Extrapolation Algorithms for Solving Mixed Boundary Integral Equations of the Helmholtz Equation by Mechanical Quadrature Methods

Abstract

This paper presents mechanical quadrature methods with high accuracy for solving mixed boundary integral equations of the Helmholtz equation. By estimating the range of eigenvalues for the discretization matrix of the integral equations and applying the collectively compact convergent theory, we prove the stability and convergence of numerical solutions, which is a challenging task for this method. Moreover, the asymptotic error expansions show the method is of order $h^3$. Hence, extrapolation algorithms can be introduced to achieve higher approximation accuracy degree $(\mathcal{O}(h^5))$. Meanwhile, an a posteriori asymptotic error estimate is derived, which can be used to construct self-adaptive algorithms. The numerical examples support our theoretical analysis.