3  Finite Element Method

A typical derivation of a finite element method for the interior Neumann eigenvalue problem [1-2] can be effected through the minimisation of the functional
1
2
 ó
õ

D 
 ( ( Ñj(p) )2 - k2 j2(p) ) dV .
(8)
Let ci(p)  (p Î D ÈS) (i=1,2,...,N) be a sequence of linearly independent basis functions. The velocity potential j(p) and Ñj(p) can be approximated by the series
j(p) » N
å
i=1 
 ui ci(p)    (p Î D),
(9)

Ñj(p) » N
å
i=1 
 ui Ñci(p)    (p Î D).
(10)
The substitution of these expressions into (8) reduces it to the following
( K - w2 M )u » 0  ,
(11)
where K and M are termed the mass and stiffness matrices respectively defined by
[K]ij = ó
õ

D 
 Ñci(p) Ñcj(p) dV ,  [M]ij = 1
c2
 ó
õ

D 
 ci(p) cj(p) dV ,  
(12)
and u=[u1, u2, ..., uN]T. The matrices K and M are generally sparse and structured.
Approximations [^(w)]* to w* and [^(u)]* to u* can now be achieved through solving the following
( K -
^
w
 
 2
 
 M )
^
u
 
 = 0 .
(13)
Equation (13) is a generalised linear eigenvalue problem and is hence amenable to solution through standard methods such as the QZ algorithm [25] . More specialised methods, such as iterative techniques, can be employed for solving (13), given the particular structure of the matrices and the fact that only a fraction of the full set of eigenvalues are generally required.