on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

on isospectral periodic potentials in rn ii on considere un reseau a n dimensions l dans r n et un potentiel q satisfaisant q x d q x pour d l on considere spec k q l ensemble des avec multiplicites telles que u qu u a une solution satisfaisant u x d u x exp 2i k d d l

a mixed finite element formulation for maxwell s equations in the time domain abstract a galerkin finite element solution technique for the maxwell s equations is discussed this new formulation can be viewed as a generalization of certain staggered grid finite difference schemes to arbitrary meshes it is shown that this technique is simple to implement and is more accurate as well as more cost effective than the standard equal order finite element approach numerical are presented to evaluate the performance of this new element relative to the standard element

the steiner tree problem i formulations compositions and extension of facets in this paper we give some integer programming formulations for the steiner tree problem on undirected and directed graphs and study the associated polyhedra we give some families of facets for the undirected case along with some compositions and extensions we also give a projection that relates the steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph this is used to show that the lp relaxation of the directed formulation is superior to the lp relaxation of the undirected one

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

locally conformal finite difference time domain techniques for particle in cell plasma simulation the dey mittra s dey r mitra a locally conformal finite difference time domain fdtd algorithm for modeling three dimensional perfectly conducting objects ieee microwave guided wave lett 7 273 1997 finite difference time domain partial cell method enables the modeling of irregularly shaped conducting surfaces while retaining second order accuracy we present an algorithm to extend this method to include charged particle emission and absorption in particle in cell codes several examples are presented that illustrate the possible improvements that can be realized using the new algorithm for problems relevant to plasma simulation

hp adaptive finite elements in electromagnetics a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for curvilinear elements is presented the proposed method is illustrated and verified by a series of 2d experiments which include elements with curved boundaries and nonhomogeneous media

solving group steiner problems as steiner problems abstract the generalized spanning tree or group steiner problem gsp is a generalization of the steiner problem in graphs spg one requires a tree spanning at least one vertex of each subset given in a family of vertex subsets while minimizing the sum of the corresponding edge costs specialized solution procedures have been developed for this problem in this paper we investigate the performance of a known but so far neglected transformation to the undirected steiner problem in graphs when combined with a recent metaheuristic for the spg this straightforward approach compares favorably with specialized gsp heuristics thus we set a standard for future algorithms

a method for incorporating gauss lasw into electromagnetic pic codes abstract an algorithm is presented which solves a modified set of maxwell s equations none of them elliptic in an electromagnetic pic type code the algorithm prevents large violations of gauss law by diffusing away numerical errors arising from the particle to grid apportionment method there are no restrictions on the manner in which charge and current densities are determined from the particle distribution in particular point particles and simple linear interpolation are used

dispersion reducing methods for edge discretizations of the electric vector wave equation we present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time domain electric vector wave equation on square meshes based on the mimetic finite difference mfd method we compare this strategy called m adaptation to two other discretizations also based on square meshes one is the lowest order nedelec edge element discretization the other is a modified quadrature approach gy adaptation proposed by guddati and yue for the acoustic wave equation in two dimensions all three discrete methods use the same edge based degrees of freedom while the temporal discretization is performed using the standard explicit leapfrog scheme to obtain efficient and explicit time stepping methods the three schemes are further mass lumped we perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error our results indicate that the method produced by gy adaptation and the nedelec method are both second order accurate for numerical dispersion but differ in the order of their numerical anisotropy fourth order versus second order respectively the result of m adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy numerical simulations are provided that illustrate the theoretical results

an analysis of ne de lec s method for the spatial discretization of maxwell s equations abstract in 1980 nedelec developed a family of curl and divergence conforming finite elements in r 3 he proposed the use of these elements to discretize the time dependent maxwell equations noting that the elements have the advantage that the discrete magnetic displacement can be made exactly divergence free in this paper we shall analyze a slight generalization of nedelec s scheme and prove essentially optimal order convergence estimates in a variety of situations we also demonstrate that the nedelec method can be superconvergent at certain special points and we relate the method to yee s finite difference scheme a by product of our analysis will be a convergence proof for yee s method

