3.5  Test problem - the interior Laplace problem

The interior Laplace problem involves the solution of the Laplace equation in a domain that is bounded by a closed surface in 3D or a closed boundary in 2D. An illustration of the problem is shown in figure 3.2.

Figure 3.2. An Illustration of the Laplace Problem.

3.5.1  Matrix Equivalent of the Operators

Applying collocation to generally requires that the closed boundary S is replaced by an approximate boundary S' made up of a set of n elements DS'j (j = 1,..,n) in the way described in Chapter 2. Let the points pj (j = 1,..,n) with pj D S'j be the collocation points. In this work we consider only the approximation of the boundary functions by a constant on each elment. It is helpful to adopt the following notation:

[ L ]ij = { L
e'
 
}DS'j(pi) ,
(34)

[ Mk ]ij = { M
e'
 
}DS'j(pi) ,
(35)

[ Mkt ]ij = { Mt
e'
 
}DS'j(pi; npi) ,
(36)

[ Nk ]ij = { N
e'
 
}DS'j(pi ; npi) ,
(37)
where npi is the unit outward normal to S' at pi. This gives the four n ×n matrices Lk, Mk, Mkt and Nk. The approximate boundary functions can be approximated by a vector

m = [
m'
 
(p1),...,
m'
 
(pn) ]T.

3.5.2  The Linear System

The application of collocation to the above equation give the following linear systems of approximations

( Mk - 1
2
I )  f   Lk  v
(38)
where vj = v(pj) for j = 1,...,n. Hence the primary stage of the boundary element method entails the solution of the following linear system of equations:

( Mk - 1
2
I )   ^
f
 
= aLk  v
(39)
which yields an approximations to f(pj), for j = 1,...,n.

The secondary stage of the boundary element method requires the calculation of the approximation to f(p) where p is a point in the approximate exterior domain E. For this the discrete forms are substituted into (3) to give

^
f
 
(p) = n

j = 1 
[ { M
e'
 
}DS'j(p) ^
f
 
j -{ L
e'
 
}DS'j(p) vj]  (p
E'
 
).
(40)
Note that the secondary stage requires the evaluation of only two integral operators in contrast with the primary stage which requires all four. Note also that the special evaluation techniques of subtracting out the singularity are required only for the diagonal components of the matrices in (34), (37). This latter point is a typical property of integral equation methods, the outcome of which is that the generally greater cost of evaluating the discrete forms when p lies on the element is not important when assessing the overall computational cost.


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