efficient nonlinear solvers for laplace beltrami smoothing of three dimensional unstructured grids the laplace beltrami system of nonlinear elliptic partial differential equations has utility in the generation of computational grids on complex and highly curved geometry discretization of this system using the finite element method accommodates unstructured grids but generates a large sparse ill conditioned system of nonlinear discrete equations the use of the laplace beltrami approach particularly in large scale applications has been limited by the scalability and efficiency of solvers this paper addresses this limitation by developing two nonlinear solvers based on the jacobian free newton krylov jfnk methodology a key feature of these methods is that the jacobian is not formed explicitly for use by the underlying linear solver iterative linear solvers such as the generalized minimal residual gmres method do not technically require the stand alone jacobian instead its action on a vector is approximated through two nonlinear function evaluations the preconditioning required by gmres is also discussed two different preconditioners are developed both of which employ existing algebraic multigrid amg methods further the most efficient preconditioner overall for the problems considered is based on a picard linearization numerical examples demonstrate that these solvers are significantly faster than a standard newton krylov approach a speedup factor of approximately 26 was obtained for the picard preconditioner on the largest grids studied here in addition these jfnk solvers exhibit good algorithmic scaling with increasing grid size

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

a curvilinear fully implicit conservative electromagnetic pic algorithm in multiple dimensions we extend a recently proposed fully implicit pic algorithm for the vlasov darwin model in multiple dimensions chen and chacon 2015 1 to curvilinear geometry as in the cartesian case the approach is based on a potential formulation a and overcomes many difficulties of traditional semi implicit darwin pic algorithms conservation theorems for local charge and global energy are derived in curvilinear representation and then enforced discretely by a careful choice of the discretization of field and particle equations additionally the algorithm conserves canonical momentum in any ignorable direction and preserves the coulomb gauge a 0 exactly an asymptotically well posed fluid preconditioner allows efficient use of large cell sizes which are determined by accuracy considerations not stability and can be orders of magnitude larger than required in a standard explicit electromagnetic pic simulation we demonstrate the accuracy and efficiency properties of the algorithm with numerical experiments in mapped meshes in 1d 3v and 2d 3v

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

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

a vector finite element time domain method for solving maxwell s equations on unstructured hexahedral grids in this paper the vector finite element time domain vfetd method is derived analyzed and validated the vfetd method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density the galerkin method is used to convert maxwell s equations to a coupled system of ordinary differential equations the leapfrog method is used to advance the fields in time the method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids a numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids several computational experiments are performed to determine the accuracy and efficiency of the method

a least squares finite element method for the magnetostatic problem in a multiply connected lipschitz domain a new least squares finite element method is developed for the curl div magnetostatic problem in lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials in order to deal with possibly low regularity of the solution local l 2 projectors are introduced to standard least squares formulation and applied to both curl and div operators coercivity is established by adding suitable mesh dependent bilinear terms as a result any continuous finite elements lower order elements are enriched with suitable bubbles can be employed a desirable error bound is obtained bf u bf u h 0 le c bf u tilde bf u 0 where bf u h and tilde bf u are the finite element approximation and the finite element interpolant of the exact solution bf u respectively numerical tests confirm the theoretical results

spurious solutions in mixed finite element method for maxwell s equations dispersion analysis and new basis functions the finite element method is a well known computational technique used to obtain numerical solutions to boundary value problems including maxwell s equations this paper first presents a brief description of the mathematical structure based on the de rham diagram to discretize maxwell s equations then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to evaluate the presence of spurious solutions for different basis functions these unwanted spurious solutions appear when the same order of element is used for electric and magnetic fields while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields in this work finite elements in both frequency and time domain are studied and the effects of these spurious solutions in both domains are analyzed in one and three dimensional cases