global superconvergence for maxwell s equations in this paper the global superconvergence is analysed on two schemes a mixed finite element scheme and a finite element scheme for maxwell s equations in r 3 such a supercovergence analysis is achieved by means of the technique of integral identity which has been used in the supercovergence analysis for many other equations and schemes on a rectangular mesh and then are generalized into more general domains and problems with the variable coefficients besides being more direct our analysis generalizes the results of monk

a new relaxation method for the generalized minimum spanning tree problem we consider a generalization of the minimum spanning tree problem called the generalized minimum spanning tree problem denoted by gmst it is known that the gmst problem is hard we present several mixed integer programming formulations of the problem based on a new formulation of the problem we give a new solution procedure that finds the optimal solution of the gmst problem for graphs with nodes up to 240 we discuss the advantages of our approach in comparison with earlier methods

a high order mixed vector finite element method for solving the time dependent maxwell equations on unstructured grids we present a mixed vector finite element method for solving the time dependent coupled ampere and faraday laws of maxwell s equations on unstructured hexahedral grids that employs high order discretization in both space and time the method is of arbitrary order accuracy in space and up to 4th order accurate in time making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods in addition the method correctly models both the jump discontinuities and the divergence free properties of the electric and magnetic fields is charge and energy conserving conditionally stable and free of spurious modes several computational experiments are performed to demonstrate the accuracy efficiency and benefits of the method

numerical electroseismic modeling a finite element approach abstract electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid saturated porous rock due to the electrokinetic phenomenon this work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid saturated media the model involves the simultaneous solution of biot s equations of motion and maxwell s equations in a bounded domain coupled via an electrokinetic coefficient with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries the 3d case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem apriori error estimates for a continuous time finite element procedure based on parallelepiped elements are derived with maxwell s equations discretized in space using the lowest order mixed finite element spaces of nedelec while for biot s equations a nonconforming element for each component of the solid displacement vector and the vector part of the raviart thomas nedelec of zero order for the fluid displacement vector are employed a fully implicit discrete time finite element method is also defined and its stability is demonstrated the results are also extended to the case of tetrahedral elements the 2d cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization psvtm mode and horizontally polarized shear waves coupled with the transverse electric polarization shte mode are also formulated and the corresponding finite element spaces are defined the 1d shte initial boundary value problem is also formulated and approximately solved using a discrete time finite element procedure which was implemented to obtain the numerical examples presented

elements finis mixtes incompressibles pour l equation de stokes dans ℝ3 we introduce some new families of finite element approximation for the stationary stokes and navier stokes equations in a bounded domain in 3 these elements can used tetahedrons or cubes the approximation satisfie exactly the incompressibility condition

hybrid block amr in cartesian and curvilinear coordinates mhd applications abstract we present a novel hybrid block adaptive scheme for use in solving sets of near conservation laws in general orthogonal coordinate systems the adaptive mesh refinement amr scheme is block based i e individual grids have a pre fixed number of grid cells and is implemented for any dimensionality d its hybrid character relaxes the common approach where a block that needs refinement triggers 2 d subblocks when grids are refined by a factor of 2 this introduces incomplete families in the grid hierarchy but approaches the optimal fit to developing flow features inherent in the original patch based amr strategy our hybrid block amr approach is compared with a patch based amr one both exploiting openmp parallelism the implementation is able to handle general curvilinear coordinates for which restriction and prolongation formulae are presented along with boundary treatments at singular boundaries demonstrative examples cover hydro and magnetohydrodynamic mhd applications including tests on a 2d polar grid a 2 5d spherical and a 3d cylindrical configuration the applications cover classical up to relativistic mhd simulations of particular relevance for astrophysical magnetized jet and stellar wind studies

