In computer simulation of particular systems (e.g. in physics, biology, economics), partial differential equations (PDE) are solved numerically, mainly by applying Finite Element (FE) or Finite Difference (FD) Analysis. This methods lead to algebraic systems of equations with large, sparse system matrices that have to be solved. Since direct solvers can not exploit the sparsity when the structu...

Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexi...

Nonpolynomial finite element methods for Helmholtz problems have seen much attention in recent years in the engineering and mathematics community. The idea is to use instead of standard polynomials Trefftz-type basis functions that already satisfy the Helmholtz equation, such as plane waves [17], Fourier-Bessel functions [8] or fundamental solutions [4]. To approximate the inter-element interfa...

In this paper, the ultraspherical operational matrices of derivatives are constructed. Based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems. The basic idea behind the suggested algorithms is basically built on transforming the eq...

The use of Fast Multipole Methods (FMM) combined with embedded Krylov solvers preconditioned by a sparse approximate inverse is investigated for the solution of large linear systems arising in industrial acoustic and electromagnetic simulations. We use a boundary elements integral equation method to solve the Helmholtz and the Maxwell equations in the frequency domain. The resulting linear syst...