In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features o...

The present work deals with high frequency Helmholtz equation resolution using geometrical optics. We give formulaes in dimension 2 and 3 for mixed Dirichlet, Neumann and Robin boundaries conditions.

A survey of different representations for lattice sums for the Helmholtz equation is given. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension dΛ. Lattice sums are related to, and can be calculated from, the quasi-periodic Gree...

We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ( +k)u = 0 on a Lipschitz domain for all wave number k 2C with Imk 0. Following the approach for the case of smooth boundary [3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothness...

Wavelength plays a distinguished role in classical electromagnetic and acoustic scattering. Most significant features of the far field patterns radiated by a collection of sources or scatterers are related to their sizes and relative distances, measured in wavelengths. These significant features are reflected in the invariance of the Helmholtz equation with respect to translation, and its homog...

In this paper, we extend to the time-harmonic Maxwell equations the p–version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a me...

The electrical position of the heart with reference to the electrodes used in studying its field is unknown. For reasons presented. it is more likely eccentric; hence, the equation defining the field of an eccentric dipole in a spherical medium might be useful for projected experimental studies and for better understanding of the way in which a given electrical position determines the electrode...