High Accuracy Algorithm For Steklov Eigenvalue In Boundary Integral Equations

This paper harnesses mechanical quadrature methods to obtain high accuracy ordersolutions for partial differential equations. Comparing against finite element methods andcollocation methods, the mechanical quadrature methods own high accuracy order, lowcomputing complexities and fast convergence rate. Extrapolation or splitting extrapola-tion algorithms are applied to improve the accuracy order to O(h5) or even O(h7). par-ticularly, splitting extrapolation algorithms can solve the equations in parallel, the com-puting cost can be even smaller than mechanical quadrature methods.This paper solvedpartial differential equations for Steklov engenvalues for Laplace equation and elasticityproblems, for elasticity problems with linear boundary condition, and for Laplace equa-tions with nonlinear boundary conditions. We obtain the high accuracy solutions for theseproblems. Numerical examples illustrate the effectiveness of the methods.we firstly discuss the methods for solving Steklov eigenvalue problems which areintroduced in Laplace equations and elasticity, and its application in solving differentialequations. Following potential theory, the differential equations will be transformed intoboundary integral equations. there are the logarithmic singularity and Cauchy singular-ity in these equations which can be approximated by mechanical quadrature methods,and linear equations can be obtained. According to Anselone’s collective convergence,asymptotic compact convergence and adjoint operator theory, we obtain the asymptoticexpansions of the errors with the power O(h3). The high accuracy order O(h5) and theposteriori error estimate can be achieved for the solutions by using the extrapolation algo-rithms. Then, with the property of the orthogonal and completeness of the eigenvectors,we obtain the generalized Fourier series expansion for solving differential equations.Secondly, we focus on Steklov eigenvalue problem which is defined in polygonaldomains. Following potential theory, the differential equations will be transformed intoboundary integral equations. there are singularities both for the integral kernels and har-monic functions in corners. In order to eliminate the singularities in the corners for inte-gral kernels and harmonic functions, we introduce the sinp transformation to reconstructthe integral kernels and the harmonic function. We obtain the multivariate asymptoticexpansions of the errors with the power O(h3) by the collective compact convergence and asymptotic compact convergence. The splitting extrapolation algorithms are used toimprove the accuracy to O(h0)5.Finally, we solve elasticity problems with linear boundary conditions and Laplaceequation with nonlinear boundary conditions by mechanical quadrature methods. follow-ing potential theory, elasticity problems are converted into boundary integral equationswith logarithmic singularity and Cauchy singularity, and Laplace equations are convertedinto non-linear boundary integral equations. The linear approximate equations or non-linear approximate equations can be obtained by mechanical quadrature methods. follow-ing Anselone collective compact convergence and asymptotic compact convergent the-ory, the solvability can be proofed and the approximate solution is existed HarnessingStepleman theorem for the non-linear equations.Furthermore, the error of the approxi-mate solution asymptotic expansion with odd power is also obtained. The high accuracyconvergence order O(h)5 is obtained by extrapolation algorithms.