3.1.3Discretization

The basics of Galerkin’s method consist of a clever discretization of the infinite linear space in the problem (3.5). The purpose of such a process is to create a controlled sub-space inside , which can be designed to approximate the solution of (3.6) with arbitrary precision.

Consider the finite sub-space of with dimension N. Consider also the functions as an orthogonal basis of . Hence, every function in can be written as linear combination of the basis as in

 (3.9)

The discrete formulation of the variational problem (3.5) can be written as

 (3.10)

Actually, the formulation (3.10) can be simplified. It can be proven that if satisfies (3.10) only for the basis function of , it satisfies it for every element in . To show this, replace in (3.10) by the basis projection (3.9) as in

On the assumption for j = 1..N, the equality (3.11) holds and satisfies (3.10) for every element of . Consequently, the simplified version of (3.10) can be summarized by:

 (3.12)

The formulation (3.12) is well suitable for the construction of a numerical scheme, because it restricts the search for the solution to the computation of a finite number of equations (N). Hence, the problem shifts to a more concrete perspective, and a method is required to compute from those N equations. For that purpose, consider the expansion of in the basis of , in the same way as was done before for (3.9). By substituting appropriately in (3.10) the obtained function is

In fact, the substitution of expansion in (3.12) leads to the transformation of the discrete problem in a linear system with N equations, where A, x and b are given by

 (3.14)