A boundary condition capturing method for Poisson’s equation on irregular domains. (English) Zbl 0958.65105

Interface problems have a variety of boundary conditions (or jump conditions) that needs to be enforced. The ghost fluid method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations and has been extended to treat more general discontinuities such as shocks, detonations, and deflagrations and compressible viscous flows.
In this paper, a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient Poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous. This new method is robust and easy to implement even in three spatial dimensions. Futhermore, the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient Poisson equation in the absence of interfaces allowing for straightforward application of standard ”black box” solvers.
Reviewer: L.G.Vulkov (Russe)


65N06 Finite difference methods for boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[2] Fedkiw, R.; Aslam, T.; Xu, S., The ghost fluid method for deflagration and detonation discontinuities, J. comput. phys., 154, 393, (1999) · Zbl 0955.76071
[3] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457, (1999) · Zbl 0957.76052
[4] R. Fedkiw, and, X.-D. Liu, The ghost fluid method for viscous flows, in, Progress in Numerical Solutions of Partial Differential Equations. Arcachon, France, 1998, M. Hafez, Ed.
[5] Greenbaum., A.; Mayo, A., Rapid parallel evaluation of integrals in potential theory on general three-dimensional regions, J. comput. phys., 145, 731, (1998) · Zbl 0911.65016
[6] 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., 134, 236, (1997) · Zbl 0888.76067
[7] Johansen, H.; Colella, P., A cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. comput. phys., 147, 60, (1998) · Zbl 0923.65079
[8] M. Kang, R. Fedkiw, and, X.-D. Liu, A boundary condition capturing method for multiphase incompressible flow, submitted for publication. · Zbl 1049.76046
[9] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31, (1994) · Zbl 0811.76044
[10] LeVeque, R.J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019, (1994) · Zbl 0811.65083
[11] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. numer. anal., 35, 230, (1998) · Zbl 0915.65121
[12] Li, Z., A note on immersed interface method for three dimensional elliptic equations, Comput. math. appl., 35, 9, (1996) · Zbl 0876.65074
[13] X.-D. Liu, and, T. Sideris, in preparation.
[14] Mayo, A., Fast high order accurate solutions of Laplace’s equation on irregular domains, SIAM J. sci. stat. comput., 6, 144, (1985) · Zbl 0559.65082
[15] Mayo, A., The fast solution of Poisson’s and the biharmonic equations in irregular domains, SIAM J. numer. anal., 21, 285, (1984) · Zbl 1131.65303
[16] Mayo, A., The rapid evaluation of volume integrals of potential theory on general regions, J. comput. phys., 100, 236, (1992) · Zbl 0772.65012
[17] Mayo, A.; Greenbaum, A., Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. sci. stat. comput., 13, 101, (1992) · Zbl 0752.65080
[18] McKenney, A.; Greengard, L.; Mayo, A., A fast Poisson solver for complex geometries, J. comput. phys., 118, 348, (1995) · Zbl 0823.65115
[19] Mulder, W.; Osher, S.; Sethian, J.A., Computing interface motion in compressible gas dynamics, J. comput. phys., 100, 209, (1992) · Zbl 0758.76044
[20] Peskin, C., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220, (1977) · Zbl 0403.76100
[21] Peskin, C.; Printz, B., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 33, (1993) · Zbl 0762.92011
[22] Osher, S.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[23] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[24] Unverdi, S.O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys., 100, 25, (1992) · Zbl 0758.76047
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