5.1  Elementary Formulations and Methods

The elementary formulations are those that are derived from Green's second theorem or from simple layer potential representation of the solution. Although the methods are not suitable for general computation, it is important to introduce them as the alternative methods originate with these formulae.

5.1.1  Elementary Direct Formulations

The application of Green's second theorem to the Helmholtz equation gives the following equations:
{ Mk f}S (p) - f(p) = { Lk v }S (p)    (p E) ,
{ Mk f}S (p) - 1
f(p) = { Lk v }S (p)     (p S) ,
where Lk and Mk, are defined in Section 3.1 and v(p) = [(f)/( n)]. The equations are also termed the Helmholtz formula or the Helmholtz-Kirchoff equation. The equation (5.3) is often also termed the surface Helmholtz equation. In this text the equations will be known as the elementary direct formulation.

The most straightforward approach to solving the exterior Helmholtz equation via the elementary formulations is to first find f and v on the boundary from equation (5.3). The value of f(p) for points p in the domain can then be obtained through the discretisation of equation (5.2).

5.1.2  Elementary Indirect Formulations

The elementary indirect integral equation formulations can be obtained by writing f as a single- or double-layer potential;
f(p) = {Lk s0 }S(p)    or    f(p) = {Mk s }S(p)   (p E)
where the s0 and s are source density functions defined on S. For points on the boundary the equations become boundary integral equations;
f(p) = {Lk s0 }S(p)    or    f(p) = {(Mk+ 1
I) s }S(p)   (p S),
where the jump conditions of Section 3.1 have been taken into account. In order to solve the exterior Helmholtz equation with a Dirichlet boundary condition using the above equations the first step is to solve one of the equations to obtain an approximation to either s0 or s. The solution at any point p in the domain can then be obtained by approximating the relevant equation of (5.4).

For the Neumann problem, further equations must be derived. Differentiating the equations (5.4) with respect to np, the normal to a point p S, and taking the limit as the point approaches S returns the following equations:

v(p) = { (Mkt - 1
I) s0 }S(p; np)    or    v(p) = { Nk s }S(p; np)   (p S)
The solution of one of the integral equations (5.6) returns either s0 or s and the solution in the domain can then be obtained from (5.4).

The solution of the Robin problem by elementary indirect methods can be achieved by substituting the expressions for f or v on S into the boundary condition (5.1):

a(p) {Lk s0 }S(p) +b(p) { (Mkt - 1
I) s0 }S(p;np) = f(p)  or
a(p) {(Mk+ 1
I) s }S(p;np) +b(p) { Nk s }S(p) = f(p) .
Once an approximation to the solution s0 is obtained from the first of these equations or an approximation to s is obtained from the second, the solution in the domain can be obtained through approximation of the relevant equation of (5.4) .

5.1.3  Elementary Methods

Methods that are derived straightforwardly from the integral equations of the previous Section are termed elementary methods. A typical example is the solution of the exterior Neumann problem using equations (5.2), (5.3). The application of collocation to the integral equation (5.3) reduces it to the following system of equations:

Mk - 1

= Lk ^
The Lk, Mk are n ×n matrices arising from the discretisation method outlined in Sections 1.1 and 2.3; for example the components of Lk are defined by [Lk]ij = {Lk [e*] }DS*j(pi) , where [e*] is the unit function. The vectors f and v represent the values of the boundary functions f and v at the collocation points and [^(f)] and [^(v)] are their approximations. The solution at any point p in the domain can then be approximated using
(p) = n

j = 1 
{ Mk ~
}DSj(p) ^

- { Lk ~
}DSj(p) ^

the discrete equivalent of (5.2).

Unfortunately, such methods have been found to be very unsuitable for the general computational solution of the exterior Helmholtz equation. The underlying reason for this is that for each formulation on each boundary with a given form of boundary condition the operator over which we solve is singular for certain values of k*, often termed the characteristic wavenumbers. For example the direct solution of the exterior Neumann problem requires the solution of equation (5.3), obtaining f from v. However at a set of real values k*, the eigenfrequencies of the interior Dirichlet problem, the operator Mk* - 1/2 I is singular. If k takes the value of any one of the values k* then a solution is impossible.

It might be argued that the wavenumbers for which a solution to the Helmholtz equation is sought do not coincide with any of the characteristic wavenumbers and so the problem does not arise. However, the operator over which we solve is not only singular at k = k* it is ill-conditioned for values of k in the neighbourhood of k*; the condition of the operator being approximately proportional to [1/( |k-k*|)] [4].

The condition of the operator over which a solution is obtained in integral equation methods is one of the most important factors governing the numerical error. In this case the numerical error can be characterised by [(e)/( |k-k*|)], following the profile of the condition of the operator. The e is determined by the nature of the problem and the accuracy of the boundary representation and the boundary function approximation and it tends to increase gently with k.

Given also the fact that the characteristic wavenumbers tend to cluster more and more at higher real wavenumbers, the elementary methods are generally unsatisfactory. Reports on results of implementations of elementary methods first appeared in the 1960s: Banaugh and Goldsmith [6], Chen and Schweikert [18], Chertock [21] and Brundrit [14]. All but the last of these references seem to have been unaware of the difficulties with the methods at the characteristic wavenumbers. The computational performance of elementary methods is compared with various alternative methods in Schenck [75], Meyer et al [60], Sayhi et al [74], for example. Because of the perceived computational difficulties, research has generally moved away from the elementary methods and towards alternative methods. A formal analysis of the solution of the exterior Helmholtz equation by elementary methods is given in Amini and Kirkup [4]. The convergence of the error with respect to element size and the accuracy of the representation of the boundary and boundary functions is considered in Juhl [38].

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