finite volume schemes for friedrichs systems with involutions in applications solutions of systems of hyperbolic balance laws often have to satisfy additional side conditions we consider initial value problems for the general class of friedrichs systems where the solutions are constrained by differential conditions given in the form of involutions these occur in particular in electrodynamics electro and magnetohydrodynamics as well as in elastodynamics neglecting the involution on the discrete level typically leads to instabilities to overcome this problem in electrodynamical applications it has been suggested in munz et al 2000 to solve an extended system here we suggest an extended formulation to the general class of constrained friedrichs systems it is proven for explicit finite volume schemes that the discrete solution of the extended system converges to the weak solution of the original system for vanishing discretization and extension parameter under appropriate scalings moreover we show that the involution is weakly satisfied in the limit the proofs rely on a reformulation of the extension as a relaxation type approximation and careful use of the convergence theory for finite volume methods for systems of friedrichs type numerical experiments illustrate our analytical results

average adjacencies for tetrahedral skeleton regular partitions for any conforming mesh the application of a skeleton regular partition over each element in the mesh produces a conforming mesh such that all the topological elements of the same dimension are subdivided into the same number of child elements every skeleton regular partition has associated special constitutive recurrence equations in this paper the average adjacencies associated with the skeleton regular partitions in 3d are studied in three dimensions different values for the asymptotic number of average adjacencies are obtained depending on the considered partition in contrast with the two dimensional case j comput appl math 140 2002 673 in addition a priori formulae for the average asymptotic adjacency relations for any skeleton regular partition in 3d are provided

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

indirect estimation via l w for a large class of queueing systems little s law l w helps provide a variety of statistical estimators for the long run time average queue length l and the long run customer average waiting time w we apply central limit theorem versions of little s law to investigate the asymptotic efficiency of these estimators we show that an indirect estimator for l using the natural estimator for w plus the known arrival rate is more efficient than a direct estimator for l provided that the interarrival and waiting times are negatively correlated thus extending a variance reduction principle for the gi g s model due to a m law and j s carson we also introduce a general framework for indirect estimation which can be applied to other problems besides l w we show that the issue of indirect versus direct estimation is related to estimation using nonlinear control variables we also show under mild regularity conditions that any nonlinear control variable scheme is equivalent to a linear control va

a dispersion analysis of finite element methods for maxwell s equations a dispersion analysis of six different finite element methods for approximating maxwell s equations in space is presented the study is limited to two space dimensions and triangular elements but all the methods considered can be generalised to three dimensions using prismatic elements spatial grids consisting of both equilateral and right triangles are considered

a new 3 d nonspurious discontinuous galerkin spectral element time domain dg setd method for maxwell s equations a new discontinuous galerkin spectral element time domain dg setd method for maxwell s equations based on the field variables mathbf e and mathbf b is proposed to analyze three dimensional 3 d transient electromagnetic phenomena compared to the previous setd method based on the field variables mathbf e and mathbf h the mathbf eh scheme in which different orders of interpolation polynomials for electric and magnetic field intensities are required the newly proposed method can eliminate spurious modes using basis functions with the same order interpolation for electric field intensity and magnetic flux density the mathbf eb scheme consequently it can reduce the number of unknowns and computation load domain decomposition for the mathbf eb scheme setd method is completed via the dg method in addition the mathbf eb scheme setd method is extended to the well posed time domain perfectly matched layer pml to truncate the computation domain when solving open region problems the effectiveness and advantages of the new dg setd method are validated by eigenvalue analysis and numerical results

approximating the revenue maximization problem with sharp demands we consider the revenue maximization problem with sharp multi demand in which m indivisible items have to be sold to n potential buyers each buyer i is interested in getting exactly d i items and each item j gives a benefit v ij to buyer i we distinguish between unrelated and related valuations in the former case the benefit v ij is completely arbitrary while in the latter each item j has a quality q j each buyer i has a value v i and the benefit v ij is defined as the product v i q j the problem asks to determine a price for each item and an allocation of bundles of items to buyers with the aim of maximizing the total revenue that is the sum of the prices of all the sold items the allocation must be envy free that is each buyer must be happy with her assigned bundle and cannot improve her utility we first prove that for related valuations the problem cannot be approximated to a factor o m 1 e for any e 0 unless p np and that such result is asymptotically tight in fact we provide a simple m approximation algorithm even for unrelated valuations we then focus on an interesting subclass of proper instances that do not contain buyers a priori known not being able to receive any item for such instances we design an interesting 2 approximation algorithm and show that no 2 e approximation is possible for any 0 e 1 unless p np we observe that it is possible to efficiently check if an instance is proper and if discarding useless buyers is allowed an instance can be made proper in polynomial time without worsening the value of its optimal solution

