Finite Element and Finite Difference Methods for Elliptic and Parabolic Differential Equations

Abstract

With the availability of powerful computers, the application of numerical methods to solve scientific and engineering problems is becoming the normal practice in engineering and scientific communities. Well-formed scientific theory with numerical methods may be used to study scientific and engineering problems. The numerical methods flourish where an experimental work is limited, but it may be imprudent to view a numerical method as a substitute for experimental work. The growth in computer technology has made it possible to consider the application of partial deferential equations in science and engineering on a larger scale than ever. When experimental work is cost prohibitive, well-formed theory with numerical methods may be used to obtain very valuable information. In engineering, experimental and numerical solutions are viewed as complimentary to one another in solving problems. It is common to use the experimental work to verify the numerical method and then extend the numerical method to solve new design and system. The fast growing computational capacity also make it practical to use numerical methods to solve problems even for nontechnical people. It is a common encounter that finite difference (FD) or finite element (FE) numerical methods-based applications are used to solve or simulate complex scientific and engineering problems. Furthermore, advances in mathematical models, methods, and computational capacity have made it possible to solve problems not only in science and engineering but also in social science, medicine, and economics. Finite elements and finite difference methods are the most frequently applied numerical approximations, although several numerical methods are available. Finite element method (FEM) utilizes discrete elements to obtain the approximate solution of the governing differential equation. The final FEM system equation is constructed from the discrete element equations. However, the finite difference method (FDM) uses direct discrete points system interpretation to define the equation and uses the combination of all the points to produce the system equation. Both systems generate large linear and/or nonlinear system equations that can be solved by the computer. Finite element and finite difference methods are widely used in numerical procedures to solve differential equations in science and engineering. They are also the basis for countless engineering computing and computational software. As the boundaries of numerical method applications expand to non-traditional fields, there is a greater need for basic understanding of numerical simulation.