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

stable spectral methods on tetrahedral elements a framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented although most of the developments are of a general nature an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron in the first part the question of well behaved multivariate polynomial interpolation on the tetrahedron is addressed and it is shown how to extend the electrostatic analogy of the jacobi polynomials to problems beyond the line this allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and subsequently for the formulation of stable spectral schemes on such unstructured nodal sets the second part of this work is devoted to a discussion of weakly imposed boundary conditions and energy stable schemes are formulated for a wide class of problems exemplified by advection problems advection diffusion problems and linear symmetric hyperbolic systems finally in the third part issues related to computational efficiency and implementation of the schemes are discussed the spectral accuracy of the approximation is confirmed through an example and factorization methods for the efficient computation of derivatives on the general nodal sets within the d simplex are developed ensuring that the proposed schemes are competitive with tensor product based methods in this last part we also show that the advective operator results in an cal o n 2 restriction on the time step similar to that of spectral collocation methods employing a tensor product based approximation the performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain confirming the expected accuracy and stability

an unsplit cell centered godunov method for ideal mhd we present a second order godunov algorithm for multidimensional ideal mhd our algorithm is based on the unsplit formulation of colella j comput phys 87 1990 with all of the primary dependent variables centered at the same location to properly represent the divergence free condition of the magnetic fields we apply a discrete projection to the intermediate values of the field at cell faces and apply a filter to the primary dependent variables at the end of each time step we test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions the test suite includes rotated planar linear waves mhd shock tube problems low beta flux tubes and a magnetized rotor problem for all of these cases we observe that the algorithm is second order accurate for smooth solutions converges to the correct weak solution for problems involving shocks and exhibits no evidence of instability or loss of accuracy due to the possible presence of non solenoidal fields

a simplicial gauge theory we provide an action for gauge theories discretized on simplicial meshes inspired by lattice gauge theory and finite element methods the action is discretely gauge invariant and we give a proof of consistency a discrete noether s theorem that can be applied to our setting is also proved

multiscale asymptotic method for maxwell s equations in composite materials in this paper we discuss the multiscale analysis of maxwell s equations in composite materials with a periodic microstructure the new contributions in this paper are the determination of higher order correctors and the explicit convergence rate for the approximate solutions see theorem 2 3 consequently we present the multiscale finite element method and derive the convergence result see theorem 4 1 the numerical results demonstrate that higher order correctors are essential for solving maxwell s equations in composite materials

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

superconvergence analysis for maxwell s equations in dispersive media in this paper we consider the time dependent maxwell s equations in dispersive media on a bounded three dimensional domain global superconvergence is obtained for semi discrete mixed finite element methods for three most popular dispersive media models the isotropic cold plasma the one pole debye medium and the two pole lorentz medium global superconvergence for a standard finite element method is also presented to our best knowledge this is the first superconvergence analysis obtained for maxwell s equations when dispersive media are involved

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

upper and lower bounding strategies for the generalized minimum spanning tree problem abstract we address the generalized minimum spanning tree problem gmst which requires spanning at least one vertex out of every set of disjoint vertices in a graph we show that the geometric version of this problem is np hard and we propose two stochastic heuristics the first one is a very fast randomized greedy search algorithm and the second one being a genetic algorithm also we investigate some existing integer programming formulations and present an new one a new lagrangian based lower bound is proposed and implemented to assess the performance of the heuristics computational experiments performed on a large set of randomly generated instances with up to 1000 vertices and 10 000 edges provide evidence of the good performance of the proposed heuristics

on the constructive determination of spectral invariants of the periodic schrodinger operator with smooth potentials in this paper we develop the asymptotic formulae obtained in my previous papers for the band functions and the bloch functions of the schrodinger operator with the smooth periodic potentials then using these formulae we determine constructively a family of spectral invariants of this operator from the given band functions some of these invariants generalize the well known invariants and others are entirely new the new invariants are explicitly expressed by fourier coefficients of the potential which present the possibility of determining the potential constructively by using the band functions as input data

spectral methods on arbitrary grids stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations these new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions specifically we show how to implement legendre galerkin legendre collocation and laguerre galerkin methodology on arbitrary grids

