A preconditioning-based analysis for a Bakhvalov-type mesh

Abstract

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.html