The boundary knot method (BKM) is applied to the Helmholtz equation in 2D bounded simply-connected domains, to show high accuracy of the solutions obtained by not many fundamental solutions (FS) used. When the Bessel function is chosen as the FS, the optimal polynomial convergence rates are obtained for disk domains. Moreover, the bounds of condition number (Cond) are derived for disk domains, to show a super-exponential growth via the number of FS used. The super-exponential growth of Cond is new and intriguing in the numerical methods for partial differential equations (PDE). It is also imperative for the BKM that good numerical solutions may be achieved by balancing accuracy and instability. Numerical experiments are carried out to support the analysis made. Comparisons between the BKM and the MFS are also made.

在2D有界简单连接域中，将边界结法（BKM）应用于Helmholtz方程，以显示由很少使用的基本解（FS）获得的解的高精度。当贝塞尔函数被选择为FS，针对磁盘结构域获得的最优多项式收敛速度。此外，为磁盘域导出了条件数（Cond）的界限，以通过使用的FS数显示超指数增长。Cond的超指数增长是偏微分方程（PDE）数值方法中的新奇事物。对于BKM，也必须通过平衡精度和不稳定性来获得良好的数值解。进行数值实验以支持所做的分析。BKM和MFS之间也进行了比较。