5 - The Finite Element Method

Abstract

This chapter discusses the applications of finite element method. The finite element method is generally considered to be a competitor of the finite difference methods and is used to solve as wide a range of ordinary and partial differential equations as the latter. The finite element methods are usually substantially more difficult to program, and this extra effort yields approximations that are of high order accuracy even when a partial differential equation is solved in a general multidimensional region and even when the solution varies more rapidly in certain portions of the region so that a uniform grid is not appropriate. These and other considerations have earned the finite element method great popularity both for initial value and boundary value differential equations. The most widely used form of the finite element method is the Galerkin method. It is suggested that when a second-order problem is approximated, all that is required of the Galerkin trial functions is continuity. The Galerkin method solves the approximation to the weak formulation a second-order partial differential equation, and the Galerkin equations involve only first derivatives of the trial functions.