The MATLAB functions, fbie, fbiead, and fcau, will be useful for calculating the conformal mapping and solving potential flow problems for domains of high connectivity.
The MATLAB function fbie. We use the MATLAB function gmres to solve the linear system (4.5), which can be used with the matrix-vector product function [f.sub.B] (x) defined by
A MATLAB function fbie for the fast solution of the integral equation (2.3) and the fast computation of the function h in (2.4) using the method presented in this section is shown in Figure 4.1.
To test the performance of the functions fbie and fbiead, five numerical examples are presented.
In this example, we study the effect of (a) the distance [epsilon], (b) the function [theta], and (c) the singularity subtraction in (4.1) and (5.4) on the accuracy of the functions fbie and fbiead.
The approximate solution obtained using the function fbie is denoted by [[mu].sub.n].
The maximum error norm [parallel][mu] - [[mu].sub.n][[parallel].sub.[infinity]] for the function fbie and the error [E.sub.n] for the function fbiead versus the number of nodes n in the discretization of each boundary component are displayed in Figures 8.2(a,e) for constant [theta] and in Figures 8.2(b,f) for nonconstant [theta] for [epsilon] = 0.5,0.1,0.001.
Thus, the accuracy of fbie is affected by [theta] being constant or not.
The eigenvalues of the coefficient matrices of the linear systems for both functions fbie and fbiead are strongly clustered around 1 for well-separated boundaries.
For both functions fbie and fbiead, the condition number of the coefficient matrices of the linear systems does not depend on [theta] being constant or not.
For both functions fbie and fbiead, the number of GMRES iterations does not depend on [theta] being constant or not.