A meshless method for the solution of 2D Helmholtz equation has been developed by using the Boundary Integral Equation (BIE) combined with Radial Basis Function (RBF) interpolations. BIE is applied by using the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source point always in the ...

The radiation pattern of a circular loop source of alternating current located in the lower half-space of two conductive media is considered. A solution of the linear Maxwell’s equations in the frequency domain, using the Hankel transform to modified Helmholtz wave equations, was obtained. This leads to a system of ordinary differential equations with respect to the vertical coordinate, conside...

The vertical mode expansion method (VMEM) [J. Opt. Soc. Am. A31, 293 (2014)] is a frequency-domain numerical method for solving Maxwell's equations in structures that are layered separately in a cylindrical region and its exterior. Based on expanding the electromagnetic field in one-dimensional vertical modes, the VMEM reduces the original three-dimensional problem to a two-dimensional (2D) pro...

A classical problem of free-space Green’s function G0Λ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G0Λ and for the expansion coefficients DL of G0Λ in the basis of regular (cyli...

In this paper, we prove uniform Morrey-Campanato type estimates for this equation, using a multiplier method borrowed from [9]. These bounds encode in the optimal way the decay: |u(x)| ∼ 1/|x| d−1 2 at infinity of the solution u. They imply weighted L-estimates for the solution u. We also prove a uniform L-estimate without weight for the trace of the solution on the interface, which states that...

The idea of replacing a divergence constraint by a divergence boundary condition is investigated. The connections between the formulations are considered in detail. It is shown that the most common methods of using divergence boundary conditions do not always work properly. Necessary and sufficient conditions for the equivalence of the formulations are given. ∗This work was supported by the Uni...

Approximate separable representations of Green’s functions for differential operators is a basic and an important aspect in the analysis of differential equations and in the development of efficient numerical algorithms for solving them. Being able to approximate a Green’s function as a sum with few separable terms is equivalent to the existence of low rank approximation of corresponding discre...

We present a multilevel high order ADI method for separable generalized Helmholtz equations. The discretization method we use is a onedimensional fourth order compact finite difference applied to each directional component of the Laplace operator, resulting in a discrete system efficiently solvable by ADI methods. We apply this high order difference scheme to all levels of grids, and then start...

This paper focuses on the multilevel projection method (PML) applied to numerical solution of the basic equations of uid dynamics formulated as least-squares problems for rst-order systems. The fundamental approach taken here is to pose the uid ow equations in their rst-order form, incorporate additional (usually redundant) equations where they improve the character of the formulation, and deen...