Development of a general finite difference approximation for a general domain part I: Machine transformation

The general idea of a ''numerical transform'' on a high speed computer for a general domain, abbreviated as a machine transformation, is illustrated by employing an equilateral triangle mesh plane for a general, second-order quasi-linear elliptic partial differential equation subject to a general third boundary value condition in a general domain. The feasibility of the technique for linear elliptic equations is demonstrated by two test problems, for which the numerical solutions are compared with exact analytic solutions. A new computing technique is devised for linear elliptic equations, and possible extensions to quasi-linear boundary value problems are discussed. This method eliminates the programming difficulties of a finite-difference method near the curved boundaries, and, most important, it can be preprogrammed for a general class of domains to yield numerical solutions.