3  Discretization of the Integral Operators

The computational method is developed in a similar way to that described in reference [20]. In order to derive the discrete forms of the integral operators (2), (3), (4) and (5), G is approximated by a set of n elements G = ånj = 1 DGj . The boundary function m is replaced by its equivalent on the approximate boundary G. The function is then replaced by a constant on each element. Thus for the Lk integral operator:

{ Lk m}G (p) » { Lk ~
m
 
 }G (p) » n
å
 j = 1 
 ó
õ

DGj 
 Gk(p,q) ~
m
 
 (pj) dSq = n
å
 j = 1 
 [ ~
m
 
 (pj) { Lk ~
e
 
 }DGj (p) ] ,
(32)
where e is the unit function. The other integral operators may be discretised in a similar way.

The discrete forms are thus defined as follows:

{ Lk ~
e
 
 }DGj(p) = ó
õ

DGj 
 Gk (p,q) dSq ,
(33)

{ Mk ~
e
 
 }DGj(p) = ó
õ

DGj 
 Gk
nq
 (p,q) dSq ,
(34)

{ Mkt ~
e
 
 }DGj(p; vp) =
vp
 ó
õ

DGj 
 Gk (p,q) dSq ,
(35)

{ Nk ~
e
 
 }DGj(p; vp) =
vp
 ó
õ

DGj 
 Gk
nq
 (p,q) dSq ,
(36)

The derivative operator in (35) can always be carried inside the integral. The same is true for the operator in (36) when p does not lie on the element DGj. Thus we may write:

{ Mkt ~
e
 
 }DGj(p; vp) = ó
õ

DGj 
 Gk
vp
 (p,q) dSq ,
(37)

{ Nk ~
e
 
 }DGj(p; vp) = ó
õ

DGj 
 2 Gk
vp nq
 (p,q) dSq when  p not in D ~
G
 
j 
  .
(38)
When p not in DGj the integrals of (33), (34), (35) and (36) will all be regular and hence are amenable to standard quadrature. The same is true for the integrands of (34) and (35) when p Î DGj (though not on the edge of the element). However, the evaluation of the discrete integral operators (33) and (36) generally require special treatment when p Î DGj.

The special techniques applied in this paper involve `subtracting out' the singularity and evaluating the singular part and remaining regular part separately. The following results are immediate from the asymptotic properties of the kernel functions (28) and (31):

{ Lk ~
e
 
 }DGj (p) = { L0 ~
e
 
 }DGj (p) + ó
õ

DGj 
 ( Gk(p,q) - G0(p,q) ) dSq ,
(39)

{ Nk ~
e
 
 }DGj (p; vp) = { N0 ~
e
 
 }DGj (p; vp) - 1
2
 k2 { L0 ~
e
 
 }DGj (p) +            

ó
õ

DGj 
 ( 2 Gk
vp nq
 (p,q) - 2 G0
vp nq
 (p,q) + 1
2
 k2 G0(p,q) ) dSq ,
(40)
where, in each of (39) and (40) the explicit integral is non-singular and the remaining expressions are independent of k. Evaluation in this way requires the computation of the regular integral (amenable to standard quadrature) and the determination of the subtracted out part.

In summary, the evaluation of the integral operators requires a summation of a set of integrand values multiplied by quadrature weights. In the case when p Î DGj the evaluation of the subtracted out part is also required for the Lk and Nk operators.