  The purpose of this study is to propose a high-accuracy and fast numerical
method for the Cauchy problem of the Laplace equation. Our problem is directly
discretized by the method of fundamental solutions (MFS). The Tikhonov
regularization method stabilizes a numerical solution of the problem for given
Cauchy data with high noises. The accuracy of the numerical solution depends on
a regularization parameter of the Tikhonov regularization technique and some
parameters of MFS. The L-curve determines a suitable regularization parameter
for obtaining an accurate solution. Numerical experiments show that such a
suitable regularization parameter coincides with the optimal one. Moreover, a
better choice of the parameters of MFS is numerically observed. It is
noteworthy that a problem whose solution has singular points can successfully
be solved. It is concluded that the numerical method proposed in this paper is
effective for a problem with an irregular domain, singular points, and the
Cauchy data with high noises.
