Finite Element Analysis. Barna Szabó

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kappa u prime right-parenthesis prime"/>, c or f would not be finite in one or more points on the interval . The function f is called a forcing function.

      We seek an approximation to u in the form:

(1.7)

      where

are fixed functions, called basis functions, and aj are the coefficients of the basis functions to be determined. Note that the basis functions satisfy the zero boundary conditions.

Let us find aj such that the integral

defined by

(1.8)

      is minimum. While there are other plausible criteria for selecting aj, we will see that this criterion is fundamentally important in the finite element method. Differentiating

with respect to ai and letting the derivative equal to zero, we have:

(1.9)

      Using the product rule:

we write

      (1.10)

The underbraced terms vanish on account of the boundary conditions, see eq. (1.7). On substituting this expression into eq. (1.9), we get

      which will be written as

(1.11)

      We define

(1.12)

and write eq. (1.11) in the following form

      (1.13)

      which represents n simultaneous equations in n unknowns. It is usually written in matrix form:

(1.14)

      On solving these equations we find an approximation un to the exact solution u in the sense that un minimizes the integral

.

Example 1.1 Let

,

,

and

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