Finite Difference Methods for Boundary Value Problems: Using Interface Problems
Finite difference methods are awkward for solving boundary value problems, such as the Dirichlet problem, with general boundaries, but they are well suited for interface problems, which have prescribed jumps in an unknown across a general interface or boundary. The two problems can be connected through potential theory: The Dirichlet boundary value problem is converted to an integral equation on the boundary, and the integrals can be thought of as solutions to interface problems. Wenjun Ying et al. have developed a practical method for solving the Dirichlet problem, and more general ones, by solving interface problems with finite difference methods and iterating to mimic the solution of the integral equation. We will describe some analysis which proves that a simplified version of Ying's method works. A recent view of classical potential theory leads to a finite difference version of the theory in which, remarkably, the crude discrete operators have much of the structure of the exact operators. This simplified method produces the Shortley-Weller solution of the Dirichlet problem. Details can be found at arxiv.org/abs/1803.08532.