a two level solution approach for solving the generalized minimum spanning tree problem abstract in this paper we are addressing the generalized minimum spanning tree problem denoted by gmstp which is a variant of the classical minimum spanning tree mst problem the main characteristic of this problem is the fact that the vertices of the graph are partitioned into a given number of clusters and we are looking for a minimum cost tree spanning a subset of vertices which includes exactly one vertex from each cluster we describe a two level solution approach for solving the gmstp obtained by decomposing the problem into two logical and natural smaller subproblems an upper level global subproblem and a lower level local subproblem and solving them separately the goal of the first subproblem is to determine global trees spanning the clusters using a genetic algorithm with a diploid representation of the individuals while the goal of the second subproblem is to determine the best tree w r t cost minimization for the above mentioned global trees spanning exactly one vertex from each cluster the second subproblem is solved optimally using dynamic programming extensive computational results are reported and discussed for an often used set of benchmark instances the obtained results show an improvement in the quality of the achieved solutions and demonstrate the efficiency of our approach compared to the existing methods from the literature

preface to the special issue recent advances in numerical methods for hyperbolic partial differential equations hyperbolic partial differential equations pde are a very powerful mathematical tool to describe complex dynamic processes in science and engineering very often hyperbolic pde can be derived directly from first principles in physics such as the conservation of mass momentum and energy these principles are universally accepted to be valid and can be used for the simulation of a very wide class of different problems ranging from astrophysics rotating gas clouds the merger of a binary neutron star system into a black hole and the associated generation and propagation of gravitational waves over geophysics generation and propagation of seismic waves after an earthquake landslides avalanches tidal waves storm surges flooding and morphodynamics of rivers to engineering turbulent flows over aircraft and the associated noise generation and propagation rotating flows in turbo machinery multi phase liquid gas flows in internal combustion engines and computational biology blood flow in the human cardio vascular system it is of course impossible to cover all the above topics in a single special issue however all these apparently different applications have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations possibly containing also higher order derivative terms non conservative products and nonlinear potentially stiff source terms from the mathematical point of view the major difficulties in these systems arise due to the inherent nonlinearities and the formation of non smooth solution features such as e g shock waves the construction of robust and accurate numerical methods for such type of problems is even after decades of successful research an ongoing quest this quest is fueled by recent advances in the development of novel methods with promising additional properties such as e g high spatial order on unstructured meshes algorithmic simplicity for modern multi core architectures and automatic mesh and or trial function adaptation the final goal is to construct methods that efficiently produce reliable results for such type of problems this special issue is dedicated to recent advances in numerical methods for such nonlinear systems of hyperbolic pde and tries to cover a wide spectrum of different problems and numerical approaches

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

error estimates for a numerical scheme for ferromagnetic problems we propose a finite element method for approximating the nonlinear equations describing the electromagnetic field in a ferromagnetic material using energy arguments we prove an optimal convergence rate for the method assuming a sufficiently smooth electromagnetic field we also verify that the finite element solution satisfies various conservation conditions appropriate to the continuous problem finally we provide some numerical examples

a continuity preserving and divergence cleaning algorithm based on purely and damped hyperbolic maxwell equations in inhomogeneous media in the numerical solution of maxwell s equations for dynamic electromagnetic fields the two gauss s laws are usually not considered since they are a natural consequence of faraday s and maxwellampre s laws if the charge conservation law is satisfied however when the charge conservation law is not satisfied the numerical errors of gauss s laws will increase unbounded leading to numerical instability or breakdown unfortunately the charge conservation law can be easily violated in self consistent waveparticle simulations in the meantime the violation of gauss s laws will also result in an increased error in the normal flux continuity in the simulations of pure electromagnetic problems the satisfaction of tangential field continuity across a material interface is sufficient to yield accurate numerical results however in a self consistent waveparticle simulation the normal components are as important as the tangential components since they are critical in predicting the particle kinetics in this paper a divergence cleaning method is presented to enforce gauss s laws and normal flux continuity by introducing auxiliary variables and damping terms into maxwell s equations in inhomogeneous media which yield hyperbolic and mixed hyperbolicparabolic maxwell equations the numerical solution schemes of the resulting purely and damped hyperbolic maxwell equations are introduced under the framework of the discontinuous galerkin time domain method it is shown through several numerical examples that the proposed method is able to preserve the continuity conditions for both tangential field components and normal flux components and is effective in cleaning the numerical errors in both gauss s laws

