From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations (BIEs) with the logarithmic singularity. In this paper, mechanical quadrature methods (MQMs) are presented to obtain the eigensolutions that are used to solve Laplace’s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone’s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm(EA), the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.