Journal of Computational Physics
Journal of Computational Physics
Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces
Journal of Computational Physics
Journal of Computational Physics
Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities
Journal of Computational Physics
A kernel-free boundary integral method for elliptic boundary value problems
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
MIB method for elliptic equations with multi-material interfaces
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
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We have developed a first-order stable Cartesian grid discretization that uses only interior grid points for inhomogeneous anisotropic elliptic operators subject to Neumann boundary conditions on a bounded nonrectangular geometry in three dimensions. For this discretization method, a necessary and sufficient condition depending on the mesh size h for the existence of this first-order stable scheme at a regular (i.e., interior) grid point is found in terms of the anisotropy matrix. For this discretization method, a way to analyze the existence of a first-order stable scheme at an irregular (i.e., boundary) grid point is also given. The arguments are identical to those for the two-dimensional case [M. A. Dumett and J. P. Keener, A Numerical Method for Solving Anisotropic Elliptic Boundary Value Problems on an Irregular Domain in 2D, manuscript]; only the details change. Unlike in [M. A. Dumett and J. P. Keener, A Numerical Method for Solving Anisotropic Elliptic Boundary Value Problems on an Irregular Domain in 2D, manuscript], a discussion of Dirichlet and Robin boundary conditions is also included. In particular, it is shown that the Gerschgorin condition does not impose sign restrictions on irregular grid points stencil coefficients as in the Neumann case.