The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

We consider the initial boundary value problem for the heat equation in a region with in nite and nite boundaries (direct problem) and the related problem to reconstruct the nite boundary from Cauchy data on the in nite boundary (inverse problem). The numerical solution of the direct problem is realized by a boundary integral equation method. For an approximate solution of the inverse problem w...

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. We analyze the mathematical relation between the h-h/2-error estimator from [8], the two-level error estimator from [15], and the averaging error estimator from [3]. We essentially show that all of these are equivalent, and we extend the analysis of [15] to cover adaptive mesh-r...

We consider solving the sound-soft scattering problem for the Helmholtz equation with the h– version of the boundary element method using the standard second-kind combined-field integral equations. We obtain sufficient conditions for the relative best approximation error to be bounded independently of k. For certain geometries, these rigorously justify the commonly-held belief that a fixed numb...

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu,...

We consider a class of potential problems on a periodic half-space for the modeling of electrified oil films, which are used in the development of novel switchable liquid optical devices (diffraction gratings). A boundary integral formulation which reduces the problem to the study of the oil-air interface alone is derived and solved in a highly efficient manner using the Nyström method. The oil...

We propose a discrete collocation method for the boundary integral equations which arise from solving Laplace's equation u = 0. The Laplace's equation is deened on connected regions D in R 3 with a smooth boundary S. The piecewise polynomial interpolation in the parametrization variables along with the collocation method is used, and a numerical integration scheme for collocation integrals is g...