Abstract
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.
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