Novel superconvergence analysis of a low order FEM for nonlinear time-fractional Joule heating problem. (English) Zbl 07827663
Summary: The aim of this paper is to develop and investigate a fully-discrete scheme with conforming \(P_1\) element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical \(L_1\) method. First, a novel superclose estimate in the \(H^1\)-norm is derived rigorously with some new analysis techniques under low regularity of the solutions \(u^n\), \(\phi^n\in L^\infty(0,T;H^3(\Omega))\) rather than \(u^n\in L^\infty(0,T;H^4(\Omega))\) and \(\phi^n\in L^\infty(0,T;H^3(\Omega)\cap W^{2,\infty}(\Omega))\) required in the previous studies. Then, the global superconvergence result is deduced by interpolated post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis. It should be mentioned that the analysis and results presented herein are also valid to some other known conforming and nonconforming finite elements.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35Kxx | Parabolic equations and parabolic systems |
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
Keywords:
time-fractional Joule heating problem; conforming \(P_1\) element; backward Euler fully-discrete scheme; new superclose and superconvergence analysisReferences:
| [1] | Allegretto, W.; Xie, H., Existence of solutions for the time-dependent thermisteor equations, IMA J Appl Math, 48, 3, 271-281, 1992 · Zbl 0754.35170 |
| [2] | Miller, K. S.; Rose, B., An introduction to fractional calculus and fractional differential equations, 1993, John Wiley: John Wiley New York · Zbl 0789.26002 |
| [3] | Akrivis, G.; Larsson, S., Linearly implicit finite element methods for the time-dependent Joule heating problem, BIT, 45, 429-442, 2005 · Zbl 1113.65091 |
| [4] | Allegretto, W.; Yan, N., A posteriori error analysis for FEM of Thermistor problems, Int J Numer Anal Model, 3, 413-436, 2006 · Zbl 1125.49025 |
| [5] | Gao, H. D., Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations, J Sci Comput, 58, 627-647, 2014 · Zbl 1305.65200 |
| [6] | Gao, H. D., Unconditional optimal error estimates of BDF-Galerkin FEMs for nonlinear Thermistor equations, J Sci Comput, 66, 504-527, 2016 · Zbl 1364.65180 |
| [7] | Li, B. Y.; Gao, H. D.; Sun, W. W., Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear Thermistor equations, SIAM J Numer Anal, 52, 933-954, 2014 · Zbl 1298.65160 |
| [8] | Yuan, G.; Liu, Z., Existence and uniqueness of the \(C^\alpha\) solution for the Thermistor problem with mixed boundry value, SIAM J Math Anal, 25, 4, 1157-1166, 1994 · Zbl 0808.35064 |
| [9] | Yue, X. Y., Numerical analysis of nonstationary Thermistor problem, J Comput Math, 12, 213-223, 1994 · Zbl 0805.65129 |
| [10] | Shi, D. Y.; Yang, H. J., Superconvergent estimates of conforming finite element method for nonlinear time-dependent Joule heating equations, Numer Meth Partial Differential Equ, 34, 1, 336-356, 2018 · Zbl 1390.65056 |
| [11] | Shi, D. Y.; Yang, H. J., Superconvergence analysis of nonconforming FEM for nolinear time-dependent Thermistor problem, Appl Math Comput, 347, 15, 210-224, 2019 · Zbl 1429.65231 |
| [12] | Shi, D. Y.; Yang, H. J., Superconvergence analysis of finite element method for time-fractional Thermistor problem, Appl Math Comput, 323, 31-42, 2018 · Zbl 1427.65259 |
| [13] | Shi, X. Y.; Lu, L. Z.; Wang, H. J., New superconvergence estimates of FEM for time-dependent Joule heating problem, Comput Math Appl, 111, 91-97, 2022 · Zbl 1524.65597 |
| [14] | Lin, Q.; Yan, N. N., Construction and analysis of high efficient finite elements, 1996, Hebei University Press: Hebei University Press Baoding, (in Chinese) |
| [15] | Sun, Z. Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl Numer Math, 56, 193-209, 2006 · Zbl 1094.65083 |
| [16] | Lin, Y.; Li, X.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J Comput Phys, 225, 2, 1533-1552, 2007 · Zbl 1126.65121 |
| [17] | Lin, Y.; Li, X.; Xu, C., Finite difference/spectral approximations for the fractional cable equation, Math Comp, 80, 1369-1396, 2011 · Zbl 1220.78107 |
| [18] | Thomee, V., Galerkin finite element methods for parabolic problems, 2006, Springer: Springer Heidelberg · Zbl 1105.65102 |
| [19] | Rannacher, R.; Turek, S., Simple nonconforming quadrilateral Stokes element, Numer Meth Partal Differential Equ, 8, 2, 97-111, 1992 · Zbl 0742.76051 |
| [20] | Zhang, S. H.; Shi, X. Y.; Shi, D. Y., Unconditional superconvergence analysis of nonconforming \(E Q_1^{r o t}\) finite element method for the nonlinear coupled predator-prey equations, Appl Numer Math, 185, 571-591, 2023 · Zbl 07699025 |
| [21] | Shi, D. Y.; Wang, J. J.; Yan, F. N., Unconditional superconvergence analysis for nonlinear parabolic equation with \(E Q_1^{r o t}\) nonconforming finite element, J Sci Comput, 70, 1, 85-111, 2017 · Zbl 1368.65162 |
| [22] | Shi, D. Y.; Zhang, Y. D., Approximation of nonconforming quasi-Wilson element for sine-Gordeon equations, J Comput Math, 31, 3, 271-282, 2013 · Zbl 1289.65228 |
| [23] | Shi, D. Y.; Ma, H., Unconditional superconvergence analysis of a modified nonconforming energy stable BDF2 FEM for Sobolev equaiton with Burgers’ type nonlearity, Comm Nonlinear Sci Numer Simul, 126, Article 107440 pp., 2023 · Zbl 07758892 |
| [24] | Shi, D. Y.; Pei, L. F., Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations, Appl Math Comput, 219, 9447-9460, 2013 · Zbl 1288.65145 |
| [25] | Zhang, S. H.; Shi, X. Y.; Shi, D. Y., Nonconforming modified Quasi-Wilson finite element method for convection-diffusion-reaction equation, Comm Nonlinear Sci Numer Simul, 125, Article 107333 pp., 2023 · Zbl 1520.65069 |
| [26] | Bi, C. J.; Ginting, V., Global superconvergence and a posteriori error estimates of the finite element method for second-order quasilinear elliptic problems, J Comput Appl Math, 260, 78-90, 2014 · Zbl 1293.65147 |
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