a finite volume method for the maxwell equations in the time domain a finite volume scheme for the maxwell equations is proposed using collocated nonorthogonal curvilinear grid arrangements with the advantage that all components of the electric and magnetic field are stored at the same computational location and time it is based on construction principles of high resolution schemes for hyperbolic conservation laws and takes into account the local wave propagation the implementation of boundary conditions based on the characteristic theory is described a new divergence correction technique is proposed to preserve locally the charge conservation this is important if the maxwell solver is used within an electromagnetic particle in cell pic code the accuracy and efficiency of this finite volume framework is demonstrated with problems of electromagnetic wave propagation and of the self consistent motion of charged particles

eigenvalue approximation by a mixed method for resonant inhomogeneous cavities with metallic boundaries for an inhomogeneous cavity bounded by a perfect conductor we prove that the approximation of the eigenvalues for the maxwell problem leads to a second order rate of convergence when using mixed finite elements if the cavity has a disconnected boundary the problem has null eigenvalues we verify the existence of null eigenvalues for the approximate problem they do not mix with the others that still converge at the same rate

generalized spanning trees abstract in this paper we propose a definition for the generalized minimal spanning tree gmst of a graph the gmst requires spanning at least one node out of every set of disjoint nodes node partition in a graph the analysis of the gmst problem is motivated by real life agricultural settings related to construction of irrigation networks in desert environments we prove that the gmst problem is np hard and examine a number of heuristic solutions for this problem computational experiments comparing these heuristics are presented

phase accuracy comparisons and improved far field estimates for 3 d edge elements on tetrahedral meshes abstract edge element methods have proved very effective for 3 d electromagnetic computations and are widely used on unstructured meshes however the accuracy of standard edge elements can be criticised because of their low order this paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3 d edge element approximations in comparison to other methods scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements and a new formulation of the far field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations

electromagnetics via the taylor galerkin finite element method on unstructured grids traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences these so called fdtd finite difference time domain methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid an unstructured grid allows one to place the grid points anywhere one chooses so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh in this paper we present a technique for solving maxwell s equations on an unstructured grid based on the taylor galerkin finite element method we present several numerical examples which reveal the fundamental accuracy and adaptability of the method although our examples are in two dimensions the techniques and results generalize readily to 3d

on the constructive determination of the periodic potentials from the bloch eigenvalues in this paper we consider the three dimensional schrodinger operator with a periodic relative to a lattice of potential q we construct a set d of trigonometric polynomials such that d is dense in where s 3 in the topology any element q of the set d can be determined constructively and uniquely modulo inversion and translation q x q x q x q x where from the given bloch eigenvalues of the schrodinger operator with the potential q

an efficient vector finite element method for nonlinear electromagnetic modeling we have developed a mixed vector finite element method vfem for maxwell s equations with a nonlinear polarization term the method allows for discretization of complicated geometries with arbitrary order representations of the b and e fields in this paper we will describe the method and a series of optimizations that significantly reduce the computational cost additionally a series of test simulations will be presented to validate the method finally a nonlinear waveguide mode mixing example is presented and discussed

a positive muscl hancock scheme for ideal magnetohydrodynamics we present a highly robust second order accurate scheme for the euler equations and the ideal mhd equations the scheme is of predictor corrector type with a muscl scheme following as a special case the crucial ingredients are an entropy stable approximate riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure for multidimensional mhd a new discrete form of the powell source terms is vital to ensure the stability properties the numerical examples show that the scheme has superior stability compared to standard schemes while maintaining accuracy in particular the method can handle very low values of pressure i e low plasma b or high mach numbers and low mass densities

