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...

This article reviews several fast algorithms for boundary integral equations. After a brief introduction of the boundary integral equations for the Laplace and Helmholtz equations, we discuss in order the fast multipole method and its kernel independent variant, the hierarchical matrix framework, the wavelet based method, the high frequency fast multipole method, and the recently proposed multi...

The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations u...

We propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is split into one–way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed ...

where p and q are points on S, G is the free-space Green’s function, ∂/∂nq denotes normal differentiation at q, and uinc is the incident field. The Helmholtz integral equations are familiar boundary integral equations, for which there is a complete theoretical framework (Kleinman and Roach, 1974; Colton and Kress, 1983). Many examples of their numerical treatment, usually using boundary element...

In this thesis, we propose and analyze a fast method for computing the solution of the Helmholtz equation in a bounded domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is first split into one–way wave equations which are then solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the soluti...

The inverse problem under consideration is to reconstruct the shape information of obstacles or inclusions embedded in the (inhomogeneous) background medium from boundary measurements of propagating waves. This article is a survey of enclosure-type methods implementing exponential complex geometrical optics waves as boundary illumination. The equations for acoustic waves, electromagnetic waves ...

and a0 is a constant, b is a continuous periodic function of variables x and y, a1 and g are continuous periodic functions. This integral equation appears in exterior boundary value problems for the two-dimensional Helmholtz equation (see [11,15– 17,27]). Constructions of conformal mapping also lead to integral equations of this kind (see [22]). In recent years, the Helmholtz equation and its n...

Novel non-hyper-singular [i.e., only strongly-singular] boundary-integral-equations for the gradients of the acoustic velocity potential, involving only O(r−2) singularities at the surface of a 3-D body, are derived, for solving problems of acoustics governed by the Helmholtz differential equation. The gradients of the fundamental solution to the Helmholtz differential equation for the velocity...

This paper serves as a survey of enclosure-type methods used to determine the obstacles or inclusions embedded in the background medium from the near-field measurements of propagating waves. A type of complex geometric optics waves that exhibits exponential decay with distance from some critical level surfaces (hyperplanes, spheres or other types of level sets of phase functions) are sent to pr...