LITMIR.BIZ

Figure 1.14 The ratio

Table 1.6 Example: p‐Convergence of the 24th eigenvalue in Example 1.16.

Example 1.16 Let us consider the problem in Example 1.15 modified so that μ is a piecewise constant function defined on a uniform mesh of 5 elements such that on elements 1, 3 and 5, on elements 2 and 4. In this case the exact eigenfunctions are not smooth and the exact eigenvalues are not known explicitly.

At there are 24 degrees of freedom. Suppose that the 24th eigenvalue is of interest. If we increase p uniformly then this eigenvalue converges to 98.312. The results of computation are shown in Table 1.6.

      Any eigenvalue can be approximated to an arbitrary degree of precision on a suitably defined mesh and uniform increase in the degrees of freedom. When κ and/or are discontinuous functions then the points of discontinuity must be node points.

      Observe that the numerically computed eigenvalues converge monotonically from above. This follows directly from the fact that the eigenfunctions are minimizers of the Rayleigh quotient.

Exercise 1.21 Prove eq. (1.143).

      Exercise 1.22 Find the eigenvalues for the problem of Example 1.15 using the generalized formulation and the basis functions , (). Assume that κ and are constants and . Let . Explain what makes this choice of basis functions very special. Hint: Owing to the orthogonality of the basis functions, only hand calculations are involved.

1.8 Other finite element methods

      Up to this point we have been concerned with the finite element method based on the generalized formulation, called the principle of virtual work. There are many other finite element methods. All finite element methods share the following attributes: