos, and R. Vulanović, Uniform pointwise convergence on Shishkin-type meshes for quasilinear convection-diffusion problems, SIAM Journal on Numerical Analysis 38 (2001), 897–912.
[5] T. A. Nhan and V. Q. Mai, A preconditioning-based analysis for a Bakhvalov-type mesh, ANZIAM Journal 62 (2022), C146–C162, Proceedings of CTAC-2020.
[6] T. A. Nhan, M. Stynes, and R. Vulanović, Optimal uniform-convergence results for convection-diffusion problems in one dimension using preconditioning, Journal of Computational and Applied Mathematics 338 (2018), 227–238.
[7] T. A. Nhan and R. Vulanović, Preconditioning and uniform convergence for convection-diffusion problems discretized on Shishkin-type meshes, Advances in Numerical Analysis 2016 (2016), Article ID 2161279.
[8] T. A. Nhan and R. Vulanović, Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh, Electronic Transactions on Numerical Analysis 51 (2019), 315–330.
[9] T. A. Nhan and R. Vulanović, The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems, Numerical Algorithms 87 (2021), 203–221, https://doi.org/10.1007/s11075-020-00964-z.
[10] H.-G. Roos, A note on the conditioning of upwind schemes on Shishkin meshes, IMA Journal of Numerical Analysis 16 (1996), 529–538.
[11] H.-G. Roos and T. Linß, Sufficient condition for uniform convergence on layer-adapted grids, Computing 63 (1999), no. 1, 27–45.
[12] R. Vulanović, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. u Novom Sadu Zb. Rad. Prir. Mat. Fak. Ser. Mat. 13 (1983), 187–201.
[13] R. Vulanović, A priori meshes for singularly perturbed quasilinear two-point boundary value problems, IMA Journal of Numerical Analysis 21 (2001), 349–366.
[14] R. Vulanović and T. A. Nhan, Uniform convergence via preconditioning, International Journal of Numerical Analysis and Modeling, Series B 5 (2016), 347–356.
[15] R. Vulanović and T. A. Nhan, Robust hybrid schemes of higher order for singularly perturbed convection-diffusion problems, Applied Mathematics and Computation 386 (2020), 125495, https://doi.org/10.1016/j.amc.2020.125495.
[16] R. Vulanović and T. A. Nhan, An Improved Kellogg-Tsan Solution Decomposition in Numerical Methods for Singularly Perturbed Convection-Diffusion Problems, Applied Numerical Mathematics 170 (2021), 128–145.
- Additional Files
- Published
- 25-03-2023
- Issue
- Vol. 2 No. 1 (2024)
- Section
- Research Article
- License
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Copyright (c) 2024 Thái Anh Nhan, Relja Vulanovic
This work is licensed under a Creative Commons Attribution 4.0 International License.
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