Five-Order Algorithms for Solving Laplace's Steklov Eigenvalue on Polygon by Mechanical Quadrature Methods.

By the potential theory, Steklov eigenvalue problems of Laplace equation on polygon are converted into boundary integral equations(BIEs). In this paper, the singularities at corners and in the integral kernels are studied to obtain five order approximate solution. Firstly,a sin transformation is used to deal with the boundary condition. Secondly, a Sidi's quadrature formula is introduced to approximate the logarithmic singularity integral operator with O(h³) approximate accuracy order. Then a similar approximate equation is also constructed for the logarithmic singular operator, which is based on coarse grid with mesh width 2h. So an extrapolation algorithm is applied to approximate the logarithmic operator and the accuracy order is improved to O(h5). Moreover, the accuracy order is based on fine grid h. Furthermore, an asymptotic expansion with odd powers of the errors is presented with convergence rate O(h5). The efficiency of the algorithms is illustrated by the example. [ABSTRACT FROM AUTHOR]