Support-Operator Finite-Difference Algorithms for General Elliptic Problems

Abstract

An algorithm is developed for discretizing boundary-value problems given by a general linear elliptic second order partial-differential equation with general mixed or Robin boundary conditions in general logically rectangular grids. The continuum problem can be written as an operator equation where the operator is self adjoint and positive definite. The discrete approximations have the same property. Consequently, the matrices for the discrete problem are symmetric and positive definite. Also, the scheme has a nearest neighbor stencil. Consequently, the most powerful linear solvers can be applied. In smooth grids, the algorithm produces second-order accurate solutions. It is the generality of the problem (general matrix coefficients, general boundary conditions, general logically rectangular grids) that makes finding such an algorithm difficult. The algorithm, which is a combination of the method of support operators and the mapping method, overcomes certain difficulties of the individual methods, producing a high-quality algorithm for solving general elliptic problems.