Introduction to Finite Element Methods

In a di erential equations course, one typically studies a few classes of problems for which there are closed form solutions, such as ordinary, linear di erential equations with constant coe cients. Most problems of interest in real world simulation problems are much more complex, however, involving domains of two or more dimensions or nonlinear e ects, yielding partial di erential equations or nonlinear di erential equations, respectively. Many phenomena in nature, such as heat conduction or convection, stress in mechanical structures, electromagnetic elds, and uid mechanics are described by partial di erential equations involving rst and second spatial derivatives and time derivatives of multidimensional functions. Examples of such equations are Maxwell's equations, the heat equation, and the Schrodinger wave equation. Other physical phenomena, such as thermal radiation, are described by integral equations or integro-di erential equations. Thermal radiation is the study of heat transfer by radiation. The character of the equations governing thermal radiation depends on whether the medium (volume) through which the radiation passes is a participating medium, which emits, scatters, or absorbs radiation, or a non-participating medium. An example of a participating medium is fog, and an example of a non-participating medium is a vacuum. Unlike conduction and convection, in which all ow of heat is determined by local phenomena (such as conductivity, pressure, or temperature di erences in an in nitesimal neighborhood), thermal radiation is determined by non-local phenomena. The radiation incident on a surface is a function of the integral of the radiation on other surfaces. For these reasons, thermal radiation in a non-participating medium is governed by an integral equation, and thermal radiation in a participating medium is governed by a integro-di erential equation. Neither di erential equations, integral equations, nor integro-di erential equations can be solved symbolically or analytically, in general. Nevertheless, we want to simulate these