6.2  Application of Collocation

By approximating the operators in the boundary integral equations in the way described in Section 3.3, they can each be reduced to an eigenvalue problem of the form (6.4). The first step is to approximate the boundary S by a set of n panels S* = j = 1n DS*j, as described in Chapter 2. The integral operators are approximated by the technique outlined in Chapter 3.

6.2.1  Indirect Method

For example the application of collocation to indirect integral equations gives an eigenvalue problem of the form (6.7) with Ak = Lk for the Dirichlet boundary condition and Ak = Mkt +1/2 I for the Neumann boundary condition. For the more general Robin condition (6.2) the eigenvalue problem is of the form (6.4) with
Ak = Da Lk + Db (Mkt + 1
2
I)
where the matrices Lk and Mk are as defined by the techniques employed in Section 1.1 and Chapter 3, Da, Db are diagonal matrices with [Da]ii = a(pi), [Db]ii = b(pi) and the pi are the collocation points.

6.2.2  Direct Method

For the Dirichlet boundary condition the discrete equivalent is identical to that of the indirect method stated earlier; Ak = Lk. For the Neumann problem and Ak = Mk +1/2 I. For the more general Robin condition (6.2) the eigenvalue problem cannot be straightforwardly put in the form (6.4). It can be written in the form
(Mk + 1
2
I) f = Lk v
(6.10)
with
Da f +Db v = 0 .
(6.11)
After rearrangement, these equations can take the form (6.4).
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