discontinuous galerkin algorithms for fully kinetic plasmas abstract we present a new algorithm for the discretization of the non relativistic vlasov maxwell system of equations for the study of plasmas in the kinetic regime using the discontinuous galerkin finite element method for the spatial discretization we obtain a high order accurate solution for the plasma s distribution function time stepping for the distribution function is done explicitly with a third order strong stability preserving runge kutta method since the vlasov equation in the vlasov maxwell system is a high dimensional transport equation up to six dimensions plus time we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution including the use of a reduced high order basis set a series of benchmarks from simple wave and shock calculations to a five dimensional turbulence simulation are presented to verify the efficacy of our set of numerical methods as well as demonstrate the power of the implemented features

peter monk s contributions to numerical analysis and maxwell s equations abstract peter monk has greatly contributed to the numerical analysis of partial differential equations and integral equations his name is mainly connected to maxwell s equations still his work has been fundamental in the advancement of numerical methods also for several other applications in this article i will try to discuss some of peter s contributions to these fields

a tabu search heuristic for the generalized minimum spanning tree problem this paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem gmstp known to be np hard given a graph whose vertex set is partitioned into clusters the gmstp consists of designing a minimum cost tree spanning all clusters an attribute based tabu search heuristic employing new neighborhoods is proposed an extended set of tsplib test instances for the gmstp is generated and the heuristic is compared with recently proposed genetic algorithms the proposed heuristic yields the best results for all instances moreover an adaptation of the tabu search algorithm is proposed for a variation of the gmstp in which each cluster must be spanned at least once

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

a mixed method for approximating maxwell s equations a semidiscrete mixed finite element approximation to the time dependent maxwell s equations on a bounded smooth domain is analyzed a variational problem for the electric and magnetic fields in which the boundary conditions are enforced naturally is derived then a general convergence result for mixed methods is proven and it is shown how this result may be used to prove various error estimates when nedelec s curl conforming finite elements are used jz3 with connected boundary f and unit outward normal n let e x and 1 x denote respectively the dielectric constant and magnetic permeability of the medium occupying q let a x denote the conductivity of the medium then if e x t and h x t denote respectively the electric and magnetic fields maxwell s equations 10 state that

overlapping schwarz methods for maxwell s equations in three dimensions two level overlapping schwarz methods are considered for finite element problems of 3d maxwell s equations nedelec elements built on tetrahedra and hexahedra are considered once the relative overlap is fixed the condition number of the additive schwarz method is bounded independently of the diameter of the triangulation and the number of subregions a similar result is obtained for a multiplicative method these bounds are obtained for quasi uniform triangulations in addition for the dirichlet problem the convexity of the domain has to be assumed our work generalizes well known results for conforming finite elements for second order elliptic scalar equations

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

ideal glm mhd about the entropy consistent nine wave magnetic field divergence diminishing ideal magnetohydrodynamics equations abstract the paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics first we show how to extend the ideal magnetohydrodynamics mhd equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics as a byproduct of these derivations we show that not all of the commonly used divergence cleaning extensions of the ideal mhd equations are thermodynamically consistent secondly we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy it includes a mechanism to control the discrete divergence error of the magnetic field by construction and is galilean invariant we implement the new high order mhd solver in the adaptive mesh refinement code flash where we compare the divergence cleaning efficiency to the constrained transport solver available in flash unsplit staggered mesh scheme

incompressible finite element methods for navier stokes equations with nonstandard boundary conditions in r 3 application des methodes a elements finis incompressibles aux equations de navier stokes avec des conditions aux limites non standards dans r 3 utilisation du potentiel vecteur de la vorticite et de la pression estimations d erreurs

