On a weakly singular integral equation and its application

AbstractA direct function theoretic method is applied to solve a weakly singular integral equation whose kernel involves logarithmic singularity. This method avoids the occurrence of strong singularity. The solution of this integral equation is then applied to re-investigate the well known problem of water wave scattering by a partially immersed vertical barrier

An improvement of the product integration method for a weakly singular Hammerstein equation

We present a new method to solve nonlinear Hammerstein equations with weakly singular kernels. The process to approximate the solution, followed usually, consists in adapting the discretization scheme from the linear case in order to obtain a nonlinear system in a finite dimensional space and solve it by any linearization method. In this paper, we propose to first linearize, via Newton method, the nonlinear operator equation and only then to discretize the obtained linear equations by the product integration method. We prove that the iterates, issued from our method, tends to the exact solution of the nonlinear Hammerstein equation when the number of Newton iterations tends to infinity, whatever the discretization parameter can be. This is not the case when the discretization is done first: in this case, the accuracy of the approximation is limited by the mesh size discretization. A Numerical example is given to confirm the theorical result

ZZ-type aposteriori error estimators for adaptive boundary element methods on a curve

In the context of the adaptive finite element method (FEM), ZZ-error estimators named after Zienkiewicz and Zhu are mathematically well-established and widely used in practice. In this work, we propose and analyze ZZ-type error estimators for the adaptive boundary element method (BEM). We consider weakly-singular and hyper-singular integral equations and prove, in particular, convergence of the related adaptive mesh-refining algorithms

Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

In this paper, we consider the boundary integral equation (BIE) method for solving the exterior Neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the Fredholm integral equations of the first kind. The innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper-singular boundary integral operators (BIO) associated with the time-harmonic elastic and thermoelastic wave equations. With the help of the new regularized formulations, we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations. The accuracy of the regularized formulations is demonstrated through numerical examples using the Galerkin boundary element method (BEM).Comment: 24 pages, 6 figure