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

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

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

prediction of evanescent coupling efficiency in two parallel silica nanowires abstract in this study the efficiency of evanescent wave coupling between two air clad silica nanowires in single mode operation is numerically predicted in time domain using the finite difference method to this end a three dimensional scheme developed for solving the maxwell s equations on staggered grids will be employed the electric and magnetic field solutions are sought subjected to the zero divergence condition or gauss s law in discrete level in addition we aim to conserve hamiltonians existing in ideal maxwell s equations at all times by adopting the explicit second order accurate symplectic partitioned runge kutta temporal scheme moreover all the spatial derivative terms in the faraday s and ampere s equations are approximated by the proposed scheme which can render not only fourth order spatial accuracy but also minimize the discrepancy between the exact and the derived numerical phase velocities

a parallel finite volume runge kutta algorithm for electromagnetic scattering a 3d explicit finite volume algorithm has been developed to simulate scattering from complex geometries on parallel computers using structured body conformal curvilinear grids most simulations for practical 3d geometries require a large number of grid points for adequate spatial resolution making them suitable to parallel computation the simulations have been carried out using a multi block zonal approach in the message passing paradigm on the sp 2 each zone is placed on a separate processor and inter processor communication is carried out using the message passing library interface mpl mpi integration of maxwell s equations is performed using the four stage runge kutta time integration method on a dual grid this method of integrating on a staggered grid gives enhanced dissipative and dispersive characteristics a scattered field formulation has been used and the liao boundary condition is used at the outer nonreflecting boundary the far zone transformation has also been implemented efficiently using specialized mpl functions to evaluate the far zone scattering results results show extremely good comparisons for scattering from the sphere and the ogive with the exact solution and standard fdtd type algorithms comparisons for nonaxisymmetric targets like the nasa almond with experimental data has also been found to be extremely good

scalable preconditioned conjugate gradient inversion of vector finite element mass matrices mass matrices arise in the numerical solution of time dependent partial differential equations by the galerkin method since these systems must be inverted at each time step rapid inversion algorithms for these systems are important when nodal finite elements are used as basis functions it is known that the mass matrices can be consistently approximated by a diagonal matrix or solved by a scalable conjugate gradient method this may not be the case for other basis functions in this paper we show that the preconditioned conjugate gradient method is scalable when used to invert mass matrices that arise from vector finite element basis functions these basis functions are particularly important for solving maxwell s equations on unstructured grids by the galerkin method

combination can be hard approximability of the unique coverage problem we prove semilogarithmic inapproximability for a maximization problem called unique coverage given a collection of sets find a subcollection that maximizes the number of elements covered exactly once specifically assuming that mathrm np not subseteq operatorname bptime 2 n varepsilon for an arbitrary varepsilon 0 we prove o 1 log sigma n inapproximability for some constant sigma sigma varepsilon we also prove o 1 log 1 3 varepsilon n inapproximability for any varepsilon 0 assuming that refuting random instances of 3sat is hard on average and we prove o 1 log n inapproximability under a plausible hypothesis concerning the hardness of another problem balanced bipartite independent set we establish an omega 1 log n approximation algorithm even for a more general budgeted setting and obtain an omega 1 log b approximation algorithm when every set has at most b elements we also show that our inapproximability results extend to envy free pricing an important problem in computational economics we describe how the budgeted unique coverage problem motivated by real world applications has close connections to other theoretical problems including max cut maximum coverage and radio broadcasting

accurate solutions of maxwell s equations around pec corners and highly curved surfaces using nodal finite elements a method is presented for computing accurate solutions of maxwell s equations in the presence of perfect electrical conductors pecs with sharp corners and highly curved surfaces using conventional nodal finite elements and a scalar vector s v potential formulation this technique approximates the pec with an impedance boundary condition ibc where the impedance is small critically it couples both potentials through this boundary condition rather than setting the scalar potential to zero this permits cancellation of the tangential components of the vector potential resulting in an accurate normal electric field the cause for the inaccuracies that nodal methods experience in the presence of sharp pec corners or highly curved pec surfaces is elucidated it is then shown how the inclusion of the scalar potential cures these deficiencies permitting accurate solutions spectral analysis of the resulting finite element matrices are shown validating the boundary conditions used examples are presented comparing a benchmark solution conventional pec and ibc boundary conditions and the new s v potential ibc on a pec wedge and pec ellipse in both cases the new s v ibc produces superior results

