Finite Element Analysis. Barna Szabó

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style="font-size:15px;">      2 A formulation cannot be meaningful unless all indicated operations are defined. In the case of eq. (1.5) this means that and are finite on the interval . In the case of eq. (1.11) the integralmust be finite which is a much less stringent condition. In other words, eq. (1.8) is meaningful for a larger set of functions u than eq. (1.5) is. Equation (1.5) is the strong form, whereas eq. (1.11) is the generalized or weak form of the same differential equation. When the solution of eq. (1.5) exists then un converges to that solution in the sense that the limit of the integral is zero.

      3 The error depends on the span and not on the choice of basis functions.

1.2 Generalized formulation

      We have seen in the foregoing discussion that it is possible to approximate the exact solution u of eq. (1.5) without knowing u when

. In this section the formulation is outlined for other boundary conditions.

      The generalized formulation outlined in this section is the most widely implemented formulation; however, it is only one of several possible formulations. It has the properties of stability and consistency. For a discussion on the requirements of stability and consistency in numerical approximation we refer to [5].

      1.2.1 The exact solution

      If eq. (1.5) holds then for an arbitrary function

, subject only to the restriction that all of the operations indicated in the following are properly defined, we have

(1.17)

      Using the product rule;

we get

therefore eq. (1.17) is transformed to:

(1.18)

We introduce the following notation:

      (1.19)

      where

is a bilinear form. A bilinear form has the property that it is linear with respect to each of its two arguments. The properties of bilinear forms are listed Section A.1.3 of Appendix A. We define the linear form:

(1.20)

The forcing function

may be a sum of forcing functions:

, some or all of which may be the Dirac delta function⁴ multiplied by a constant. For example if

then

(1.21)

The properties of linear forms are listed in Section A.1.2. Note that

in eq. (1.21) is a linear form only if v is continuous and bounded.

      The definitions of

and

are modified depending on the boundary conditions. Before proceeding further we need the following definitions.

      1 The energy norm is defined by(1.22) where I represents the open interval . This notation should be understood to mean that if and only if x satisfies the condition to the right of the bar (). This notation may be shortened

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