A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh

Charuka Dilhara Wickramasinghe

doi:doi:10.11648/j.ajam.20231106.14

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A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh

Charuka Dilhara Wickramasinghe

Published in American Journal of Applied Mathematics (Volume 11, Issue 6)

Received: 3 July 2023 Accepted: 24 July 2023 Published: 23 December 2023

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Abstract

This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter ε multiplying the highest derivative. These differential equations have various real-world applications in engineering and physics. For example, these equations can be used to describe the behavior of diffusing chemical species, the behavior of viscous flows with both convection and diffusion effects, or the heat transfer in microfluidic channels or electronic chips, among others. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring ε multiplying the highest derivative. In the presence of the small parameter ε, singularly perturbed problems frequently involve boundary layers, where the solution varies rapidly in a small region near the boundaries. It is widely recognized that numerical solutions of higher-order problems are significantly more challenging than those of lower- order problems. To address both of these issues, we first decompose the fourth-order problem into a system of second-order problem. Then, we propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtaind from the second order system is equal to the fourth order problem. Our solver is both direct and highly accurate, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

Published in	American Journal of Applied Mathematics (Volume 11, Issue 6)
DOI	10.11648/j.ajam.20231106.14
Page(s)	130-144
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Shishkin Mesh, Finite Element Algorithm, Boundary Layers, Convection-Diffusion Problems

References

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Dilhara Wickramasinghe, C. (2023). A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. American Journal of Applied Mathematics, 11(6), 130-144. https://doi.org/10.11648/j.ajam.20231106.14

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ACS Style

Dilhara Wickramasinghe, C. A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. Am. J. Appl. Math. 2023, 11(6), 130-144. doi: 10.11648/j.ajam.20231106.14

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AMA Style

Dilhara Wickramasinghe C. A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. Am J Appl Math. 2023;11(6):130-144. doi: 10.11648/j.ajam.20231106.14

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@article{10.11648/j.ajam.20231106.14,
  author = {Charuka Dilhara Wickramasinghe},
  title = {A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh},
  journal = {American Journal of Applied Mathematics},
  volume = {11},
  number = {6},
  pages = {130-144},
  doi = {10.11648/j.ajam.20231106.14},
  url = {https://doi.org/10.11648/j.ajam.20231106.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231106.14},
  abstract = {This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter ε multiplying the highest derivative. These differential equations have various real-world applications in engineering and physics. For example, these equations can be used to describe the behavior of diffusing chemical species, the behavior of viscous flows with both convection and diffusion effects, or the heat transfer in microfluidic channels or electronic chips, among others. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring ε multiplying the highest derivative. In the presence of the small parameter ε, singularly perturbed problems frequently involve boundary layers, where the solution varies rapidly in a small region near the boundaries. It is widely recognized that numerical solutions of higher-order problems are significantly more challenging than those of lower- order problems. To address both of these issues, we first decompose the fourth-order problem into a system of second-order problem. Then, we propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtaind from the second order system is equal to the fourth order problem. Our solver is both direct and highly accurate, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.
},
 year = {2023}
}

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TY  - JOUR
T1  - A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh
AU  - Charuka Dilhara Wickramasinghe
Y1  - 2023/12/23
PY  - 2023
N1  - https://doi.org/10.11648/j.ajam.20231106.14
DO  - 10.11648/j.ajam.20231106.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 130
EP  - 144
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20231106.14
AB  - This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter ε multiplying the highest derivative. These differential equations have various real-world applications in engineering and physics. For example, these equations can be used to describe the behavior of diffusing chemical species, the behavior of viscous flows with both convection and diffusion effects, or the heat transfer in microfluidic channels or electronic chips, among others. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring ε multiplying the highest derivative. In the presence of the small parameter ε, singularly perturbed problems frequently involve boundary layers, where the solution varies rapidly in a small region near the boundaries. It is widely recognized that numerical solutions of higher-order problems are significantly more challenging than those of lower- order problems. To address both of these issues, we first decompose the fourth-order problem into a system of second-order problem. Then, we propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtaind from the second order system is equal to the fourth order problem. Our solver is both direct and highly accurate, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

VL  - 11
IS  - 6
ER  -

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Author Information

Charuka Dilhara Wickramasinghe

Department of Mathematics, University of Kentucky, Lexington, USA

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APA Style

Dilhara Wickramasinghe, C. (2023). A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. American Journal of Applied Mathematics, 11(6), 130-144. https://doi.org/10.11648/j.ajam.20231106.14

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ACS Style

Dilhara Wickramasinghe, C. A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. Am. J. Appl. Math. 2023, 11(6), 130-144. doi: 10.11648/j.ajam.20231106.14

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AMA Style

Dilhara Wickramasinghe C. A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh. Am J Appl Math. 2023;11(6):130-144. doi: 10.11648/j.ajam.20231106.14

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@article{10.11648/j.ajam.20231106.14,
  author = {Charuka Dilhara Wickramasinghe},
  title = {A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh},
  journal = {American Journal of Applied Mathematics},
  volume = {11},
  number = {6},
  pages = {130-144},
  doi = {10.11648/j.ajam.20231106.14},
  url = {https://doi.org/10.11648/j.ajam.20231106.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231106.14},
  abstract = {This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter ε multiplying the highest derivative. These differential equations have various real-world applications in engineering and physics. For example, these equations can be used to describe the behavior of diffusing chemical species, the behavior of viscous flows with both convection and diffusion effects, or the heat transfer in microfluidic channels or electronic chips, among others. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring ε multiplying the highest derivative. In the presence of the small parameter ε, singularly perturbed problems frequently involve boundary layers, where the solution varies rapidly in a small region near the boundaries. It is widely recognized that numerical solutions of higher-order problems are significantly more challenging than those of lower- order problems. To address both of these issues, we first decompose the fourth-order problem into a system of second-order problem. Then, we propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtaind from the second order system is equal to the fourth order problem. Our solver is both direct and highly accurate, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.
},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - A Finite Element Method for Singularly Perturbed Convection Reaction Diffusion Problems on Shishkin Mesh
AU  - Charuka Dilhara Wickramasinghe
Y1  - 2023/12/23
PY  - 2023
N1  - https://doi.org/10.11648/j.ajam.20231106.14
DO  - 10.11648/j.ajam.20231106.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 130
EP  - 144
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20231106.14
AB  - This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter ε multiplying the highest derivative. These differential equations have various real-world applications in engineering and physics. For example, these equations can be used to describe the behavior of diffusing chemical species, the behavior of viscous flows with both convection and diffusion effects, or the heat transfer in microfluidic channels or electronic chips, among others. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring ε multiplying the highest derivative. In the presence of the small parameter ε, singularly perturbed problems frequently involve boundary layers, where the solution varies rapidly in a small region near the boundaries. It is widely recognized that numerical solutions of higher-order problems are significantly more challenging than those of lower- order problems. To address both of these issues, we first decompose the fourth-order problem into a system of second-order problem. Then, we propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtaind from the second order system is equal to the fourth order problem. Our solver is both direct and highly accurate, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

VL  - 11
IS  - 6
ER  -

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