Barna Szabó - Finite Element Analysis

If the Babuška‐Brezzi condition is not satisfied then there will be at least some for which as even though there may be for which as . Examples are presented in [6]. In general it is difficult, or may even be impossible, to separate those for which the method works well from those for which it does not. The Babuška‐Brezzi condition guarantees that the condition number of the stiffness matrix will not become too large as i increases.

Remark 1.15Any implementation of the finite element method must be shown to satisfy the Babuška‐Brezzi condition otherwise there will be some input data for which the method will fail even though it may work well for other input data. The formulation based on the principle of virtual work satisfies the Babuška‐Brezzi condition.

Exercise 1.23Show that the finite element method based on the principle of virtual work satisfies the Babuška‐Brezzi condition.

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.2.2and 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 ).