Читать книгу Finite Element Analysis - Barna Szabó - Страница 52

Consider writing eq. (1.5) in the following form:

(1.156)

(1.157)

and assume that the boundary conditions are .

In the following we will use the one‐dimensional equivalent of the notation introduced in sections A.2.2 and A.2.3. Multiply eq. (1.156) by and eq. (1.157) by , integrate by parts and sum the resulting equations to obtain:

(1.158)

We define the bilinear form:

(1.159)

and the linear form

(1.160)

The problem is now stated as follows: Find , such that

(1.161)

The finite element problem is formulated as follows: Find where is a subspace of and where is a subspace of such that

(1.162)

We now ask: In what sense will be close to ? The answer is that there is a constant C, independent of the finite element mesh and , such that

(1.163)

provided, however, that and were properly selected.

For example, let S be the space defined in eq. (1.61) with , . The space has the dimension . For consider three choices:

1  is the set of functions which are constant on each finite element. has the dimension .

2  is the space S defined in (3.11) with , (dimension ).

3  is the set of functions which are linear on every element and discontinuous at the nodes (dimension ).

For these choices of the mixed formulation leads to systems of linear equations with , and unknowns, respectively. In the cases and , a constant C exists such that the inequality (1.163) holds for all , and . In the case of , however, such a constant does not exist. This means that no matter how large C is, there exist some and and mesh so that the inequality (1.163) is not satisfied. On the other hand, there will be and for which the inequality is satisfied and therefore the finite element solutions will converge to the underlying exact solution.