Abstract
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.
Highlights
Laplace EquationThe partial differential equation under analysis is the Laplace Equation. The Laplace Equation is given by: u=0
There were three parameters taken into consideration; the number of nodes, the types of domains and the boundary conditions imposed on the system
We determined that when the domain had an increased number of meshed triangular elements, the accuracy of the finite element approximate solution increased
Summary
The partial differential equation under analysis is the Laplace Equation. The Laplace Equation is given by: u=0. U(x,y) represents a potential function whose gradient is the velocity vector for a 2 dimensional fluid flow [4]. Since this equation is related to the equilibrium of systems, various boundary conditions determine different applications of the system. In terms of fluid mechanics, where u is the velocity potential, the Neumann boundary conditions can represent solid surfaces where there is no fluid flow across these walls and the Dirichlet conditions may represent openings in the domain where there is a constant horizontal velocity [4][5].
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