Geodesic
Adaptive grids and high-order schemes for solving singularly-perturbed problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 77-92

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Layer-resolving grids remain an important element of comprehensive software codes when solving real-life problems with layers of singularities as they can substantially enhance the efficiency of computer-resource utilization. This paper describes an explicit approach to generating layer-resolving grids which is aimed at application of difference schemes of an arbitrary order. The approach proposed is based on estimates of derivatives of solutions to singularly-perturbed problems and is a generalization of the approach developed for the first order schemes. The layer-resolving grids proposed are suitable to tackle problems with exponential-, power-, logarithmic-, and mixed-type boundary and interior layers. Theoretical results have been confirmed by the numerical experiments on a number of test problems with such layers; the results were compared to those obtained with difference schemes of different orders of accuracy. 
@article{SJVM_2021_24_1_a6,
     author = {V. D. Liseikin and V. I. Paasonen},
     title = {Adaptive grids and high-order schemes for solving singularly-perturbed problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {77--92},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/}
}
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V. D. Liseikin; V. I. Paasonen. Adaptive grids and high-order schemes for solving singularly-perturbed problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 77-92. http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/