Finite Element Analysis. Barna Szabó

      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

With these data the exact solution of eq. (1.5) is

Figure 1.1 Exact and approximate solutions for the problem in Example 1.1.

      We seek an approximation to u in the form:

      On computing the elements of

,

and

we get