Galerkin method of approximation — irreducible and mixed forms

Abstract

Equations can be presented in a strong form as a set of partial differential equations or alternatively in terms of a variational principle or weak form expressed as an integral over the domain of interest. This chapter uses the weak form to construct approximate solutions based on the finite element method. This results in a Galerkin method for which general properties are well known. This chapter presents a full summary of the basic steps to construct a solution for the transient problem. It emphasizes on the differences between linear and non-linear effects as well as the numerical procedures used to establish the final discrete form of the equations that is the form used in computer analysis. It also considers both irreducible and mixed forms of approximation. The mixed forms are introduced to overcome deficiencies arising in use of low order elements based on irreducible forms. This chapter considers a mixed form appropriate for use in problems in which near incompressible behavior can occur. This chapter closes by applying the methods developed for the equations of solid mechanics to that for thermal analysis based on a non-linear form of the quasi-harmonic equation.