High-Order Approximation Of Hypersingular Integrals And Its Application In Electromagnetic Scattering

Hypersingular integrals widely exist in many scientific and engineering problems such as electromagnetic scattering,fracture mechanics,elasticity,electron optics,etc.The key to the above problems is the accurate approximation of hypersingular integrals.With the further research of scholars,Gauss method,Newton-Cotes method,transformation method,extrapolation method and other quadrature methods gradually appeared.According to the different positions of singular points,the methods for solving hypersingular integrals can be divided into two categories: grid-type method and nodal-type method.For the gridtype method,the singular point is required to be inside two grid points,while the nodal method requires the singular point to coincide with a grid point.When using the numerical method to solve the problem,we need to discretize the interval of the hypersingular integral.The coincidence of the singular point and the discrete point is inevitable,which causes difficulties in the calculation of hypersingular integration.In this paper,we propose a piecewise quartic spline quadrature method for hypersingular integrals where the singular points coincide with the grid points,which achieves fourthorder convergence at any point in the interval.Firstly,the integral interval is segmented according to the positions of the singular point,and the quartic spline interpolation polynomial is constructed.By using piecewise quartic spline interpolation polynomial to replace the integrand in the integral,we can obtain the approximate quadrature formula and the error estimate,which is proved theoretically that piecewise quartic spline quadrature rule can reach fourth-order approximation.Then,we analyze the stability of the piecewise quartic spline quadrature rules.The rounding error increases linearly,and this method has good stability.Finally,several numerical examples are given to verify the effectiveness and convergence order of the quadrature rule proposed in this paper.This method is a nodal-type method,which can solve the situation that the singular points coincide with grid points.Since this rule requires no information about the derivatives of the integrand function,we can easily apply it to solve many practical problems.We successfully apply the piecewise quartic spline quadrature rule to solve the electromagnetic scattering from cavities.The cavity scattering is an unbounded domain problem,which is reduced to a bounded domain problem with nonlocal boundary conditions by introducing artificial boundary conditions.The boundary conditions contain hypersingular integral operators.Combining with a fourth-order scheme for the discretization of the Helmholtz equation,we apply the piecewise quartic spline quadrature rule to deal with the hypersingular integral operator on the aperture.The convergence order of the numerical solution reaches fourth order in the whole computational domain of the cavity.Numerical experiments show that the piecewise quartic spline quadrature method can efficiently solve the high-order approximation of hypersingular integrals in nonlocal boundary conditions for scattering problems involving single cavities,partially covered cavities and multiple cavities.