Abstract:
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.
Shown here: Solution of the Poisson equation with a source and a sink subject to no-flux Neumann boundary conditions, on a domain interior to a sphere and exterior to an ellipsoid, solved on a 65 x 65 x 65 grid using the immersed interface method.