This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green’s functions, Green’s identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the ...

The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems from bounded scatterers. This paper presents the directional preconditioner for the linear systems of the boundary integral method in two dimensions. This new preconditioner builds a data-sparse approximation of the integral operator, transforms it into a sparse linear system, and comput...

We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nyström’s method for discretization, and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals. The overall asymptotic complexity of our method is O(N), where N is the number of discretization points on the ...

This paper is concerned with the application of boundary integral equation method to the electromagnetic scattering of a perfect conductor in the three dimensional space. A collocation method is employed for the magnetic eld integral equation and error estimates are derived. Fareld patterns and radar cross sections are computed for various wave numbers in the case of sphere. Numerical experimen...

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on ...

A boundary element formulation for the solution of multiple moving boundary problems is presented and tested herein. A heat transfer problem involving heating of solid, melting of solid and vaporisation of liquid is considered. Numerical results show that the boundary element method is more suitable and more accurate than both finite difference and finite element methods for this kind of problems

A time-domain, finite element–boundary integral (FE–BI) method is presented for analyzing electromagnetic (EM) scattering from two-dimensional (2-D) inhomogeneous objects. The scheme’s finite-element component expands transverse fields in terms of a pair of orthogonal vector basis functions and is coupled to its boundary integral component in such a way that the resultant finite element mass ma...

We investigate a three-dimensional mathematical thermoelastic scattering problem from an open surface which will be referred to as a screen. Under the assumption of the local "nite energy of the uni"ed thermoelastic scattered "eld, we give a weak model on the appropriate Sobolev spaces and derive equivalent integral equations of the "rst kind for the jump of some trace operators on the open sur...

We design a boundary integral method for time-dependent, threedimensional, doubly periodic water waves and prove that it converges with O(h3) accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transported according to a computed velocity. An integral equation arising from potential theory is solved for the normal velocity. A new method is deve...

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filte...