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**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.

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)

### MSC:

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 |

### Keywords:

Poisson’s equations; interface problems; capturing method; irregular domains; discontinuous coefficients; discontinuous solution
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\textit{X.-D. Liu} et al., J. Comput. Phys. 160, No. 1, 151--178 (2000; Zbl 0958.65105)

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