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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]

Laplace equation with linear boundary condition will be converted into a boundary integral equation(BIE) with logarithmic singularity following potential theory. In this paper, a Sidi quadrature formula is introduced to approximate the logarithmic singularity integral operator with O(h³) approximate accuracy order. 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. The convergence and stability are proved based on Anselone's collective compact and asymptotic compact theory. Furthermore, an asymptotic expansion with odd powers of the errors is presented with convergence rate O(h5). Using h5-Richardson extrapolation algorithms(EAs), not only the approximation accuracy order can be improved to O(h7), but also an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. numerical examples are shown to verify its efficiency. [ABSTRACT FROM AUTHOR]

The paper presents a new stability analysis for the finite difference method (FDM) for biharmonic equations, based on the effective condition number. Upper bounds of the effective condition number and the simplest effective condition number are derived. In general cases, the bound of the effective condition number is O(h-3.5 ), which are smaller than Cond. = O(h-4), where h is the minimal meshspace of the difference grids. Surprisingly, the bounds of the effective condition number are only O(1) for homogeneous boundary conditions. Namely, these new stability analysis are more valid than previous stability analysis. Numerical experiments are provided to verify the stability analysis. [ABSTRACT FROM AUTHOR]

In this paper, we research the standard 13-point difference scheme for solving the biharmonic equation. Existence and convergence of finite difference methods(FDM) solutions are obtained by estimating the lower bounds of the minimum eigenvalues of the discrete matrix and making use of the Taylor series expansions, respectively. The accuracy are proved to be O(h²). Moreover, the extrapolation techniques are used to improve the high accuracy of the solutions. For the given examples, numerical results show that the errors O(h4) is obtained from the first level extrapolation. The very accurate solutions with the errors O(10-8) are obtained by the third level extrapolation techniques under the homogeneous essential boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we research the standard 13-point difference scheme for solving the biharmonic equation. Existence and convergence of finite difference methods(FDM) solutions are obtained by estimating the lower bounds of the minimum eigenvalues of the discrete matrix and making use of the Taylor series expansions, respectively. The accuracy are proved to be O(h²). Moreover, the extrapolation techniques are used to improve the high accuracy of the solutions. For the given examples, numerical results show that the errors O(h4) is obtained from the first level extrapolation. The very accurate solutions with the errors O(10-8) are obtained by the third level extrapolation techniques under the homogeneous essential boundary conditions. [ABSTRACT FROM AUTHOR]