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BIT Numerical Mathematics

, Volume 45, Issue 3, pp 429–442 | Cite as

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

  • G. Akrivis
  • S. Larsson
Article

Abstract

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions.

Key words

error estimate finite element multistep semi-implicit 

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References

  1. 1.
    G. Akrivis and M. Crouzeix, Linearly implicit methods for nonlinear parabolic equations, Math. Comp., 73 (2004), pp. 613–635.Google Scholar
  2. 2.
    G. Akrivis, M. Crouzeix, and Ch. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp., 68 (1998), pp. 457–477.Google Scholar
  3. 3.
    G. Akrivis, M. Crouzeix, and Ch. Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations, Numer. Math., 82 (1999), pp. 521–541.Google Scholar
  4. 4.
    G. Cimatti, Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor, Ann. Mat. Pura Appl., (4) 162 (1992), pp. 33–42.Google Scholar
  5. 5.
    J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numér., 17 (1983), pp. 17–33.Google Scholar
  6. 6.
    J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numér., 17 (1983), pp. 249–265.Google Scholar
  7. 7.
    C. M. Elliott and S. Larsson, A finite element model for the time-dependent Joule heating problem, Math. Comp., 64 (1995), pp. 1433–1453.Google Scholar
  8. 8.
    A. Sunmonu, Galerkin method for a nonlinear parabolic-elliptic system with nonlinear mixed boundary conditions, Numer. Methods Partial Differ. Equations, 9 (1993), pp. 235–259.Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of IoanninaIoanninaGreece
  2. 2.Department of MathematicsChalmers University of Technology and Göteborg UniversityGöteborgSweden

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