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# 2.3 Galerkin's Method

A weak formulation of ( ) for is given by, (2.7) is called test function. In this case we have an inner product between a scalar and a vector function on both sides of equation (2.7).

If the solution of the problem (2.1) in the space exists, than it is possible to represent this solution as the sum of an infinite series with the weighted basis functions of the space .

The central idea of Galerkin's method is to seek the solution of the problem given by (2.1), (2.2), and (2.3) not represented through the basis functions of the infinite space but through the basis functions of the finite space ( dim ) [9,10]. Such a solution can only be an approximation of the real solution of (2.1).
We choose linear independent functions , which span the space , e.g. (2.8)

where .
Now we can write equation (2.7) in the space , (2.9)

Trivially , and we can write, for (2.10)

Considering that (2.11)

we see that inner product in (2.10) transforms the differential operator into the operator (2.12)

and we write (2.10) as a system of nonlinear algebraic equations for and (2.13)

This equation is solved by means of Newton's method (see Section 2.6). The solutions of (2.13) are used to construct approximation of the functions from using (2.8).
The value for and (2.14)

is called residuum.    Next: 2.4 Time Dependent Problems Up: 2. Finite Element Method Previous: 2.2 Rayleigh-Ritz Method

H. Ceric: Numerical Techniques in Modern TCAD