The comparative analysis of the two dimensional Laplace equation using the Galerkin finite element method with the exact solution for various domains (circular and rectangular) with triangular elemental meshing

摘要

Laplace Equation is used in many areas of studies such as potential theory and the fundamental forces of nature, Newtonian theory of gravity and electrostatics. It is used in Probability theory and Markov Chains as well as potential flows in fluid mechanics. Laplace Equation is used in various research areas and for this reason, to determine an accurate solution to this equation is of importance. In this study, the Finite Element Method is used to approximate the solution of the 2D Laplace Equation for two regions, circular and rectangular domains. These are compared to the exact solutions of the systems subjected to various physical restrictions; boundary conditions. This was done by using a mesh generator in Matlab, Distmesh, to obtain a mesh of triangular elements and then using Matlab to plot the exact and the approximated solutions as well as to determine the errors; norm. For these domains, the number of elements in the mesh was incremented and it was noted there was a convergence of the approximated to the actual solution. The boundary conditions were altered to observe the changes in the regions' field variable distribution (intensity values) of the Matlab plots.