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

least squares finite element method and preconditioned conjugate gradient solution a least squares variational procedure for first order systems of differential equations and an approximate formulation based on finite elements are developed error estimates a condition number bound and analysis of weighting factors are given steepest descent and conjugate gradient solution procedures are examined and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size numerical studies of rates of convergence for a test problem are given

a finite element method for first order elliptic systems in three dimensions abstract we consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions we also discuss div curl type systems in three independent variables both a least squares method and a new subdomain collocation least squares method are considered the analysis yield optimal asymptotic h 1 and l 2 error estimates

evidence of exponential convergence in the computation of maxwell eigenvalues abstract this paper describes a finite element approach for the approximation of maxwell eigenvalues for three dimensional cavity problems hierarchical h 1 and h curl conforming discretisations are employed in conjunction with a technique that reduces the size of the problem to be solved numerical results are given on both polyhedral and curvilinear domains these results indicate that exponential rates of convergence can be obtained with the appropriate combination of h and p refinements

a nodal based finite element approximation of the maxwell problem suitable for singular solutions a new mixed finite element approximation of maxwell s problem is proposed its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term which yields a very weak control on the divergence of the unknown the method is shown to be stable and convergent in the natural h rm mathbf curl 0 omega norm for this unknown in particular convergence also applies to singular solutions for which classical nodal based interpolations are known to suffer from spurious convergence upon mesh refinement

high order numerical method for magnetohydrodynamic control of shock induced separation magnetic fields significantly affect the flow of ionized gases and may be exploited for the control of separation transition and turbulence the wave structure of ionized gas high speed flows under the influence of magnetic fields is more involved than the waves of ordinary gas dynamics and high resolution is required to accurately compute the complex wave interactions of such flows with discontinuities numerical solutions of the nonlinear ideal and viscous magnetohydrodynamic equations are obtained with a high order accurate shock capturing scheme the numerical method is tested for flows containing strong magnetohydrodynamic discontinuities and smooth flow features to ensure that it maintains high order accuracy for the smooth parts of the flow it preserves numerical stability and eliminates nonphysical features that result from the violation of the divergence free condition for the magnetic field problems with exact solutions such as the convection of smooth waves caused by magnetic fields are also solved numerically to validate the method the numerical method is applied to simulate separation control at the interaction region of an oblique shock with a laminar boundary layer under the influence of magnetic fields

elimination of spurious modes in finite element analysis abstract a common cause of spurious non physical modes that arise in either finite difference or finite element derived eigenvalue problems is identified it is shown that these modes are the result of an excessively flexible system the flexibility is removed by constraining the problem an analogy is drawn between electromagnetic wave propagation and acoustic wave propagation in liquids to show that the spurious modes encountered in both cases are due to fundamentally the same cause results are presented to show that constraining the problem yields a significant reduction in the number of spurious modes a big improvement in the quality of the eigenvector of the physical modes and only a marginal increase in the error in the eigenvalue for the low order modes

eigenmode solution of 2 d and 3 d electromagnetic cavities containing absorbing materials using the jacobi davidson algorithm eigenmodes of electromagnetic cavities containing absorbing dielectric materials are determined using an adaptation of the jacobi davidson technique to solve discrete matrix eigenequations derived from maxwell s equations the discretisations obtained using finite difference and finite integration methods give rise to non hermitian matrices having complex eigenvalues and the jacobi davidson method is shown to be applicable even when very low q cavity eigenmodes exist in the presence of highly lossy dielectric or permeable materials examples are given of eigensolutions for both 2 d cylindrically symmetric and 3 d electromagnetic operators

