Solutions To a Parabolic Equation Based on Polynomials P_1,P_2 and P_3 in the Square Domain Using Galerkin Method
Keywords:
Parabolic Equations, Finite Element, Square Domain
Abstract
In this work, we provide a finite element method (FEM) that is based on linear, quadratic and cubic polynomials for the purpose of solving the two-dimensional parabolic heat problem in the square domain. In light of this, the discrete issue is governed by a set of linear equations, which can be solved by applying traditional methods. In these equations, the coefficients are positive-definite and symmetric at the same time. By doing numerical experiments, one can obtain error estimators that are optimal for both for L^2 and H^1 norms. For the purpose of validating the solution and developing a graphical representation, we make use the MATLAB R2018b software.
References
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3. Alfarawi, Suliman, Azeldin El-sawi, and Hossin Omar. "Exploring Discontinuous Meshing for CFD Modelling of Counter Flow Heat Exchanger." Journal of Advanced Research in Numerical Heat Transfer 5, no. 1 (2021): 26-34.
4. Azmi, Anis Zafirah. "Finite Element Solution of Heat Conduction Problem." PhD diss., Universiti Teknologi Malaysia, 2010.
5. Brenner, Susanne C., L. Ridgway Scott, and L. Ridgway Scott. The mathematical theory of finite element methods. Vol. 3. New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75934-0
6. Choi, Lee Yuk, Wong Sau Keong, Ooi Hooi Woon, and Sia Chee Kiong. "Numerical Study of Two-Dimensional Transient Heat Conduction Using Finite Element Method." In Computational Methods in Engineering & Science, pp. 241-241. Springer, Berlin, Heidelberg, 2006. https://doi.org/10.1007/978-3-540-48260-4_87
7. Devi, Supriya, K. V. Nagaraja, Sarada Jayan, and T. V. Smitha. "2D Higher order triangular mesh generation in irregular domain for finite element analysis using MATLAB." In IOP Conference Series: Materials Science and Engineering, vol. 577, no. 1, p. 012132. IOP Publishing, 2019. https://doi.org/10.1088/1757-899X/577/1/012132
8. Feng, Yan-Yan, and Cun-Hai Wang. "Discontinuous finite element method applied to transient pure and coupled radiative heat transfer." International Communications in Heat and Mass Transfer 122 (2021): 105156. https://doi.org/10.1016/j.icheatmasstransfer.2021.105156
9. Fourier, Joseph. The Analytical Theory of Heat. The University Press, 1874.
10. Ghoshdastidar, Partha Sarathi. Computer Simulation of flow and heat transfer. Tata McGraw-Hill Publishing Company Limited, 1998.
11. Kailani, Nor Hafizah Ahmad. "The Application of Finite Element Method in 2d Heat Distribution Problems for Irregular Geometry." PhD diss., Universiti Teknologi Malaysia, 2014.
12. L. Chen. ifem: an innovative finite element methods package in matlab. Preprint, University of Maryland, 2008.
13. Lienhard, I. V., and H. John. A heat transfer textbook. Phlogiston Press, 2005.
14. Liu, Xu, Nan Gui, Xingtuan Yang, Jiyuan Tu, and Shengyao Jiang. "A new discrete element-embedded finite element method for transient deformation, movement and heat transfer in packed bed." International Journal of Heat and Mass Transfer 165 (2021): 120714. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120714
15. Narasimhan, Thiruppudaimarudhur N. "Fourier's heat conduction equation: History, influence, and connections." Reviews of Geophysics 37, no. 1 (1999): 151-172. https://doi.org/10.1029/1998RG900006
16. Ngarisan, Noor Syazana, Yeak Su Hoe, Nur Idalisa Norddin, and Nur Solihah Khadhiah Abdullah. "The Application of Finite Element Method to Solve Heat Transfer Problem Involving 2D Irregular Geometry." Journal of Applied Environmental and Biological Sciences 6, no. 11S (2016): 23-30.
17. Papathanasiou, Theodosios, Payam Khazaeinejad, and Hamid Bahai. "Finite element analysis of heat transfer in thin multilayered plates." In Proceedings of the 15th UK Heat Transfer Conference, UKHTC. 2017.
18. Rao, Singiresu S. The finite element method in engineering. Butterworth-heinemann, 2017. https://doi.org/10.1016/B978-0-12-811768-2.00001-8
19. Rosero, D. Garcia, and A. M. Zsaki. "Finite element mesh improvement using an a priori local p-refinement for stress analysis of underground excavations." Cogent Engineering 7, no. 1 (2020): 1769287. https://doi.org/10.1080/23311916.2020.1769287
20. Sharma, Rahul. "Impact of temperature and moisture on thermal conductivity and diffusivity of stones using one dimensional steady state heat flow: An experimental study." International Journal of Advance Research in Science and Engineering 6, no. 2 (2017): 61-67.
Published
2024-07-03
How to Cite
Abd, S. K. (2024). Solutions To a Parabolic Equation Based on Polynomials P_1,P_2 and P_3 in the Square Domain Using Galerkin Method. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 5(3), 132-143. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/639
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Articles
