Figure 1.8: The piecewise quadratic basis functions corresponding to the edge midpoints [3]

It is possible to have xTAx = 0 for a nonzero vector x. This is the case if the constant function u = 1 is contained in S. This happens if Neumann boundary conditions ∂n∂u = 0 are posed on the whole boundary ∂Ω. Then,

X

u(x, y) = 1 = φi(x, y),

i

i.e., we have xTAx = 0 for x = [1, 1, . . . , 1].

1.6.3A numerical example

We want to determine the acoustic eigenfrequencies and corresponding modes in the interior of a car. This is of interest in the manufacturing of cars, since an appropriate shape of the form of the interior can suppress the often unpleasant droning of the motor. The problem is three-dimensional, but by separation of variables the problem can be reduced to two dimensions. If rigid, acoustically hard walls are assumed, the mathematical model of the problem is again the Laplace eigenvalue problem (1.24) together with Neumann boundary conditions. The domain is given in Fig. 1.9 where three finite element triangulations are shown with 87 (grid1), 298 (grid2), and 1095 (grid3) vertices (nodes), respectively. The

Finite element method

Table 1.2: Numerical solutions of acoustic vibration problem

results obtained with piecewise linear polynomials are listed in Table 1.2. From the results we notice the quadratic convergence rate. The smallest eigenvalue is always zero. The corresponding eigenfunction is the constant function. This function can be represented exactly by the finite element spaces, whence its value is correct (up to rounding error).