Abstract
In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, San Diego (1978)
Akrivis, G., Larsson, S.: Linearly implicit finite element methods for the time-dependent Joule heating problem. BIT 45, 429–442 (2005)
Boccardo, L., Orsina, L., Poretta, A.: Existence of finite energy solutions for elliptic systems with L 1 valued nonlinearities. Math. Models Methods Appl. Sci. 18(5), 669–687 (2008)
Bradji, A., Herbin, R.: Discretization of the coupled heat and electrical diffusion problems by the finite element and the finite volume methods. IMA J. Numer. Anal. 28(3), 469–495 (2008)
Casado-Díaz, J., Chacón Rebollo, T., Girault, V., Mármol Gómez, M., Murat, F.: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1. Numer. Math. 105(3), 337–374 (2007)
Chen, L., Holst, M., Xu, J.: The finite element approximation of the nonlinear Poisson-Boltzmann equation. SIAM J. Numer. Anal. 45(6), 2298–2320 (2007)
Elliott, C.M., Larsson, S.: A finite element model for the time-dependent Joule heating problem. Math. Comput. 64, 1433–1453 (1995)
Gallouët, T., Herbin, R.: Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 7(2), 49–55 (1994)
Henneken, V.A., Tichem, M., Sarro, P.M.: In-package MEMS-based thermal actuators for micro-assembly. J. Micromech. Microeng. 16(6), 107–115 (2006)
Holst, M.J., Tsogtgerel, G., Zhu, Y.: Local convergence of adaptive methods for nonlinear partial differential equations. arXiv:1001.1382v1
Kerkhoven, T., Jerome, J.W.: L ∞ stability of finite element approximations to elliptic gradient equations. Numer. Math. 57, 561–575 (1990)
Stampacchia, G.: L probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15(1), 189–258 (1965)
Sun-Sig, B.: Elliptic equations with BMO coefficients in Lipschitz domains. Trans. Am. Math. Soc. 357(3), 1025–1046 (2004)
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Communicated by Ralf Hiptmair.
A. Målqvist work was supported by The Göran Gustafsson foundation.
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Holst, M.J., Larson, M.G., Målqvist, A. et al. Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions. Bit Numer Math 50, 781–795 (2010). https://doi.org/10.1007/s10543-010-0287-z
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DOI: https://doi.org/10.1007/s10543-010-0287-z