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

Published electronically:
May 14, 2003

MathSciNet review:
1986802

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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.

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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