Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations. (English) Zbl 1305.65200
The Joule heating equations consist of a nonlinear parabolic equation and a coupled homogeneous elliptic one. Following B. Li and W. Sun [Int. J. Numer. Anal. Model. 10, No. 3, 622–633 (2013; Zbl 1281.65122)], the author considers a semi-decoupled scheme using Euler backward differences in time and Galerkin finite elements in space and mentions also a fully decoupled approach for the numerical solution of the system along with Dirichlet boundary conditions in \(\mathbb R^2\) and \(\mathbb R^3\). For the first method, and employing a boundedness result of Y.-Z. Chen and L.-C. Wu [Second order elliptic equations and elliptic systems. Transl. from the Chinese by Bei Hu. Providence, RI: American Mathematical Society (1998; Zbl 0902.35003)], for the solution of the Dirichlet problem of the Poisson problem in \(\mathbb R^3\), he shows unconditional stability (by comparing with the solution of a corresponding time-discrete system – the Rothe method) and gets optimal error estimates in \(L_2\) and \(H^1\). Numerically, he illustrates his theoretical results presenting numerical experiments for the unit square and unit circle, and for the unit ball. His numerical results also exhibit the unconditional stability of the fully decoupled scheme.
Reviewer: Gisbert Stoyan (Budapest)
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 80M10 | Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlinear parabolic system; linearized semi-implicit scheme; Galerkin method; unconditional convergence; optimal error estimate; Joule heating equations; backward differences; finite elements; Poisson problem; Rothe method; numerical experimentSoftware:
FEniCSReferences:
| [1] | Achdou, Y., Guermond, J.L.: Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 37, 799-826 (2000) · Zbl 0966.76041 · doi:10.1137/S0036142996313580 |
| [2] | Akrivis, G., Larsson, S.: Linearly implicit finite element methods for the time dependent Joule heating problem. BIT 45, 429-442 (2005) · Zbl 1113.65091 · doi:10.1007/s10543-005-0008-1 |
| [3] | Allegretto, W., Xie, H.: Existence of solutions for the time dependent thermistor equation. IMA. J. Appl. Math. 48, 271-281 (1992) · Zbl 0754.35170 · doi:10.1093/imamat/48.3.271 |
| [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] | Allegretto, W., Lin, Y., Ma, S.: Existence and long time behavior of solutions to obstacle thermistor equations. Discrete Contin. Dyn. Syst. Ser. A 8, 757-780 (2002) · Zbl 1062.35187 · doi:10.3934/dcds.2002.8.757 |
| [6] | Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002) · Zbl 1012.65115 · doi:10.1007/978-1-4757-3658-8 |
| [7] | 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) · Zbl 0769.35059 · doi:10.1007/BF01759998 |
| [8] | Chen, Y.Z., Wu, L.C.: Second order elliptic equations and elliptic systems, Translations of Mathematical Monographs 174, AMS, USA (1998) · Zbl 0902.35003 |
| [9] | Chen, Z., Hoffmann, K.-H.: Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity. Adv. Math. Sci. Appl. 5, 363-389 (1995) · Zbl 0846.65051 |
| [10] | Deng, Z., Ma, H.: Optimal error estimates of the Fourier spectral method for a class of nonlocal, nonlinear dispersive wave equations. Appl. Numer. Math. 59, 988-1010 (2009) · Zbl 1167.65051 · doi:10.1016/j.apnum.2008.03.042 |
| [11] | Douglas Jr, J., Ewing, R., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numer. 17, 249-265 (1983) · Zbl 0526.76094 |
| [12] | Elliott, C.M., Larsson, S.: A finite element model for the time-dependent joule heating problem. Math. Comput. 64, 1433-1453 (1995) · Zbl 0846.65047 · doi:10.1090/S0025-5718-1995-1308451-4 |
