14.5 Galerkin Approximation

To transform the weak form into a linear set of equations in order to derive an approximate solution the following steps are taken.

Step 1. Element division As shown in the previous section the domain ? may be split into a number of subdomains ? _e, elements, and within each element a polynomial interpolation can be made of the function u. An example of such an element division is given in Fig. 14.4. This distribution of elements is called a mesh. Then, the integration over the domain ? can be performed by summing up the integrals over each element. Consequently, Eq. (14.14) yields:

where N _el denotes the number of elements.

Figure 14.4: Element distribution and unknowns at local and global levels.

Step 2. Interpolation Suppose that the domain ? has been divided into three linear elements, as depicted in Fig. 14.4. Then the nodal values u _i may be collected in an array :

The unknowns associated with each of the elements ? _e are collected in the arrays , such that

So, it is important to realize that each particular element array contains a subset of the total, or global, array . Within each element array a local numbering may be used, such that for this particular example with linear elements:

For instance in case of the second element the element array contains

Clearly, in the case of...