Electrostatic and magnetostatic finite-difference analysis without the 'staircase' effect

In finite difference methods, the notorious 'staircase' effect arises when a slanted or curved boundary is rendered on a cartesian grid. The new flexible local approximation methods (FLAME) approximate material interfaces algebraically, by a suitable set of basis functions, rather than by geometrically conforming meshes. For example, the approximating functions in the vicinity of spherical particles are chosen as spherical harmonics. FLAME seamlessly incorporates these local analytical approximations into the difference scheme. Even though FLAME typically operates on cartesian grids, its solution for problems with dielectric or magnetic particles can be orders of magnitude more accurate than the finite-element solution on complex meshes.