Quadrature method and extrapolation for solving a class of elastic boundary value problems

Elastic boundary value problems of the third kind are converted into boundary integral equations (BIE) with the logarithmic singularity and the Hilbert singularity. In this paper, quadrature methods are presented to deal with the logarithmic singularity and the Hilbert singularity simultaneously for solving the BIE, which possesses the high accuracies O(h³) and low computing complexities O(h^-1), where h is the mesh width. The convergence and stability are proved based on Anselone´s collective compact theory. Furthermore, the asymptotic expansions with the odd powers of the errors are presented. Using Richardson extrapolation, we can not only greatly improve the accuracy order of approximation O(h⁵), but also derive an a posteriori error estimate as a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.