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Volume 18, Issue 4
An $hp$ Finite Element Method for a Singularly Perturbed Reaction-Convection-Diffusion Boundary Value Problem with Two Small Parameters

​Irene Sykopetritou & Christos Xenophontos

Int. J. Numer. Anal. Mod., 18 (2021), pp. 481-499.

Published online: 2021-05

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  • Abstract

We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called $Spectral$ $Boundary$ $Layer$ mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings as well as compare the proposed method with others found in the literature.

  • Keywords

Singularly perturbed problem, reaction-convection-diffusion, boundary layers, $hp$ finite element method, robust exponential convergence.

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-18-481, author = {Sykopetritou , ​Irene and Xenophontos , Christos}, title = {An $hp$ Finite Element Method for a Singularly Perturbed Reaction-Convection-Diffusion Boundary Value Problem with Two Small Parameters}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {4}, pages = {481--499}, abstract = {

We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called $Spectral$ $Boundary$ $Layer$ mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings as well as compare the proposed method with others found in the literature.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19111.html} }
TY - JOUR T1 - An $hp$ Finite Element Method for a Singularly Perturbed Reaction-Convection-Diffusion Boundary Value Problem with Two Small Parameters AU - Sykopetritou , ​Irene AU - Xenophontos , Christos JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 481 EP - 499 PY - 2021 DA - 2021/05 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19111.html KW - Singularly perturbed problem, reaction-convection-diffusion, boundary layers, $hp$ finite element method, robust exponential convergence. AB -

We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called $Spectral$ $Boundary$ $Layer$ mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings as well as compare the proposed method with others found in the literature.

Sykopetritou , ​Irene and Xenophontos , Christos. (2021). An $hp$ Finite Element Method for a Singularly Perturbed Reaction-Convection-Diffusion Boundary Value Problem with Two Small Parameters. International Journal of Numerical Analysis and Modeling. 18 (4). 481-499. doi:
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