Numerical Solution of a Second Order Boundary Problem for Two Dimensions UsingGalerkin Approximations

摘要

This report outlines the development and use of the program LAPLACE. LAPLACE iscapable of solving a second order two dimensional boundary value problem, employing graphics to assist in mesh generation and solution presentation. Galerkin approximation methods, along with the development of a finite element mesh, permit the program to calculate nodal results over the domain of the problem. The use of these nodal solutions with additional subroutines allows for the computation of equipotential lines and lines perpendicular to the equipotential lines. The current format of this program solves Laplace's Equation. Nodal solutions to Laplace's Equation are calculated over the domain of the problem and used as the basis for the generation of equipotential lines and their perpendiculars. Equipotential lines are interpreted as contours and their perpendiculars represent flow lines for the solution to Laplace's Equation. These lines are used in combination to develop a flow net over the domain of the problem and this flow net is graphically displayed. This program was written with the capability of solving several types of second order two dimensional boundary value problems. The calculation of solutions to other second order two dimensional boundary value problems is accomplished by entering the appropriate functional coefficients of the differential equation into one subroutine.