1 Approximate
Methods
1.1 INTRODUCTION
Once the mathematical equations governing a problem have been
established, then excepting very simple cases, they must be solved
using an approximate method. The type of approximation used is
important as it affects the accuracy of the results and the economy of
the solution. Unfortunately it is difficult to compare different
approximating techniques as they are usually presented as totally
unrelated. The finite element method for instance, is viewed as a
virtual work method, a variational technique or a weighted residual
method, depending on the particular application under consideration
or the researcher's preference. Boundary solutions are sometimes
presented as a consequence of the reciprocity principles and only
recently they have been interpreted as weighted residual solutions. In
what follows we will present a rational classification of the different
approximate methods based on the weighted residual formulation
and on the type of approximating and weighting functions used.
Weighted residual formulations are more general than classical
variational principles which can only be applied to a restricted type of
operator. Weighted residuals can be used with complex
non-self-
adjoint operators equally well as with self-adjoint ones. Variational
techniques, finite differences, finite elements, integral type for-
mulations and many other methods can be interpreted as special cases
of weighted residual methods. The converse is not generally true.
The aim of an approximate solution is to reduce a governing
equation (or set of equations) plus boundary conditions to a system of
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