1.4  Numerical Solution of the Integral Equation

In order to develop a numerical method for the solution of integral equations like (3), a technique is applied so that the equation is simplified into a linear system of equations. Hence there is a close analogy between linear integral equations and systems of linear equations; the integral operators can be viewed as matrices, the boundary functions as vectors.

The application of such a technique transforms the equation (3) to an equation of the form

M ^
f
 
 + 1
2
 ^
f
 
 = L ^
v
 
      or    (M + 1
2
 I) ^
f
 
 = L ^
v
 
(5)
where the components of the vectors f and v represent the values of the function f(p) and f/ nq(p) at a set of points on the boundary and the ^s represent their approximations that may arise through measurement of the physical quantities or through numerical solution. L, M and I are matrices derived from the corresponding integral operators in (3) with I representing the identity matrix. A complete demonstration of how the matrix-vector equation is obtained from the integral equation in given in Section 1.7. The connection between the system of linear equations (5) and the integral equation (3) or (1) is now clear. As stated earlier, the boundary data f, f/ nq or some combination of the two functions are given and the solution of the system of linear equations (5) can be used to derive approximations to the unknown boundary data.

There are a variety of techniques for deriving the system of linear equations from a given integral equation. In general, a method can be derived by replacing the integrals in an integral equation by a quadrature formula or by a weighted residual method such as the Galerkin method. Many methods for solving integral equations can be used to develop a particular boundary element method [3].

The application of collocation to a boundary integral equation requires that the boundary is represented by a set of panels. For example a two dimensional boundary can be approximated by a set of straight lines. In order to complete the discretisation of the integral equations, the boundary functions also need to be approximated on each panel. It is the characteristics of the panel and the representation of the boundary function on the panel that together define the element in the boundary element method.

By representing the boundary functions by a characteristic form on each panel, the boundary integral equations can be simplified into the linear system of equations of the form introduced earlier. Most simply, the boundary functions can be approximated by a constant on each panel. The collocation point is at the centre of the panel (C0 collocation). The overall process is that of discretising the integral operators and the methods for carrying this out are covered in (for the more general Helmholtz equation) in [18] and [17]. For the particular Helmholtz integral operators, the methods employed are reiterated in Chapter 3.

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