We develop and apply a finite-difference method to discretize the Laplacian operator with Neumann boundary conditions on an irregular domain using a regular Cartesian grid. The method is an extension of the immersed interface method developed by LeVeque and Li [SIAM J. Numer. Anal., 31 (1994) 1019--1044]. With careful selection of stencils, the method is second order accurate and produces a matrix that is stable (diagonally semidominant). The method is illustrated on several two-dimensional problems and one three-dimensional problem.
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Fogelson et al. (2001) studied this question.
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