An ‘upwind’ finite element method for electromagnetic field problems in moving media

Abstract

Time periodic finite element solutions for sinusoidally excited electroniagnetic field problems in moving media are presented. Solutions by the Galerkin method contain spurious oscillations when the grid Peclet number is more than one. To suppress these osillations an upwind finite element method using two different time periodic test functions is introduced. One is multiplied to second and first order space derivative terms and the other to the time derivative term. Test functions are obtained from trial functions by adding or subtracting quadratic bias functions with appropriate scaling factors. Phasc diffcrcnces are considered between trial functions and bias functions. For simple interpretations of thc phase differences complex scaling factors are used. The proposed rnethod is developed to give nodally exact solutions for uniform grid spacing in one dimensional problems. Based on the one dimensional results a two dimensional upwinding scheme is also derived.

Numerical Methods for Singularly Perturbed Differential Equations

“…Applications involving (1.1) arise for example in modelling water quality problems in river networks [BBG81], simulation of oil extraction from underground reservoirs [Ew83] -both of these examples come from a consideration of the linearized Navier-Stokes equations of fluid dynamics (see [Hi88] or [KL89]), convective heat transport problems with large Peclet numbers [Ja59], electromagnetic field problems in moving media [HBS87], and semiconductor device modelling [PHS87]. A closely related problem, where one of the boundary conditions is at x = 00, appears in the study of unsteady hydromagnetic flow over a continuous moving flat surface for large suction Reynolds number [VR90].…”

Numerical solution of singularly perturbed convection–diffusion problem using parameter uniform B-spline collocation method

“…Singularly perturbed convection-diffusion problems arise in various branches of science and engineering such as modelling of water quality problems in river networks [2], fluid flow at high Reynolds numbers [10], convective heat transport problem with large Péclet numbers [11], drift diffusion equation of semiconductor device modelling [21], electromagnetic field problem in moving media [8], financial modelling [3] and turbulence model [16]. Normally, boundary and interior layers are present in the solutions of such problems, when the singular perturbation parameter ε is small.…”

Robust Numerical Methods for Singularly Perturbed Differential Equations

“…Applications involving (1.1) arise for example in the linearized Navier-Stokes equations of fluid dynamics [Hir88,KL04], simulation of oil extraction from underground reservoirs [Ewi83], convective heat transport problems with large Péclet numbers [Jak59], electromagnetic field problems in moving media [HBS87], miscible and multiphase flows [EW01], semiconductor device modelling [MRS90], and meteorology [Sal98]. We discuss this phenomenon more rigorously immediately after Theorem 2.6.…”

“…If one tries to solve (1.1) using standard numerical methods for partial differential equations, then very inaccurate solutions are obtained unless the mesh discretization used is extremely fine (see [HBS87] for an example). That is, the situation is just as for the singularly perturbed ordinary differential equations of Part I: in order to get inexpensive but accurate numerical results, it will be necessary to devise methods that can cope with boundary and interior layers.…”