| Citation: |
Bhupen Deka, Papri Roy, Naresh Kumar. WEAK GALERKIN FINITE ELEMENT METHODS COMBINED WITH CRANK-NICOLSON SCHEME FOR PARABOLIC INTERFACE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1433-1442. doi: 10.11948/20190218
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WEAK GALERKIN FINITE ELEMENT METHODS COMBINED WITH CRANK-NICOLSON SCHEME FOR PARABOLIC INTERFACE PROBLEMS
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Abstract
This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K), \; {\cal P}_{k-1}(\partial K), \; \big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results.-
Keywords:
- Parabolic /
- Interface /
- Finite element method /
- Weak Galerkin method /
- Optimal error estimates /
- Low regularity /
- Crank-Nicolson.
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References
[1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. [2] G. Chen and J. Cui, On the error estimates of a hybridizable discontinuous galerkin method for second-order elliptic problem with discontinuous coefficients, IMA Journal of Numerical Analysis, 2019. DOI: 10.1093/imanum/drz003. [3] W. Dai, H. Wang, P. M. Jordan et al., A mathematical model for skin burn injury induced by radiation heating, Int. Jour. Heat and Mass Transfer, 2008, 51(23–24), 5497–5510. doi: 10.1016/j.ijheatmasstransfer.2008.01.006 [4] B. Deka, A weak galerkin finite element method for elliptic interface problems with polynomial reduction, Numer. Math. Theor. Meth. Appl., 2018, 11(3), 655–672. doi: 10.4208/nmtma.2017-OA-0078 [5] B. Deka and P. Roy, Weak galerkin finite element methods for parabolic interface problems with nonhomogeneous jump conditions, Numer. Funct. Anal. Optim., 2019, 40(3), 259–279. doi: 10.1080/01630563.2018.1549074 [6] J. S. Gupta, A posteriori error Analysis of Finite Element Method for Parabolic Interface Problems, Ph.D. thesis, Indian Institute of Technology Guwahati, Guwahati, India, 2015. [7] T. Lin, Q. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations, 2015, 31(6), 1925–1947. doi: 10.1002/num.21973 [8] L. Mu, J. Wang, X. Ye and S. Zhao, A new weak galerkin finite element method for elliptic interface problems, J. Comput. Phys., 2016, 325, 157–173. doi: 10.1016/j.jcp.2016.08.024 [9] J. Wang and X. Ye, A weak galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 2013, 241, 103–115. doi: 10.1016/j.cam.2012.10.003 [10] J. Wang and X. Ye, A weak galerkin mixed finite element method for second order elliptic problems, Math. Comp., 2014, 83(289), 2101–2126. doi: 10.1090/S0025-5718-2014-02852-4 [11] S. Zhou, F. Gao, B. Li and Z. Sun, Weak galerkin finite element method with second-order accuracy in time for parabolic problems, Appl. Math. Lett., 2019, 90, 118–123. doi: 10.1016/j.aml.2018.10.023 -
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Citation
Bhupen Deka, Papri Roy, Naresh Kumar. WEAK GALERKIN FINITE ELEMENT METHODS COMBINED WITH CRANK-NICOLSON SCHEME FOR PARABOLIC INTERFACE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1433-1442. doi: 10.11948/20190218
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