Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$, and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$-elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.

We show the incompressible Navier–Stokes–Maxwell system with solenoidal Ohm's law can be derived from the two-fluid incompressible Navier–Stokes–Maxwell system when the momentum transfer coefficient tends to zero. The strategy is based on the decay and dissipative properties of the electromagnetic field.

This work deals with the numerical resolution of the M1-Maxwell system in the quasi-neutral regime. In this regime the stiffness of the stability constraints of classical schemes causes huge calculation times. That is why we introduce a new stable numerical scheme consistent with the transitional and limit models. Such schemes are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the Asymptotic-Preserving scheme in different regimes. The first one corresponds to a regime where electromagnetic effects are predominant. The second one on the contrary shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

The implicit 2D3V particle-in-cell (PIC) code developed to study the interaction of ultrashort pulse lasers with matter [G. M. Petrov and J. Davis, Computer Phys. Comm. 179, 868 (2008); Phys. Plasmas 18, 073102 (2011)] has been parallelized using MPI (Message Passing Interface). The parallelization strategy is optimized for a small number of computer cores, up to about 64. Details on the algorithm implementation are given with emphasis on code optimization by overlapping computations with communications. Performance evaluation for 1D domain decomposition has been made on a small Linux cluster with 64 computer cores for two typical regimes of PIC operation: “particle dominated”, for which the bulk of the computation time is spent on pushing particles, and “field dominated”, for which computing the fields is prevalent. For a small number of computer cores, less than 32, the MPI implementation offers a significant numerical speed-up. In the “particle dominated” regime it is close to the maximum theoretical one, while in the “field dominated” regime it is about 75-80 % of the maximum speed-up. For a number of cores exceeding 32, performance degradation takes place as a result of the adopted 1D domain decomposition. The code parallelization will allow future implementation of atomic physics and extension to three dimensions.

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lamé, and the semi-discrete eddy current equations.

We prove the discrete compactness property of the edge elements of any order on a classof anisotropically refined meshes on polyhedral domains. The meshes, made up oftetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl.Sci. 21 (1998) 519–549]. They are appropriately graded nearsingular corners and edges of the polyhedron.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

We develop a well-posedness theory for second order systems in bounded domains whereboundary phenomena like glancing and surface waves play an important role. Attempts havepreviously been made to write a second order system consisting of nequations as a larger first order system. Unfortunately, the resulting first order systemconsists, in general, of more than 2n equations which leads to manycomplications, such as side conditions which must be satisfied by the solution of thelarger first order system. Here we will use the theory of pseudo-differential operatorscombined with mode analysis. There are many desirable properties of this approach: (1) thereduction to first order systems of pseudo-differential equations poses no difficulty andalways gives a system of 2n equations. (2) We can localize the problem,i.e., it is only necessary to study the Cauchy problem and halfplaneproblems with constant coefficients. (3) The class of problems we can treat is much largerthan previous approaches based on “integration by parts”. (4) The relation betweenboundary conditions and boundary phenomena becomes transparent.

In this paper, we model laser-gas interactions and propagation in some extreme regimes. After a mathematical study of a micro-macro Maxwell-Schrödinger model [1] for short, high-frequency and intense laser-gas interactions, we propose to improve this model by adding a plasma equation in order to precisely take into account free electron effects. We examine if such a model can predict and explain complex structures such as filaments, on a physical and numerical basis. In particular, we present in this paper a first numerical observation of nonlinear focusing effects using an ab-initio gas representation and linking our results with existing nonlinear models.

Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex.In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes.As for Whitney edge elements of degree one, the basis is expressedonly in terms of the barycentric coordinates of the simplex.

Electro-muscular disruption (EMD) devices such as TASER M26 andX26 have been used as a less-than-lethal weapon. Such EMD devicesshoot a pair of darts toward an intended target to generate anincapacitating electrical shock. In the use of the EMD device,there have been controversial questions about its safety andeffectiveness. To address these questions, we need to investigatethe distribution of the current density J inside the targetproduced by the EMD device. One approach is to develop acomputational model providing a quantitative and reliable analysisabout the distribution of J. In this paper, we set up amathematical model of a typical EMD shock, bearing in mind that weare aiming to compute the current density distribution inside thehuman body with a pair of inserted darts. The safety issue ofTASER is directly related to the magnitude of |J| at the regionof the darts where the current density J is highlyconcentrated. Hence, fine computation of J near the dart isessential. For such numerical simulations, serious computationaldifficulties are encountered in dealing with the darts having twodifferent very sharp corners, tip of needle and tip of barb. Theboundary of a small fishhook-shaped dart inside a largecomputational domain and the presence of corner singularitiesrequire a very fine mesh leading to a formidable amount ofnumerical computations. To circumvent these difficulties, wedeveloped a multiple point source method of computing J. It hasa potential to provide effective analysis and more accurateestimate of J near fishhook-shaped darts. Numerical experimentsshow that the MPSM is just fit for the study of EMD shocks.

When it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

During the development of a parallel solver for Maxwell equations by integral formulations and Fast Multipole Method (FMM), we needed to optimize a critical part including a lot of communications and computations. Generally, many parallel programs need to communicate, but choosing explicitly the way and the instant may decrease the efficiency of the overall program. So, the overlapping of computations and communications may be a way to reduce this drawback. We will see a implementation of this techniques using dynamic and adaptive overlapping based on the EasyMSG high level C++ library over MPI, a case of SPMD programming.

The topic of this paper is the numerical analysis of timeperiodic solution for electro-magnetic phenomena. The Limit Absorption Method (LAM)which forms the basis of our study is presented. Theoreticalresults have been proved in the linear finite dimensional case. Thismethod is applied to scattering problems and transport of chargedparticles.

We consider solutions to the time-harmonic Maxwell's Equationsof a TE (transverse electric) nature. For such solutions we providea rigorous derivation of the leading order boundary perturbationsresulting from the presence of a finite number of interior inhomogeneitiesof small diameter. We expect that these formulas will form the basis for very effective computational identification algorithms, aimed at determininginformation about the inhomogeneities from electromagnetic boundary measurements.

We investigate time harmonic Maxwell equations in heterogeneous media, where thepermeability μ and the permittivity ε are piecewise constant. Theassociated boundary value problem can be interpreted as a transmission problem. Ina very natural way the interfaces can have edges and corners. We give a detaileddescription of the edge and corner singularities of the electromagnetic fields.

In this paper we are interested in the numerical modelingof absorbing ferromagnetic materialsobeying the non-linear Landau-Lifchitz-Gilbert law with respect to the propagation and scattering of electromagnetic waves.In this workwe consider the 1D problem. We first show that the corresponding Cauchy problemhas a unique global solution. We then derive a numerical scheme based on an appropriate modificationof Yee's scheme, that we show to preserve some importantproperties of the continuous model such as the conservation of the normof the magnetization and the decay of the electromagnetic energy.Stability is proved under a suitable CFL condition.Some numerical results for the 1D model are presented.

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.