Volume 6, Issue 5
Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

Mei-Ling Sun & Shan Jiang

Adv. Appl. Math. Mech., 6 (2014), pp. 604-614.

Published online: 2014-06

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  • Abstract

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L2 norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.

  • Keywords

Multiscale basis functions singular perturbation boundary layer adaptively graded meshes

  • AMS Subject Headings

35J25 65N12 65N30

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COPYRIGHT: © Global Science Press

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@Article{AAMM-6-604, author = {Mei-Ling Sun and Shan Jiang}, title = {Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {5}, pages = {604--614}, abstract = {

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L2 norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m488}, url = {http://global-sci.org/intro/article_detail/aamm/38.html} }
TY - JOUR T1 - Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes AU - Mei-Ling Sun & Shan Jiang JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 604 EP - 614 PY - 2014 DA - 2014/06 SN - 6 DO - http://dor.org/10.4208/aamm.2013.m488 UR - https://global-sci.org/intro/aamm/38.html KW - Multiscale basis functions KW - singular perturbation KW - boundary layer KW - adaptively graded meshes AB -

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L2 norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.

Mei-Ling Sun & Shan Jiang. (1970). Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes. Advances in Applied Mathematics and Mechanics. 6 (5). 604-614. doi:10.4208/aamm.2013.m488
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