Introduction

In finite element technique, the nodal equations for the field variables are obtained through an integral formulation, which may be set up through a variational principle (if one exists), or through the Galerkin's weighted residual approach. Here we shall consider the Galerkin's approach, which has a general applicability. Let us consider a general representation of a differential equation on a region

(18.1)

For the one dimensional heat conduction equation, the governing differential equation is

(18.2)

The symbol is an operator

that is operating on . The exact solution requires to satisfy Eqn. 18.1 at every points. Let us seek for an approximate solution that introduces an error called the residual

(18.3)

The approximate methods are centered around the concept of setting the residual relative to a weighting function to zero

(18.4)

The can be chosen based on the guiding philosophies of different variants of the weighted residual methods. In the Galerkin method, the are chosen from the basis functions used for constructing . We shall deal with this aspect, in detail, in the subsequent sections.