Pushed by the rising interest in high resolution requirements and high-dimensional applications, the diffusion term in the Laplacian equation drives the condition number of the associated discretized operator to undesirable sizes for standard iterative methods to converge rapidly. In addition, for realistic values of the wavenumber k(x) in (1), the Helmholtz operator H becomes indefinite, destr...

We consider in this paper approximation properties and applications of Mathieu functions. A first set of optimal error estimates are derived for the approximation of periodic functions by using angular Mathieu functions. These approximation results are applied to study the Mathieu-Legendre approximation to the modified Helmholtz equation and Helmholtz equation. Illustrative numerical results co...

Talk Abstract We present an efficient integral equation method approach to solve the forced heat equation, ut(x) − ∆u(x) = F (x, u, t), in a two dimensional, multiply connected domain, with Dirichlet boundary conditions. We first discretize in time, which is known as Rothe’s method, resulting in a non-homogeneous modified Helmholtz equation that is solved at each time step. We formulate the sol...

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) wide-angle (WA) beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator, which gives evanescent waves the desired damping. The resulting approach allows more accurate approximation...

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 analyze in detail two-grid methods for solving the 1D Helmholtz equation discretized by a standard finite-difference scheme. We explain why both basic components, smoothing and coarse-grid correction, fail for high wave numbers, and show how these components can be modified to obtain a convergent iteration. We show how the parameters of a two-step Jacobi method can be chosen to yield a stabl...

Within the framework of shifted-Laplace preconditioners [Erlangga, Vuik, Oosterlee, Appl. Numer. Math., 50(2004), pp.409–425] for the Helmholtz equation, different methods for the approximation of the inverse of a complex-valued Helmholtz operator are discussed. The performance of the preconditioner for Helmholtz problems at high wavenumbers in heterogeneous media is evaluated. Comparison with ...