APPENDIX B

 The Galerkin Method:

For most practical problems, it is impossible to determine the exact solution to the differential equations in terms of known functions, which exactly satisfies the governing equations and the boundary conditions. As an alternative, the FEM seeks an approximate solution; an explicit expression in terms of known functions, which only approximately satisfies the governing equations and the boundary conditions.

 

The FEM obtains an approximate solution by using the classical trial-solution procedure. The trial-solution procedure is composed of three principal operations, respectively Burnett (1987);

 

·        Construction of trial solution,

·        Application of optimizing criterion,

·        Estimation of the accuracy.

 

One has to determine an approximate solution , which comes close to the unknown true solution u for a given problem. The function  will be defined everywhere in terms of a finite set of mathematical basis functions  whose properties are a priori well known:

 

                                                       (B-1)

 

The coefficients  are the primary unknowns of any problem once the basis has been selected. In any practical problem, the basis in use will necessarily be finite and incomplete -i.e. incapable except in lucky cases representing the exact solution perfectly. Any numerical solution may be viewed as a two-step process. First, select a basis which is likely to fit the unknown solution for the particular problem. Second, determine the coefficients  in a reliable way.

 

Use of a finite or incomplete basis guarantees that in general a given differential equation cannot be satisfied everywhere, leaving an imbalance or residual, R, everywhere. Clearly, R depends on the selection of both the basis  and the coefficient . Problem-dependent basis selection is the first step towards a small residual. Given a finite basis, one must then settle for making R small in some average way by choosing the coefficients .

 

In the Method of Weighted Residuals (MWR), in order to determine , R is required to vanish in a weighted integral sense. In Cartesian space we have

 

                                                                (B-2)

 

for a set of distinct weighting functions , and the integration performed over the full domain in which the differential equation governs. We can use the inner product notation,< , > to indicate domain (volume) integration and the MWR is stated compactly:

 

                                                              (B-3)

 

Equivalently, “R is orthogonal to W_i “ with N basis functions  selected a priori, a choice of N independent weighting functions W_iwill determine the N unknown u_j.

 

Galerkin Method is an MWR in which weighting functions are identical to the basis functions:

 

                                                                                    (B-4)

 

This is extensively used with finite elements.