Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity

Based on potential theory, Steklov eigensolutions of elastic problems can be converted into eigenvalue problems of boundary integral equations (BIEs). The kernels of these BIEs are characterized by logarithmic and Hilbert singularities. In this article, the Nyström methods are presented for obtaining eigensolutions (λ⁽ⁱ⁾,u⁽ⁱ⁾), which have to deal with the two kinds of singularities simultaneously. The solutions possess high accuracy orders O(h³) and an asymptotic error expansion with odd powers. Using h³ ‐Richardson extrapolation algorithms, we can greatly improve the accuracy orders to O(h⁵). Furthermore, a generalized Fourier series is constructed by the eigensolutions, and then solving the elasticity displacement and traction problems involves just calculating the coefficients of the series. A class of elasticity problems with boundary Γ is solved with high convergence rate O(h⁵). The efficiency is illustrated by a numerical example. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012