Finite Element Analysis. Barna Szabó

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such that for all ” where is defined by eq. (1.29). Note that is independent of the choice of . Essential boundary conditions are enforced by restriction on the space of admissible functions.

      2 When is prescribed on a boundary then the boundary condition is called Neumann⁶ boundary condition. Assume that and are prescribed. In this case(1.31) and the generalized formulation is: “Find such that for all ” where is defined by eq. (1.25).An important special case is when and and are prescribed. In this case:(1.32) and the generalized formulation is “Find such that for all where is defined by eq. (1.23).” Since the left‐hand side is zero for (constant) the specified data must satisfy the condition(1.33)

      3 When and/or , where , , δ0 and δℓ are given real numbers, is prescribed on a boundary then the boundary condition is called a Robin⁷ 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 1.1 The solution of the generalized formulation is unique in the energy space. The proof is by contradiction: Assume that there are two solutions u₁ and u₂ in

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

.

      Summary of the main points

      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 Скачать книгу