Finite Element Analysis

The nite element method is a computational technique for obtaining approximate solutions to the partial dierential equations that arise in scientic and engineering applications. Rather than approximating the partial dierential equation directly as with, e.g., nite dierence methods, the nite element method utilizes a variational problem that involves an integral of the dierential equation over the problem domain. This domain is divided into a number of subdomains called nite elements and the solution of the partial dierential equation is approximated by a simpler polynomial function on each element. These polynomials have to be pieced together so that the approximate solution has an appropriate degree of smoothness over the entire domain. Once this has been done, the variational integral is evaluated as a sum of contributions from each nite element. The result is an algebraic system for the approximate solution having a nite size rather than the original innite-dimensional partial dierential equation. Thus, like nite dierence methods, the nite element process has discretized the partial dierential equation but, unlike nite dierence methods, the approximate solution is known throughout the domain as a pieceise polynomial function and not just at a set of points.