An iterative solver for the finite‐difference frequency‐domain (FDFD) method for the simulation of materials with negative permittivity

A stable iterative solver for the simulation of optical waves in metals using finite difference frequency domain (FDFD) method is presented. The corresponding discretization of Maxwell's equations enables simulating electromagnetic waves in structures when materials with negative permittivity are involved. Convergence of the iterative solver is proved for positive and negative permittivities. Numerical results are presented for a thin‐film silicon solar cell structure containing silver back contact. Copyright © 2010 John Wiley & Sons, Ltd.     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

The boundary element method in contiguous domains

As an example of a problem involving two finite contiguous domains, the boundary element method is used to compute the eddy current density in a conductor subjected to a transverse magnetic field.     

Global and domain-decomposed mixed methods for the solution of Maxwell's equations with application to magnetotellurics

A collection of global and domain decomposition mixed finite element schemes for the approximate solution of the harmonic Maxwell's equations on a bounded domain with absorbing boundary conditions at the artificial boundaries are presented. The numerical procedures allow us to solve efficiently the direct problem in magnetotellurics consisting of determining the electromagnetic scattered field in a two–dimensional earth model of arbitrary conductivity properties. Convergence results for the numerical procedures are derived. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 407–437, 1998     

Finite element approximation of coupled seismic and electromagnetic waves in fluid‐saturated poroviscoelastic media

This work presents a collection of global and iterative finite element procedures for the numerical approximation of coupled seismic and electromagnetic waves in 2D bounded fluid‐saturated porous media, with absorbing boundary conditions at the artificial boundaries. The equations being analyzed are the coupled Biot's equations of motion and Maxwell equations in the diffusive range. Both seismoelectric and electroseismic coupling are simultaneously included and analyzed in the model. The case of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM‐mode) is analyzed in detail, including the derivation of a priori error estimates on the global finite element procedure and results on the convergence of a domain decomposition iterative algorithm. Later, the corresponding results for the case of horizontally polarized shear waves coupled with the transverse electric polarization (SHTE‐mode) are stated. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

Error estimate and regularity for the compressible Navier‐Stokes equations by Newton's method

The finite element discretization error estimate and H¹ regularity are shown for the solution generated by Newton's method to the stationary compressible Navier‐Stokes equations by interpreting Newton's method as an equivalent iterative method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 511–524, 2003     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

Mixed and hybrid formulations for the three‐dimensional magnetostatic problem

We propose mixed and hybrid formulations for the three‐dimensional magnetostatic problem. Such formulations are obtained by coupling finite element method inside the magnetic materials with a boundary element method. We present a formulation where the magnetic field is the state variable and the boundary approach uses a scalar Dirichlet‐Neumann map to describe the exterior domain. Also, we propose a second formulation where the magnetic induction is the state variable and a vectorial Dirichlet‐Neumann map is used to describe the outer field. Numerical discretizations with “edge” and “face” elements are proposed, and for each discrete problem we study an “inf‐sup” condition. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 85–104, 2002     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Effect of cold plasma permittivity on the radiation of the dominant TEM‐wave by an impedance loaded parallel‐plate waveguide radiator

The effects of wall impedances on the radiation of the dominant transverse electromagnetic wave by an impedance loaded parallel‐plate waveguide radiator immersed in a cold plasma have been analyzed. The solution to the governing mathematical model in cold plasma is determined while using the Wiener–Hopf technique. It is observed that the amplitude of the radiated field increases with increasing permittivity of the plasma. The work presented may be of great interest to quantify the effects of ionosphere plasma on the communicating signals between Earth station and an artificial satellite in the Earth's atmosphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

In this paper, we study a class of tuned preconditioners that will be designed to accelerate both the DACG–Newton method and the implicitly restarted Lanczos method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large‐scale scientific computations. These tuning strategies are based on low‐rank modifications of a given initial preconditioner. We present some theoretical properties of the preconditioned matrix. We experimentally show how the aforementioned methods benefit from the acceleration provided by these tuned/deflated preconditioners. Comparisons are carried out with the Jacobi–Davidson method onto matrices arising from various large realistic problems arising from finite element discretization of PDEs modeling either groundwater flow in porous media or geomechanical processes in reservoirs. The numerical results show that the Newton‐based methods (which includes also the Jacobi–Davidson method) are to be preferred to the – yet efficiently implemented – implicitly restarted Lanczos method whenever a small to moderate number of eigenpairs is required. Copyright © 2016 John Wiley & Sons, Ltd.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation

We consider the time‐dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal‐based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi‐discretized in space to get bounds on stabilization parameters where we distinguish equal‐order elements and Taylor‐Hood elements. Different numerical experiments are performed to illustrate our theoretical findings.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.     

Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media

We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell's equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order formulation for the electric field with an ordinary differential equation for the electric polarization added as an auxiliary constraint. The finite element method uses linear finite elements in space for the electric field as well as the electric polarization, and a theta scheme for the time discretization. Numerical experiments suggest the method is unconditionally stable for both Debye and Lorentz models. We compare the stability and phase error properties of the method presented here with those of finite difference methods that have been analyzed in the literature. We also conduct numerical simulations that verify the stability and dispersion properties of the scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015