| [13] | Ewing, R.E., Wheeler, M.F.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17, 351-365 (1980) · Zbl 0458.76092 · doi:10.1137/0717029 |
| [14] | He, Y.: The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77, 2097-2124 (2008) · Zbl 1198.65222 · doi:10.1090/S0025-5718-08-02127-3 |
| [15] | 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) · Zbl 0694.76014 · doi:10.1137/0727022 |
| [16] | Holst, M.J., Larson, M.G., Malqvist, A., Axel, R.: Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions. BIT 50, 781-795 (2010) · Zbl 1205.80084 · doi:10.1007/s10543-010-0287-z |
| [17] | 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) · Zbl 1439.74421 · doi:10.1137/110854813 |
| [18] | Kellogg, B., Liu, B.: The analysis of a finite element method for the Navier-Stokes equations with compressibility. Numer. Math. 87, 153-170 (2000) · Zbl 0961.76043 · doi:10.1007/s002110000172 |
| [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) · Zbl 1281.65122 |
| [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) · Zbl 1311.76067 |
| [21] | Li, B.: Mathematical modeling, analysis and computation for some complex and nonlinear flow problems. PhD Thesis, City University of Hong Kong, Hong Kong (2012) · Zbl 1439.74421 |
| [22] | Logg, A., Mardal, K.-A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012). doi:10.1007/978-3-642-23099-8 · Zbl 1247.65105 · doi:10.1007/978-3-642-23099-8 |
| [23] | Ma, H., Sun, W.: Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. Anal. 39, 1380-1394 (2001) · Zbl 1008.65070 · doi:10.1137/S0036142900378327 |
| [24] | Mu, M., Huang, Y.: An alternating Crank-Nicolson method for decoupling the Ginzburg-Landau equations. SIAM J. Numer. Anal. 35, 1740-1761 (1998) · Zbl 0914.65129 · doi:10.1137/S0036142996303092 |
| [25] | Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733-737 (1966) · Zbl 0163.29905 |
| [26] | 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) · Zbl 1332.76034 · doi:10.1007/s00211-011-0402-3 |
| [27] | Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006) · Zbl 1105.65102 |
| [28] | Tourigny, Y.: Optimal \[H^1\] H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal. 11, 509-523 (1991) · Zbl 0737.65095 · doi:10.1093/imanum/11.4.509 |
| [29] | Wang, K., Wang, H.: An optimal-order error estimate to ELLAM schemes for transient advection-diffusion equations on unstructured meshes. SIAM J. Numer. Anal. 48, 681-707 (2010) · Zbl 1226.65084 · doi:10.1137/080723946 |
| [30] | Wu, H., Ma, H., Li, H.: Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation. SIAM J. Numer. Anal. 41, 659-672 (2003) · Zbl 1050.65083 · doi:10.1137/S0036142901399781 |
| [31] | Yue, X.: Numerical analysis of nonstationary thermistor problem. J. Comput. Math. 12, 213-223 (1994) · Zbl 0805.65129 |
| [32] | Yuan, G.: Regularity of solutions of the thermistor problem. Appl. Anal. 53, 149-155 (1994) · Zbl 0843.35046 · doi:10.1080/00036819408840253 |
| [33] | Yuan, G., Liu, Z.: Existence and uniqueness of the \[C^\alpha\] Cα solution for the thermistor problem with mixed boundary value. SIAM J. Math. Anal. 25, 1157-1166 (1994) · Zbl 0808.35064 · doi:10.1137/S0036141092237893 |
| [34] | Zhao, W.: Convergence analysis of finite element method for the nonstationary thermistor problem. Shandong Daxue Xuebao 29, 361-367 (1994) · Zbl 0834.65096 |
| [35] | Zhou, S., Westbrook, D.R.: Numerical solutions of the thermistor equations. J. Comput. Appl. Math. 79, 101-118 (1997) · Zbl 0885.65147 · doi:10.1016/S0377-0427(96)00166-5 |
| [36] | Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10, 229-240 (1973) · Zbl 0285.65067 · doi:10.1137/0710022 |
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