Optical diffraction for periodical interface belongs to relatively fewer exploited application of boundary integral equations method. Our contribution presents the formulation of diffraction problem based on vector tangential fields, for which the periodical Green function of Helmholtz equation is of key importance. There are discussed properties of obtained boundary operators with singular kernel and a numerical implementation is proposed.
In this paper, we introduce a new approach for the convergence problem of optimized Schwarz methods by studying a generalization of these methods for a semilinear elliptic equation. We study the behavior of the algorithm when the overlapping length is large.
We consider the homogeneous time-dependent Oseen system in the whole space . The initial data is assumed to behave as , and its gradient as , when tends to infinity, where is a fixed positive number. Then we show that the velocity decays according to the equation , and its spatial gradient decreases with the rate , for tending to infinity, uniformly with respect to the time variable . Since these decay rates are optimal even in the stationary case, they should also be the best possible...
Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on , and meshes. It is shown that the combination FEM yields (up to a factor ) the same order of accuracy in the associated...
A proof is given of the following theorem: infinitely differentiable solenoidal vector - functions are dense in the space of functions, which are solenoidal in the distribution sense only. The theorem is utilized in proving the convergence of a dual finite element procedure for Dirichlet, Neumann and a mixed boundary value problem of a second order elliptic equation.