DOI QR코드

DOI QR Code

A FREQUENCY-DOMAIN METE10D FOR FINITE ELEMENT SOLUTIONS OF PARABOLIC PROBLEMS

  • Published : 2002.11.01

Abstract

We introduce and analyze a frequency-domain method for parabolic partial differential equations. The method is naturally parallelizable. After taking the Fourier transformation of given equations in the space-time domain into the space-frequency domain, we propose to solve an indefinite, complex elliptic problem for each frequency. Fourier inversion will then recover the solution in the space-time domain. Existence and uniqueness as well as error estimates are given. Fourier invertibility is also examined. Numerical experiments are presented.

Keywords

parabolic problems;finite element methods;parallel algorithm;Fourier transform

References

  1. Math. Mod. Meth. Appl. Sci. v.4 Approximation of scalar waves in the space-frequency domain J. Douglas, Jr.;J. E. Santos;D. Sheen https://doi.org/10.1142/S0218202594000297
  2. Trans. Amer. Math. Soc. v.346 An elliptic regularity estimate for a problem arising from the frequency domain treatment of waves X. Feng;D. Sheen https://doi.org/10.2307/2154856
  3. Comput. Methods Appl. Mech. Engng. v.187 no.1-2 Frequency domain formulation of linearized Navier-Stokes equations C.-O. Lee;J. Lee;D. Sheen https://doi.org/10.1016/S0045-7825(99)00132-2
  4. Sobolev Spaces R. A. Adams
  5. Mathematical Analysis and Numerical Methods for Science and Technology v.2 Functional and variational methods R. Dautray;J.-L. Lions;R. Dautray(ed.);J.-L. Lions(ed.)
  6. Numerical Solutions of Partial Differential Equations by the Finite Element Method C. Johnson
  7. Quelques methodes de resolution des problems aux linites non lineaires J. L. Lions
  8. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics v.15 S. C. Brenner;L. R. Scott
  9. Elliptic Problem Solvers Ⅱ The (new) Yale sparse matrix package S. C. Eisenstat;H. E. Elman;M. H. Schultz;A. H. Sherman;G. Birkhoff(ed.);
  10. Handbook of Numerical Analysis v.2 Finite element Method Ph. Ciarlet;J. L Lions(ed.);Ph. Ciarlet(ed.)
  11. Math. Comp. v.69 A parallel method for time-discretization of parabolic problems based on counter integral representation quadrature D. Sheen;I. H. Sloan;V. Thomee https://doi.org/10.1090/S0025-5718-99-01098-4
  12. Comput. Methods Appl. Mech. Engng. v.169 no.1-2 A frequency-domain parallel method for the numerical approximation of parabolic problems C.-O. Lee;J. Lee;D. Sheen;Y. Yeom https://doi.org/10.1016/S0045-7825(98)00168-6
  13. Finite Element Methods for Navoer-Stokes Equations, Theory and Algorithms V. Girault;P.-A. Raviart
  14. The Finite Element Method for Elliptic Problems Ph. Ciarlet
  15. Les Methodes Directes en Theorie des Equations Elliptiques J. Necas
  16. Math. Mod. Meth. Appl. Sci. v.3 Frequency domain treatment of one-dimensional scalar waves J. Douglas, Jr.;J. E. Santos;D. Sheen;L. S. Bennethum https://doi.org/10.1142/S0218202593000102
  17. Boundary Value Problems in Non-Smooth Domains P. Grisvard
  18. Finite Elecments, Theory, fast solvers, and applications in solid mechanis D. Braess