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Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem

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Abstract

In this paper, a linearized Galerkin fully-discrete scheme is proposed and investigated for the time-dependent nonlinear thermistor problem, where the temporal direction and the spatial direction are approximated by the semi-implicit backward Euler scheme and the standard bilinear finite element method, respectively. The unconditionally superclose and superconvergent error estimates are derived without any time step-size restrictions based on two crucial issues. One is that a time-discrete (elliptic) system is introduced to split the error function as the temporal error function plus the spatial error function. The other is that the high accuracy estimation of the bilinear element and the superclose error estimate between the numerical solution and the Ritz projection of the solution of the time-discrete system are employed to deal with the coupled nonlinear term skillfully, which is a key role to bound the numerical solution in \(L^{\infty }\)-norm. Finally, numerical results are provided to confirm the theoretical analysis and show the unconditional stability of the linearized scheme.

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References

  1. Allegretto, W., Xie, H.: Existence of solutions for the time-dependent thermistor equations. IMA J. Appl. Math. 48, 271–281 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allegretto, W., Lin, Y., Ma, S.: Existence and long time behavior of solutions to obstacle thermistor equations. Discrete Contin. Dyn. Syst. 8, 757–780 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cimatti, G.: Existence of weak solutions for the nonstationary problem fo the Joule heating of a conductor. Ann. Mat. Pura Appl. 162, 33–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yuan, G.: Regularity of solutions of the thermistor problem. Appl. Anal. 53, 149–156 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yuan, G., Liu, Z.: Existence and uniqueness of the \(C^{\alpha }\) solution for the thermistor problem with mixed boundary value. SIAM J. Math. Anal. 25, 1157–1166 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wu, X., Xu, X.: Existence for the thermoelastic thermistor problem. J. Math. Anal. Appl. 319, 124–138 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Elliott, C.M., Larsson, S.: A finite element model for the time-dependent Joule heating problem. Math. Comp. 64, 1433–1453 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhao, W.: Finite difference method and its convergence error analyes for thermistor problem. Apll. Math. J. Chinese Univ. 14, 349–358 (1999)

    Article  Google Scholar 

  9. Zhao, W.: Convergence analysis of finite element method for the nonstationary thermistor problem. Shandong Daxue Xuebao. 29, 361–367 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Zhou, S., Westbrook, D.R.: Numerical solutions of the thermistor equations. J. Comput. Appl. Math. 79, 101–118 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yue, X.: Numerical analysis of nonstationary thermistor problem. J. Comput. Math. 12, 213–223 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Arkivis, G., Larsson, S.: Linearly implicit finite element methods for the time-dependent Joule heating problem. BIT Numer. Math. 45, 429–442 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shi, D.Y., Yang, H.J.: Superconvergent estimates of conforming finite element method for nonlinear time-dependent Joule heating equations. Numerical Methods Partial Differential Equations. 34(1), 336–356 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shi, D.Y., Yang, H.J.: Superconvergence analysis of nonconforming FEM for nonlinear time-dependent thermistor problem. Appl. Math. Comput. 347, 210–224 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Gao, H.: Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations. J. Sci. Comput. 58, 627–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, B., Gao, H., Sun, W.: Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933–954 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gao, H.: Unconditional optimal error estimates of BDF-Galerkin FEMs for nonlinear Thermistor equations. J. Sci. Comput. 66, 504–527 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gao, H., Li, B., Sun, W.: Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity. SIAM J. Numer. Anal. 52, 1183–1202 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gao, H.: Optimal error estimates of a linearized backward Euler Galerkin FEM for the Landau-Lifshitz equation. SIAM J. Numer. Anal. 52, 2574–2593 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. An, R.: Optimal error estimates of linearized Crank-Nicolson Galerkin method for Landau-Lifshitz equation. J. Sci. Comput. 69, 1–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gao, H., Li, B., Sun, W.: Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon. Numer. Math. 136, 383–409 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Si, Z., Wang, J., Sun, W.: Unconditional stability and error estimates of modified characteristics FEMs for the Navier-Stokes equations. Numer. Math. 134, 139–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, D., Wang, J.: Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system. J. Sci. Comput. 72, 892–915 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, W., Wang, J.: Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D. J. Comput. Appl. Math. 317, 685–699 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, J.: A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci Comput. 60, 390–407 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shi, D., Yang, H.: Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation. Appl. Math. Comput. 310, 40–47 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Shi, D., Wang, J., Yan, F.: Unconditional superconvergence analysis for nonlinear parabolic equation with \(EQ_1^{rot}\) nonconforming finite element. J. Sci. Comput. 70, 85–111 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shi, D., Wang, J.: Unconditional superconvergence analysis of a Crank-Nicolson Galerkin FEM for nonlinear Schrödinger equation. J. Sci. Comput. 72, 1093–1118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shi, D., Yan, F., Wang, J.: Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation. Appl. Math. Comput. 274, 182–194 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Shi, D., Wang, J.: Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation. Appl. Math. Comput. 294, 216–226 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Cai, W.T., Wang, J.L., Wang, K.: Convergence analysis of Crank-Nicolson Galerkin-Galerkin FEMs for miscible displacement in porous media. J. Sci. Comput. 83, Article number: 25(2020)

  35. Wang, J.: Unconditional stability and convergence of Crank-Nicolson Galerkin FEMs for a nonlinear Schrödinger-Helmholtz system. Numer. Math. 139, 479–503 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, W.W., Wu, C.D.: New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media. Math. Comp. 90, 81–102 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, Y., Guan, Z., Nie, Y.F.: Unconditionally optimal error estimates of a linearized weak Galerkin finite element method for semilinear parabolic equations. Adv. Comput. Math. 48, 47 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  38. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Acamemic Press, (2003)

  39. Thomee, V.: Galerkin Finite element methods for parabolic problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  40. Ciarlet, P.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  41. Brenner, S., Scoot, L.: The mathematical theory of finite element methods. Springer, New York (2002)

    Book  Google Scholar 

  42. Yan, N.: Superconvergence analysis and a posteriori error estimation in finite element methods, Science Press (2008)

  43. Shi, D.Y., Wang, P.L., Zhao, Y.M.: Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation. Appl. Math. Lett. 38, 129–134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, Y.Z., Wu, L.C.: Second order elliptic equations and elliptic systems, Transl. Math. Monogr. 174, AMS, Providence, RI (1998)

  46. Lin, Q., Lin, J.F.: Finite element methods: accuracy and improvement. Science Press, Beijing (2006)

    Google Scholar 

  47. Chen, Z.X.: Finite element methods and their applications. Springer, Berlin (2005)

    MATH  Google Scholar 

Download references

Funding

This work is supported by National Natural Science Foundation of China (Nos. 12101568, 12071443).

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Authors and Affiliations

  1. School of Mathematics, Zhengzhou University of Aeronautics, 450046, Zhengzhou, People’s Republic of China

    Huaijun Yang

  2. School of Mathematics and Information Sciences, Yantai University, 264005, Yantai, People’s Republic of China

    Dongyang Shi

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  1. Huaijun Yang
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Correspondence to Dongyang Shi.

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Yang, H., Shi, D. Unconditionally superconvergent error estimates of a linearized Galerkin finite element method for the nonlinear thermistor problem. Adv Comput Math 49, 33 (2023). https://doi.org/10.1007/s10444-023-10038-y

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