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.     

Finite element study of time‐dependent Maxwell's equations in dispersive media

In this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain

The solution of Maxwell's equations in a non‐convex polyhedral domain is less regular than in a smooth or convex polyhedral domain. In this paper we show that this solution can be decomposed into the orthogonal sum of a singular part and a regular part, and we give a characterization of the singular part. We also prove that the decomposition is linked to the one associated to the scalar Laplacian. Copyright © 1999 John Wiley & Sons, Ltd.     

On well‐posedness for some thermo‐piezoelectric coupling models

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost‐effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well‐posedness of such models. There is a need to extend the use of such sensors into high‐temperature environments, and so this paper shows, for a broad class of models, the well‐posedness of the magneto‐electro‐thermo‐elastic problem. Because of its widespread use in the literature, we also show the well‐posedness of the quasi‐electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.     

Exterior problem of magnetic field in Darwin model and its numerical solution

In this paper, we consider the solutions of magnetic field in the Darwin model to the Maxwell's equations in 2D unbounded domain. We first deduce the variational formulation and prove the well‐posedness of the weak solution, and then prove the existence and uniqueness of the infinite element solution. Error estimate and the numerical examples are provided. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Analysis of a FitzHugh–Nagumo–Rall model of a neuronal network

Pursuing an investigation started in (Math. Meth. Appl. Sci. 2007; 30 :681–706), we consider a generalization of the FitzHugh–Nagumo model for the propagation of impulses in a network of nerve fibres. To this aim, we consider a whole neuronal network that includes models for axons, somata, dendrites, and synapses (of both inhibitory and excitatory type). We investigate separately the linear part by means of sesquilinear forms, in order to obtain well posedness and some qualitative properties. Once they are obtained, we perturb the linear problem by a nonlinear term and we prove existence of local solutions. Qualitative properties with biological meaning are also investigated. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of the energy‐conserved S‐FDTD scheme for variable coefficient Maxwell's equations in disk domains

In this paper, we analyze the energy‐conserved splitting finite‐difference time‐domain (FDTD) scheme for variable coefficient Maxwell's equations in two‐dimensional disk domains. The approach is energy‐conserved, unconditionally stable, and effective. We strictly prove that the EC‐S‐FDTD scheme for the variable coefficient Maxwell's equations in disk domains is of second order accuracy both in time and space. It is also strictly proved that the scheme is energy‐conserved, and the discrete divergence‐free is of second order convergence. Numerical experiments confirm the theoretical results, and practical test is simulated as well to demonstrate the efficiency of the proposed EC‐S‐FDTD scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

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     

Asymptotic behaviour of the energy for electromagnetic systems in the presence of small inhomogeneities

In this article we consider solutions to the time-harmonic and time-dependent Maxwell's systems with piecewise constant coefficients with a finite number of small inhomogeneities in ?³. In time-harmonic case and for such solutions, we derive the asymptotic expansions due to the presence of small inhomogeneities embedded in the entire space. Further, we analyse the behaviour of the electromagnetic energy caused by the presence of these inhomogeneities. For a general time-dependent case, we show that the local electromagnetic energy, trapped in the total collection of these well-separated inhomogeneities, decays towards zero as the shape parameter decreases to zero or as time increases.     

A macroscopic model for an intermediate state between type‐I and type‐II superconductivity

A vectorial nonlocal and nonlinear parabolic problem on a bounded domain for an intermediate state between type‐I and type‐II superconductivity is proposed. The domain is for instance a multiband superconductor that combines the characteristics of both types. The nonlocal term is represented by a (space) convolution with a singular kernel arising in Eringen's model. The nonlinearity is coming from the power law relation by Rhyner. The well‐posedness of the problem is discussed under low regularity assumptions and the error estimate for a semi‐implicit time‐discrete scheme based on backward Euler approximation is established. In the proofs, the monotonicity methods and the Minty–Browder argument are used. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1551–1567, 2015     

Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity

In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ? → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

New results on global asymptotic stability for a class of delayed Nicholson's blowflies model

This paper is concerned with a class of non‐autonomous delayed Nicholson's blowflies model, which is defined on the positive function space. Under proper conditions, we employ a novel proof to establish several criteria on the global asymptotic stability of zero equilibrium point for this model. Moreover, we give an example and its numerical simulation to illustrate our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

A priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain

The equations for the evolution of electromagnetic fields in chiral media, in the time domain, are nonlocal in time. In this work we study the validity of a singular limit (local in time) approximation for these nonlocal in time equations, by estimating the size of the difference of the fields as predicted by both models. In particular, we establish an a priori estimate for this difference, depending on the time horizon, properties of the domain, spatial properties of the initial data and the source terms and the chirality measure β of the approximating model.     

Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves

A domain integral method employing a specific Green's function (i.e. incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab‐like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially‐varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two‐layer slab. Copyright © 2006 John Wiley & Sons, Ltd.     

Fully discrete A‐ϕ finite element method for Maxwell's equations with nonlinear conductivity

This article is devoted to the study of a fully discrete A ‐ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of an inverse initial boundary-value problem for the full time-dependent Maxwell's equations in the presence of imperfections of small volume

We consider the numerical solution, in a three-dimensional bounded domain, of the inverse problem for identifying the location of small electromagnetic imperfections in a medium with homogeneous background. Our numerical algorithm is based on the coupling of a discontinuous Galerkin method for the time-dependent Maxwell's equations, on the exact controllability method and on a Fourier inversion. Several numerical results are given with one and two imperfections and the robustness and accuracy of the numerical method used for the dynamic detection problem are shown.