Chapter 3 - Weak Forms and Finite Element Approximation: 1-D Problems

Abstract

The differential equations in Chapter 2 are specialized to one dimension in this chapter. For this case we recast the equilibrium equation and continuity equation in an alternate weak form from which accurate approximate solutions may be achieved. The chapter shows how the equations may be converted to a weak form using a simple direct approach in which each differential equation is multiplied by an arbitrary function and integrated over its domain. Using integration by parts allows direct introduction of the some boundary condition. We use the weak form to construct approximations by the weighted residual-Galerkin method. This solution procedure requires expressing the solution dependent variable as a series of specified functions multiplied by time dependent parameters. A finite element solution is then merely one very convenient way to express the set of specified functions that can be defined on individual elements as shape functions. Problems that involve the minimum number of dependent variables are defined as irreducible those with additional variables are called mixed.