a block preconditioner for an exact penalty formulation for stationary mhd the magnetohydrodynamics mhd equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas this system of non self adjoint nonlinear pdes couples the navier stokes equations for fluids and maxwell s equations for electromagnetics there has been recent interest in fully coupled solvers for the mhd system because they allow for fast steady state solutions that do not require pseudo time stepping when the fully coupled system is discretized the strong coupling can make the resulting algebraic systems difficult to solve requiring effective preconditioning of iterative methods for efficiency in this work we consider a finite element discretization of an exact penalty formulation for the stationary mhd equations posed in two dimensional domains this formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a lagrange multiplier we consider extending block preconditioning techni

high order well balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres wave propagation in idealized stellar atmospheres is modeled by the equations of ideal mhd together with the gravity source term the waves are modeled as small perturbations of isothermal steady states of the system we consider a formulation of ideal mhd based on the godunov powell form with an embedded potential magnetic field appearing as a parameter the equations are discretized by finite volume schemes based on approximate riemann solvers of the hll type and upwind discretizations of the godunov powell source terms local hydrostatic reconstructions and suitable discretization of the gravity source term lead to a well balanced scheme i e a scheme which exactly preserves a discrete version of the relevant steady states higher order of accuracy is obtained by employing suitable minmod eno and weno reconstructions based on the equilibrium variables to construct a well balanced scheme the resulting high order well balanced schemes are validated on a suite of numerical experiments involving complex magnetic fields the schemes are observed to be robust and resolve the complex physics well

a convergence analysis of yee s scheme on nonuniform grids the yee scheme is the principal finite difference method used in computing time domain solutions of maxwell s equations on a uniform grid the method is easily seen to be second order convergent in space this paper shows that the yee scheme is also second order convergent on a nonuniform mesh despite the fact that the local truncation error is nodally only of first order

two coupled particle finite volume methods using delaunay voronoi meshes for the approximation of vlasov poisson and vlasov maxwell equations this paper deals with the approximation of vlasov poisson and vlasov maxwell equations we present two coupled particle finite volume methods which use the properties of delaunay voronoi meshes these methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices

numerical approximation of the maxwell equations in inhomogeneous media by ap1conforming finite element method the aim of this paper is to present an extension of thep1conforming finite element method for the time dependent maxwell equations that has been previously exposed we shall consider inhomogeneous media with the piecewise constant dielectric and magnetic parameters and precisely the interface of two such media by analogy with the idea developed we propose a method based on the dualization and then the approximation of the interface conditions in a way consistent with the one derived for the fields and the lagrange multipliers of the divergence constraints inside the domain

semi discrete galerkin approximation method applied to initial boundary value problems for maxwell s equations in anisotropic inhomogeneous media in this paper the semi discrete galerkin approximation of initial boundary value problems for maxwell s equations is analysed for the electric field a hyperbolic system of equations is first derived the standard galerkin method is applied to this system and a priori error estimates are established for the approximation

gauss s law satisfying energy conserving semi implicit particle in cell method abstract the energy conserving semi implicit method ecsim introduced by lapenta 2017 has many advantageous properties compared to the classical semi implicit and explicit pic methods most importantly energy conservation eliminates the growth of the finite grid instability we have implemented ecsim in a different and more efficient manner than the original approach more importantly we have addressed two major shortcomings of the original ecsim algorithm there is no mechanism to enforce gauss s law and there is no mechanism to reduce the numerical oscillations of the electric field a classical approach to satisfy gauss s law is to modify the electric field and its divergence using either an elliptic or a parabolic hyperbolic correction based on the generalized lagrange multiplier method this correction however violates the energy conservation property and the oscillations related to the finite grid instability reappear in the modified ecsim scheme we invented a new alternative approach the particle positions are modified instead of the electric field in the correction step displacing the particles slightly does not change the energy conservation property while it can satisfy gauss s law by changing the charge density we found that the new gauss s law satisfying energy conserving semi implicit method gl ecsim produces superior results compared to the original ecsim algorithm in some simulations however there are still some numerical oscillations present in the electric field we attribute this to the simple finite difference discretization of the energy conserving implicit electric field solver we modified the spatial discretization of the field solver to reduce these oscillations while only slightly violating the energy conservation properties we demonstrate the improved quality of the gl ecsim method with several tests

