Barna Szabó - Finite Element Analysis

3 When and/or , where , , δ0 and δℓ are given real numbers, is prescribed on a boundary then the boundary condition is called a Robin 7 boundary condition. Assume, for example, that and are prescribed. In that case(1.34) and the generalized formulation is: “Find such that for all where is defined by eq. (1.23).”

These boundary conditions may be prescribed in any combination. The Neumann and Robin boundary conditions are called natural boundary conditions. Natural boundary conditions cannot be enforced by restriction. This is illustrated in Exercise 1.3.

The generalized formulation is stated as follows: “Find such that for all ”. The space X is called the trial space, the space Y is called the test space. We will use this notation with the understanding that the definitions of X , Y , and depend on the boundary conditions. It is essential for analysts to understand and be able to precisely state the generalized formulation for any set of boundary conditions.

Under frequently occurring special conditions the mathematical problem can be formulated on a subdomain and the solution extended to the full domain by symmetry, antisymmetry or periodicity. The symmetric, antisymmetric and periodic boundary conditions will be discussed in Chapter 2.

Theorem 1.1The solution of the generalized formulation is unique in the energy space. The proof is by contradiction: Assume that there are two solutions u 1and u 2in that satisfy

Using property 1 of bilinear forms stated in the appendix, Section A.1.3, we have

Selecting we have . That is, in energy space. Observe that when and where C is an arbitrary constant, then .

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

Exercise 1.3Consider 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.2in 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 1.4Show that defined on by eq. (1.20)satisfies the properties of linear forms listed in Section A.1.2if f is square integrable on I . This is a sufficient but not necessary condition for to be a linear form.