University of Khartoum

A Finite Element Method for Solving Laplace’s Equation in 3- dimensions

A Finite Element Method for Solving Laplace’s Equation in 3- dimensions

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Title: A Finite Element Method for Solving Laplace’s Equation in 3- dimensions
Author: Ahmed, Seif eldin Ali
Abstract: Finite Element Methods (FEMs) are originated from the need for solving complex problems emerging as a result of modeling real life problems. FEMs are considered as numerical techniquesfor finding approximate solutions of partial differential equations as well as integral equations. In this thesis, a detailed algorithm based on FEMs is given to solve Laplaceequation in three dimensional Cartesian coordinates. The details are meant to pave the way of having close understanding to the steps used in the different aspects of the implementation. The domain of definition is divided into tetrahedron elements with fournodes. The shape functions are linear. The variational formulation of Laplaceequation is solved to find an approximate solution based on the shape functions.Four Matlab functions are used to implement the method. Reasonableresults are obtained when applying the method on test Problems. The Gauss–Seidel iterative method is used in solving the resulting linear systems. Graphs for the solutions are plotted and compared to those obtained using the classical separation of variables method. It is found that the results obtained from the two methods are almost the same, within reasonable tolerance ranging from 0.0001 to 0.001.
Description: 185 page
URI: http://khartoumspace.uofk.edu/123456789/23207


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