Eigensolution Method and Extrapolation for Solving Potential Problems

By potential theorem, the fundamental boundary eigenproblem problems are converted into boundary integral equations(BIE) with the logarithmic singularity. In this paper, mechanical quadrature methods (MQM) are presented to obtain the eigensolutions (λ
_l
, ũ
_l
) and the eigensolutions are used to solve the potential problems, which possesses the high accuracies O(h
³
) and low computing complexities O(h
^-1
). The convergence and stability are proved based on Anselone's collective compact theory. Using Richardson extrapolation, we can not only greatly improve the accuracy order to O(h
⁵
) of approximation, but also derive an a posteriori error estimate as a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.