Finite Element Analysis. Barna Szabó

The solution of this problem is

,

. These coefficients, together with the basis functions, define the approximate solution un. The exact and approximate solutions are shown in Fig. 1.1.

      The choice of basis functions

      By definition, a set of functions

,

are linearly independent if

      implies that

for

. It is left to the reader to show that if the basis functions are linearly independent then matrix

is invertible.

      Given a set of linearly independent functions

,

, the set of functions that can be written as

      is called the span and

are basis functions of S.

      We could have defined other polynomial basis functions, for example;

      (1.15)

      When one set of basis functions

can be written in terms of another set

in the form:

      (1.16)

      where

is an invertible matrix of constant coefficients then both sets of basis functions are said to have the same span. The following exercise demonstrates that the approximate solution depends on the span, not on the choice of basis functions.

Exercise 1.1 Solve the problem of Example 1.1 using the basis functions

,

and show that the resulting approximate solution is identical to the approximate solution obtained in Example 1.1. The span of the basis functions in this exercise and in Example 1.1 is the same: It is the set of polynomials of degree less than or equal to 3 that vanish in the points

and

.

      Summary of the main points

      1 The definition of the integral by eq. (1.8) made it possible to find an approximation to the exact solution u of eq. (1.5) without knowing u.

      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