arbitrary order edge elements for electromagnetic scattering simulations using hybrid meshes and a pml the use of arbitrary order edge elements for the simulation of two dimensional electromagnetic scattering problems on hybrid meshes of triangles and quadrilaterals is described single frequency incident waves generated by a source in the far field are considered and the solution is determined in the frequency domain for numerical simulation the solution domain is truncated at a finite distance from the perfectly conducting scatterer and the non reflecting boundary condition at the truncated boundary is imposed by the use of a perfectly matched layer pml several examples are included to demonstrate the performance of the proposed procedure copyright 2002 john wiley sons ltd

handling the divergence constraints in maxwell and vlasov maxwell simulations the aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the vlasov maxwell equations and the maxwell equations with given sources these methods can be classified in two types field correction methods and source correction methods the field correction methods introduce new unknowns in the equations for which additional boundary conditions are in some cases non trivial to find the source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete gauss law that needs to be defined in accordance with the discretization of the maxwell propagation operator

an algorithm for finding the periodic potential of the three dimensional schrodinger operator from spectral invariants in this paper we investigate the three dimensional schr dinger operator with a periodic relative to a lattice of potential q a special class v of the periodic potentials is constructed which is easily and constructively determined from the spectral invariants first we give an algorithm for the unique determination of the potential q v of the three dimensional schr dinger operator from the spectral invariants that were determined constructively from the given bloch eigenvalues then we consider the stability of the algorithm with respect to the spectral invariants and bloch eigenvalues finally we prove that there are no other periodic potentials in the set of large class of functions whose bloch eigenvalues coincides with the bloch eigenvalues of q v

short note an efficient solver for the equations of resistive mhd with spatially varying resistivity we regularize the variable coefficient helmholtz equations arising from implicit time discretizations for resistive mhd in a way that leads to a symmetric positive definite system uniformly in the time step standard centered difference discretizations in space of the resulting pde leads to a method that is second order accurate and that can be used with multigrid iteration to obtain efficient solvers

edge element methods for maxwell s equations with strong convergence for gauss laws in this paper we propose and investigate some edge element approximations for three maxwell systems in three dimensions the stationary maxwell equations the time harmonic maxwell equations and the time dependent maxwell equations these approximations have three novel features first the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of nite element meshes including the stationary maxwell equations second they ensure the optimal strong convergence of the gauss laws in some appropriate norm in addition to the standard optimal convergence in energy norm under the general weak regularity assumptions that hold for both convex and non convex polyhedral domains and for the discontinuous coecients that may have large jumps across the interfaces between dierent media finally no saddle point discrete systems are needed to solve for the stationary maxwell equations unlike most existing edge element schemes

nonlinear magnetohydrodynamics simulation using high order finite elements a conforming representation composed of 2d finite elements and finite fourier series is applied to 3d nonlinear non ideal magnetohydrodynamics using a semi implicit time advance the self adjoint semi implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments growth rates for resistive tearing modes with experimentally relevant lundquist number are computed accurately with time steps that are large with respect to the global alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree p two or larger an error diffusion method controls the generation of magnetic divergence error convergence studies show that this approach is effective for continuous basis functions with p 2 where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity is computed accurately with p 3 without mesh alignment a simulation of tearingmode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3d magnetic topologies

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported

implicit runge kutta methods and discontinuous galerkin discretizations for linear maxwell s equations in this paper we consider implicit runge kutta methods for the time integration of linear maxwell s equations we first present error bounds for the abstract cauchy problem which respect the unboundedness of the differential operators using energy techniques the error bounds hold for algebraically stable and coercive methods such as gauss and radau collocation methods the results for the abstract evolution equation are then combined with a discontinuous galerkin discretization in space using upwind fluxes for the case that permeability and permittivity are piecewise constant functions we show error bounds for the full discretization where the constants do not deteriorate if the spatial mesh width tends to zero