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Volume 4, Issue 1
An $\varepsilon$-Uniform Finite Element Method for Singularly Perturbed Two-Point Boundary Value Problems

Q. S. Song, G. Yin & Z. Zhang

Int. J. Numer. Anal. Mod., 4 (2007), pp. 127-140.

Published online: 2007-04

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  • Abstract

This work develops an $\varepsilon$-uniform finite element method for singularly perturbed two-point boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (without boundary layers). Using this fact, an effective $\varepsilon$-uniform approximation out of boundary layer is proposed by adding one point only in the element that contains the boundary layer. The thickness of the boundary layer need not be known a priori. Numerical results are carried out and compared to the Shishkin mesh for demonstration purpose.

  • Keywords

finite element method, singular perturbation, $\varepsilon$-uniform approximation, layer-adapted mesh, Shishkin mesh.

  • AMS Subject Headings

65N30, 65L10, 65L60, 34D15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-4-127, author = {Q. S. Song, G. Yin and Z. Zhang}, title = {An $\varepsilon$-Uniform Finite Element Method for Singularly Perturbed Two-Point Boundary Value Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2007}, volume = {4}, number = {1}, pages = {127--140}, abstract = {

This work develops an $\varepsilon$-uniform finite element method for singularly perturbed two-point boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (without boundary layers). Using this fact, an effective $\varepsilon$-uniform approximation out of boundary layer is proposed by adding one point only in the element that contains the boundary layer. The thickness of the boundary layer need not be known a priori. Numerical results are carried out and compared to the Shishkin mesh for demonstration purpose.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/855.html} }
TY - JOUR T1 - An $\varepsilon$-Uniform Finite Element Method for Singularly Perturbed Two-Point Boundary Value Problems AU - Q. S. Song, G. Yin & Z. Zhang JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 127 EP - 140 PY - 2007 DA - 2007/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/855.html KW - finite element method, singular perturbation, $\varepsilon$-uniform approximation, layer-adapted mesh, Shishkin mesh. AB -

This work develops an $\varepsilon$-uniform finite element method for singularly perturbed two-point boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (without boundary layers). Using this fact, an effective $\varepsilon$-uniform approximation out of boundary layer is proposed by adding one point only in the element that contains the boundary layer. The thickness of the boundary layer need not be known a priori. Numerical results are carried out and compared to the Shishkin mesh for demonstration purpose.

Q. S. Song, G. Yin and Z. Zhang. (2007). An $\varepsilon$-Uniform Finite Element Method for Singularly Perturbed Two-Point Boundary Value Problems. International Journal of Numerical Analysis and Modeling. 4 (1). 127-140. doi:
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