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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 2/2016

23.04.2015

Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

verfasst von: Huadong Gao

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2016

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Abstract

In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal \(L^2\) error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal \(L^2\) error estimate for \(r\)-th order FEMs \((r \ge 1)\). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF–Galerkin FEMs.
Vorheriger Artikel Spectral Method for Fourth-Order Problems on Quadrilaterals
Nächster Artikel Local Error Estimates for the SUPG Method Applied to Evolutionary Convection–Reaction–Diffusion Equations

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Literatur
1.
Akrivis, G., Crouzeix, M.: Linearly implicit methods for nonlinear parabolic equations. Math. Comput. 73, 613–635 (2003)CrossRefMathSciNet
2.
Akrivis, G., Larsson, S.: Linearly implicit finite element methods for the time dependent Joule heating problem. BIT 45, 429–442 (2005)CrossRefMathSciNetMATH
3.
Allegretto, W., Xie, H.: Existence of solutions for the time dependent thermistor equation. IMA J. Appl. Math. 48, 271–281 (1992)CrossRefMathSciNetMATH
4.
Allegretto, W., Yan, N.: A posteriori error analysis for FEM of thermistor problems. Int. J. Numer. Anal. Model. 3, 413–436 (2006)MathSciNetMATH
5.
Allegretto, W., Lin, Y., Ma, S.: Existence and long time behavior of solutions to obstacle thermistor equations. Discret. Contin. Dyn. Syst. Ser. A 8, 757–780 (2002)CrossRefMathSciNetMATH
6.
Baker, G., Dougalis, V., Karakashian, O.: On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comput. 39, 339–375 (1982)CrossRefMathSciNetMATH
7.
Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)CrossRefMATH
8.
Byun, S., Wang, L.: Elliptic equations with measurable coefficients in Reifenberg domains. Adv. Math. 225, 2648–2673 (2010)CrossRefMathSciNetMATH
9.
Cimatti, G.: Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor. Ann. Mat. Pura Appl. 162, 33–42 (1992)CrossRefMathSciNetMATH
10.
Chen, Y.-Z., Wu, L.-C.: Second order elliptic equations and elliptic systems. In: Translation of Mathematical Monographs. Translated from the 1991 Chinese Original by Bei Hu, vol. 174. American Mathematical Society, Providence, RI (1998)
11.
Chen, W., Gunzburger, M., Sun, D., Wang, X.: Efficient and long time accurate second-order methods for Stokes-Darcy system. SIAM J. Numer. Anal. 51, 2563–2584 (2013)CrossRefMathSciNetMATH
12.
Elliott, C.M., Larsson, S.: A finite element model for the time-dependent joule heating problem. Math. Comput. 64, 1433–1453 (1995)CrossRefMathSciNetMATH
13.
14.
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)CrossRefMathSciNetMATH
15.
Garvie, M., Trenchea, C.: A three level finite element approximation of a pattern formation model in developmental biology. Numer. Math. 127, 397–422 (2014)CrossRefMathSciNetMATH
16.
Gear, C.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1971)MATH
17.
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)CrossRefMATH
18.
Heywood, J., 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)CrossRefMathSciNetMATH
19.
Holst, M., Larson, M., Malqvist, A., Axel, R.: Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions. BIT 50, 781–795 (2010)CrossRefMathSciNetMATH
20.
Hou, Y., Li, B., Sun, W.: Error analysis of splitting Galerkin methods for heat and sweat transport in textile materials. SIAM J. Numer. Anal. 51, 88–111 (2013)CrossRefMathSciNetMATH
21.
Lambert, J.: Numerical Methods in Ordinary Differential Systems: the Initial Value Problems. Wiley, Chichester (1991)
22.
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)CrossRefMathSciNetMATH
23.
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)MathSciNetMATH
24.
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)CrossRefMathSciNetMATH
25.
Liu, J.: Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains. SIAM J. Numer. Anal. 51, 743–772 (2013)CrossRefMathSciNetMATH
26.
Logg, A., Mardal, K., Wells, G. (eds.): Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)MATH
27.
Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa 3(20), 733–737 (1966)MathSciNet
28.
Simader, C.G.: On Dirichlet’s Boundary Value Problem: LP-Theory Based on a Generalizayion of garding’s Inequality. Lecture Notes in Mathematics, vol. 268. Springer, Berlin (1972)
29.
Sun, W., Sun, Z.: Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials. Numer. Math. 120, 153–187 (2012)CrossRefMathSciNetMATH
30.
Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)MATH
31.
Yue, X.: Numerical analysis of nonstationary thermistor problem. J. Comput. Math. 12, 213–223 (1994)MathSciNetMATH
32.
33.
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)CrossRefMathSciNetMATH
34.
Zhao, W.: Convergence analysis of finite element method for the nonstationary thermistor problem. Shandong Daxue Xuebao 29, 361–367 (1994)MATH
35.
36.
Metadaten
Titel
Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations
verfasst von
Huadong Gao
Publikationsdatum
23.04.2015
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 2/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-015-0032-5

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