A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation

Jake Leonard Nkeck

doi:doi:10.11648/j.acm.20261504.12

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A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation

Jake Leonard Nkeck

Published in Applied and Computational Mathematics (Volume 15, Issue 4)

Received: 4 June 2026 Accepted: 15 June 2026 Published: 12 August 2026

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Abstract

The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.

Published in	Applied and Computational Mathematics (Volume 15, Issue 4)
DOI	10.11648/j.acm.20261504.12
Page(s)	133-143
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), 2026. Published by Science Publishing Group

Keywords

Heat Equation, Crank-nicolson Finite Element Methods, Time Singularities

References

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[3]	Grisvard, P. (1985). Elliptic Problems in Nonsmooth Domains. Pintman Advanced Publishing Program, Boston. https://doi.org/10.1137/1.9781611972030
[4]	Koslov, V. A., Maz'ya, V. G., Rosmann, J. (2001). Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. AMS Mathematical Surveys and Monographs Rhode Island. https://doi.org/10.1090/surv/085
[5]	Shu-Yu, H. (2006). Removable Singularities of Semilinear Parabolic Equations. Advances in Differential Equations 15, 137-158. https://doi.org/10.57262/ade/1355854766
[6]	Lesnic, D., Elliott, L., Ingham, B. D. (1995). Treatment of singularities in time-dependent problems using boundary element methods. Engineering Analysis with Boundary Elements 16, 65-70. https://doi.org/10.1016/j.jsv.2003.09.059
[7]	Kan, T., Takahashi, J.(2016). Time-dependent singularities in semilinear parabolic equations: Behavior at the singularities. J. Differential Equations, 260, 7278-7319. https://doi.org/10.1016/j.jde.2016.01.026
[8]	Brenner, S. C., Scott, L. R.(2008). The Mathematical Theory of Finite Element Methods. Springer Science+Business Media, New York. https://doi.org/10.1007/978-0-387-75934-0
[9]	Ciarlet, P. (2002). The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Amsterdam. https://doi.org/10.1137/1.9780898719208
[10]	Steinbach, O.(2015). Space-time finite element methods for parabolic problems. Computational Methods in Applied Mathematics 15(4), 551-566. https://doi.org/10.1515/cmam-2015-0026
[11]	Thom'ee, V. (2006). Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Heidelberg. https://doi.org/10.1137/1028090
[12]	Jung, M., Langer, U. (2013). Methode der finiten Elemente f"ur Ingenieure. Springer Science+Business Media. https://doi.org/10.1007/978-3-658-01101-7
[13]	Kondratiev, V. (1967). Boundary Values problems for elliptic equations in domains with conical or angular points. Translated. Moscow Mathematical Society 16, 227-313.
[14]	Nkemzi, B., Tanekou, S. (2018). Predictor-corrector p - and hp -versions of the finite element method for Poisson's equation in polygonal domains. Computer Methods in Applied Mechanics and Engineering 333, 74-93. https://doi.org/10.1016/j.cma.2018.01.027
[15]	Nkemzi, B., Nkeck, J. (2020). A predictor-corrector finite element method for Maxwell's equations in polygonal domains. Mathematical Problems in Engineering, 1-13. https://doi.org/10.1155/2020/3502513
[16]	Nkemzi, B. (2006).On the solution of Maxwell's equations in polygonal domains. Mathematical Methods in Applied Science 29, 1053-1080 https://doi.org/10.1002/mma.717
[17]	Renardy, M., and Rogers, R. C. (2004). An Introduction to Partial Differential Equations. (Second ed.) Springer, New York. https://doi.org/10.1007/b97427
[18]	Adams, R. A. (1975). Sobolev Spaces. Academic Press, San Francisco.
[19]	Mclean, W. (2000). Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge. https://doi.org/10.2307/3621632
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APA Style

Nkeck, J. L. (2026). A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Applied and Computational Mathematics, 15(4), 133-143. https://doi.org/10.11648/j.acm.20261504.12

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ACS Style

Nkeck, J. L. A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Appl. Comput. Math. 2026, 15(4), 133-143. doi: 10.11648/j.acm.20261504.12

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AMA Style

Nkeck JL. A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Appl Comput Math. 2026;15(4):133-143. doi: 10.11648/j.acm.20261504.12

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@article{10.11648/j.acm.20261504.12,
  author = {Jake Leonard Nkeck},
  title = {A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {133-143},
  doi = {10.11648/j.acm.20261504.12},
  url = {https://doi.org/10.11648/j.acm.20261504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.12},
  abstract = {The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.},
 year = {2026}
}

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TY  - JOUR
T1  - A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation
AU  - Jake Leonard Nkeck
Y1  - 2026/08/12
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.12
DO  - 10.11648/j.acm.20261504.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 133
EP  - 143
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.12
AB  - The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.
VL  - 15
IS  - 4
ER  -

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Author Information

Jake Leonard Nkeck

Department of Mathematics, University of Buea, Buea, Cameroon

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http://orcid.org/0000-0003-0492-8247

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Nkeck, J. L. (2026). A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Applied and Computational Mathematics, 15(4), 133-143. https://doi.org/10.11648/j.acm.20261504.12

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ACS Style

Nkeck, J. L. A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Appl. Comput. Math. 2026, 15(4), 133-143. doi: 10.11648/j.acm.20261504.12

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AMA Style

Nkeck JL. A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation. Appl Comput Math. 2026;15(4):133-143. doi: 10.11648/j.acm.20261504.12

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@article{10.11648/j.acm.20261504.12,
  author = {Jake Leonard Nkeck},
  title = {A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {133-143},
  doi = {10.11648/j.acm.20261504.12},
  url = {https://doi.org/10.11648/j.acm.20261504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.12},
  abstract = {The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation
AU  - Jake Leonard Nkeck
Y1  - 2026/08/12
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.12
DO  - 10.11648/j.acm.20261504.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 133
EP  - 143
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.12
AB  - The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.
VL  - 15
IS  - 4
ER  -

Copy | Download