Читать книгу Finite Element Analysis - Barna Szabó - Страница 20

The exact solution of the generalized formulation is called the generalized solution or weak solution whereas the solution that satisfies equation (1.5) is called the strong solution. The generalized formulation has the following important properties:

1 The exact solution, denoted by , exists for all data that satisfy the conditions where α and β are real numbers, and f is such that satisfies the definitive properties of linear forms listed in Section A.1.2 for all . Note that κ, c and f can be discontinuous functions.

2 The exact solution is unique in the energy space, see Theorem 1.1.

3 If the data are sufficiently smooth for the strong solution to exist then the strong and weak solutions are the same.

4 This formulation makes it possible to find approximations to with arbitrary accuracy. This will be addressed in detail in subsequent sections.

Exercise 1.2 Assume that and are given. State the generalized formulation.

Exercise 1.3 Consider the sequence of functions

illustrated in Fig. 1.2. Show that converges to in the space as . For the definition of convergence refer to Section A.2 in the appendix.

This exercise illustrates that restriction imposed on (or higher derivatives of u) at the boundaries will not impose a restriction on . Therefore natural boundary conditions cannot be enforced by restriction. Whereas all functions in are continuous and bounded, the derivatives do not have to be continuous or bounded.

Exercise 1.4 Show that defined on by eq. (1.20) satisfies the properties of linear forms listed in Section A.1.2 if f is square integrable on I. This is a sufficient but not necessary condition for to be a linear form.

Figure 1.2 Exercise 1.3: The function .

Remark 1.1 defined on by eq. (1.20) satisfies the properties of linear forms listed in Section A.1.2 if the following inequality is satisfied:

(1.35)