A preconditioning-based analysis for a Bakhvalov-type mesh

A new preconditioning-based parameter-uniform convergence analysis is presented for one-dimensional singularly perturbed convection-diffusion problems discretized by an upwind difference scheme on a Bakhvalov-type mesh. The proof technique utilizes the classical convergence principle: uniform stability and uniform consistency imply uniform convergence, which can only be used after applying an appropriate preconditioner to the discrete operator. References N. S. Bakhvalov. The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys. 9.4 (1969), pp. 139–166. doi: 10.1016/0041-5553(69)90038-X. I. P. Boglaev. Approximate solution of a non-linear boundary value problem with a small parameter for the highest-order differential. USSR Comput. Math. Math. Phys. 24.6 (1984), pp. 30–35. doi: 10.1016/0041-5553(84)90005-3. T. Linß. Layer-adapted meshes for reaction-convection-diffusion problems. Vol. 1985. Lecture Notes in Mathematics. Springer-Verlag, 2010. doi: 10.1007/978-3-642-05134-0. T. Linß, H.-G. Roos, and R. Vulanović. Uniform pointwise convergence on Shishkin-type meshes for quasi-linear convection-diffusion problems. SIAM J. Numer. Anal. 38.3 (2000), pp. 897–912. doi: 10.1137/S0036142999355957. V. D. Liseikin. Layer resolving grids and transformations for singular perturbation problems. De Gruyter, 2001. doi: 10.1515/9783110941944. T. A. Nhan, M. Stynes, and R. Vulanović. Optimal uniform-convergence results for convection-diffusion problems in one dimension using preconditioning. J. Comput. Appl. Math. 338 (2018), pp. 227–238. doi: 10.1016/j.cam.2018.02.012. T. A. Nhan and R. Vulanović. Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh. Electron. Trans. Numer. Anal. 51 (2019), pp. 315–330. doi: 10.1553/etna_vol51s315 T. A. Nhan and R. Vulanović. The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems. Numer. Alg. 87 (2021), pp. 203–221. doi: 10.1007/s11075-020-00964-z H.-G. Roos and T. Linß. Sufficient conditions for uniform convergence on layer-adapted grids. Computing 63.1 (1999), pp. 27–45. doi: 10.1007/s006070050049. H.-G. Roos and M. Stynes. Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Meth. Appl. Math. 15.4 (2015), pp. 531–550. doi: 10.1515/cmam-2015-0011. H.-G. Roos, M. Stynes, and L. Tobiska. Robust numerical methods for singularly perturbed differential equations. Vol. 24. Springer Series in Computational Mathematics. Springer-Verlag, 2008. doi: 10.1007/978-3-540-34467-4 G. I. Shishkin. A difference scheme for a singularly perturbed equation of parabolic type with discontinuous boundary conditions. USSR Comput. Math. Math. Phys. 28.6 (1988), pp. 32–41. doi: 10.1016/0041-5553(88)90039-0. M. Stynes. Steady-state convection-diffusion problems. Acta Numer. 14 (2005), pp. 445–508. doi: 10.1017/S0962492904000261. R. Vulanović and T. A. Nhan. Robust hybrid schemes of higher order for singularly perturbed convection-diffusion problems. Appl. Math. Comput. 386 (2020), p. 125495. doi: 10.1016/j.amc.2020.125495. R. Vulanović and T. A. Nhan. Uniform convergence via preconditioning. Int. J. Numer. Anal. Model. Ser. B 5.4 (2014), pp. 347–356. url: www.global-sci.org/intro/article_detail/ijnamb/239.htm