Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of the ghost fluid method for elliptic equations with interfaces


Authors: Xu-Dong Liu and Thomas C. Sideris
Journal: Math. Comp. 72 (2003), 1731-1746
MSC (2000): Primary 65N12, 35J25
DOI: https://doi.org/10.1090/S0025-5718-03-01525-4
Published electronically: May 14, 2003
MathSciNet review: 1986802
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. It is shown that the application of the ghost fluid method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the ghost fluid method is proved to be convergent.


References [Enhancements On Off] (What's this?)

  • 1. Bramble, J. and King, J. A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. vol. 6, pp. 109-138, (1996). MR 98e:65094
  • 2. Ciarlet, P.G. The finite element method for elliptic problems. North-Holland, New York (1978). MR 58:25001
  • 3. Chen, Z. and Zou, J. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. vol. 79, pp. 175-202 (1998). MR 99d:65313
  • 4. Ewing, R. Li, Z., Lin, T. and Lin, Y. The immersed finite volume element methods for the elliptic interface problems. Math. Comput. Simulation vol. 50 pp. 63-76 (1999). MR 2000g:65118
  • 5. Fedkiw, R., Aslam, T., and Xu, Shaojie. The Ghost Fluid Method for Deflagration and Detonation Discontinuities. J. Comput. Phys. vol. 154, pp. 393-427 (1999). MR 2000e:76096
  • 6. Fedkiw, R., Aslam, T., Merriman, B., and Osher, S. A Non-Oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method). J. Comput. Phys. vol. 152, pp. 457-492 (1999). MR 2000c:76061
  • 7. Fedkiw, R. and Liu, X.-D. The Ghost Fluid Method for Viscous Flows, Progress in Numerical Solutions of Partial Differential Equations. Arcachon, France, edited by M. Hafez, July 1998.
  • 8. Hou, T., Li, Z., Osher, S., Zhao, H. A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow. J. Comput. Phys., vol. 134, pp. 236-252 (1997). MR 98d:76128
  • 9. Johansen, H. and Colella, P. A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains. J. Comput. Phys., vol. 147, pp. 60-85 (1998). MR 99m:65231
  • 10. LeVeque, R.J. and Li, Z. The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources. SIAM J. Numer. Anal. vol. 31, pp. 1019-1044 (1994). MR 95g:65139
  • 11. Li, Z. A Fast Iterative Algorithm for Elliptic Interface Problems. SIAM J. Numer. Anal., vol. 35, no. 1, pp. 230-254, (1998). MR 99b:65126
  • 12. Li, Z., Lin, T., and Wu, X. New Cartesian grid methods for interface problems using finite element formulation. Preprint.
  • 13. Liu, X.-D., Fedkiw, R., and Kang, M. A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains. J. Comput. Phys. vol. 160, pp. 151-178 (2000). MR 2001a:65152
  • 14. Mulder, W., Osher, S., and Sethian, J.A. Computing Interface Motion in Compressible Gas Dynamics. J. Comput. Phys. vol. 100, pp. 209-228 (1992). MR 93a:76077
  • 15. Peskin, C. Numerical Analysis of Blood Flow in the Heart. J. Comput. Phys., vol. 25, pp. 220-252 (1977). MR 58:9389
  • 16. Peskin, C. and Printz, B. Improved Volume Conservation in the Computation of Flows with Immersed Elastic Boundaries., J. Comput. Phys., vol. 105, pp. 33-46 (1993). MR 93k:76081

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 35J25

Retrieve articles in all journals with MSC (2000): 65N12, 35J25


Additional Information

Xu-Dong Liu
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: xliu@math.ucsb.edu

Thomas C. Sideris
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: sideris@math.ucsb.edu

DOI: https://doi.org/10.1090/S0025-5718-03-01525-4
Received by editor(s): August 21, 2001
Received by editor(s) in revised form: May 3, 2002
Published electronically: May 14, 2003
Additional Notes: Research partially supported by the National Science Foundation: DMS-9805546 (X.-D.L.) and DMS-9800888 (T.C.S.)
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society