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

on social envy freeness in multi unit markets abstract we consider a market setting in which buyers are individuals of a population whose relationships are represented by an underlying social graph given buyers valuations for the items being sold an outcome consists of a pricing of the objects and an allocation of bundles to the buyers an outcome is social envy free if no buyer strictly prefers the bundles of her neighbors in the social graph we focus on the revenue maximization problem in multi unit markets in which there are multiple copies of the same item being sold and each buyer is assigned a subset of identical items we consider four different cases that arise when considering two different buyers valuations i e single minded or general and by adopting two different forms of pricing that is item or bundle pricing for all the above cases we show the hardness of the revenue maximization problem and give corresponding approximation results all our approximation bounds are optimal or nearly optimal moreover under the assumption of social graphs of bounded treewidth we provide an optimal allocation algorithm for general valuations with item pricing finally we determine optimal bounds on the corresponding price of envy freeness that is on the worst case ratio between the maximum revenue that can be achieved without envy freeness constraints and the one obtainable in case of social relationships some of our results close hardness open questions or improve already known ones in the literature concerning the standard setting without sociality

finite element transport modeling using analytic element flow solutions 1 finite element methods for solute transport simulation typically use a discrete representation of the flow domain obtained from a finite element solution of the associated groundwater flow problem velocity saturated thickness and components of the dispersion coefficient tensor are represented as a set of nodal and or element averaged values in contrast the analytic element method aem provides continuous mesh independent solutions for these variables in this paper a set of techniques for using two dimensional aem flow solutions as the basis of finite element solute transport models is introduced first a general aem based discretization approach is presented that addresses the existence of curved boundaries singularities and discontinuities in vertically averaged concentration second residual integration methods that handle continuous parameters with internal and boundary singularities are developed and evaluated third an approach is introduced for handling internal discontinuities in concentration across certain analytic elements this new approach uses a nonstandard mesh topology and a new formulation for internal coupled boundary conditions the aem based transport simulation methods introduced in this paper are demonstrated to be robust and accurate for a variety of test problems

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

short note boundary conditions for maxwell solvers which is solved in implicit plasma simulations compared with the standard helmholtz equation 4 p68 c t 2 replaces c and eq 1 is solved each time step instead of for each value of for e with s given the source term s is neither divergence nor curl free in general specifically we ask under what conditions charge conservation is satisfied namely do numerical solutions of eq 1 satisfy poisson s equation

a p adaptive method for electromagnetic wave propagation summary the discontinuous galerkin dg finite element method is used for the numerical solution of the three dimensional maxwell equations control of errors in the numerical level for the divergence free constraint of the magnetic field can be obtained through the use of divergence free vector bases in this work the so called perfectly hyperbolic ph formulation of the maxwell equations is employed to retain both divergence free magnetic field and in the presence of charges to satisfy the gauss constraint for the electric field at the numerical level for both approaches it is found that higher order approximations have favorable effect on the preservation of the divergence constraints and that the ph formulations retains these errors to a lower level it is shown that high order accuracy in space and time is achieved in unstructured meshes using implicit time marching for nonuniform meshes local resolution refinement is employed using p type adaptivity to ensure accurate electromagnetic wave propagation thus the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated this article is protected by copyright all rights reserved

a finite element method for magnetohydrodynamics abstract this paper presents a finite element method for the solution of 3d incompressible magnetohydrodynamic mhd flows two important issues are thoroughly addressed first appropriate formulations for the magnetic governing equations and the corresponding weak variational forms are discussed the selected b q formulation is conservative in the sense that the local divergence free condition of the magnetic field is accounted for in the variational sense a galerkin least squares variational formulation is used allowing equal order approximations for all unknowns in the second issue a solution algorithm is developed for the solution of the coupled problem which is valid for both high and low magnetic reynolds numbers several numerical benchmark tests are carried out to assess the stability and accuracy of the finite element method and to test the behavior of the solution algorithm

a quadratic plane triangular element immune to quadratic mesh distortions under quadratic displacement fields abstract a quadratic plane triangular element is developed extending the concepts of us quad8 element the element belongs to the broad class of petrov galerkin formulation which is characterized by the choice of test and trial functions from different spaces the performance of the proposed element has been studied for typical static and free vibration problems the element exhibits a high degree of tolerance to mesh distortions as compared to the classical isoparametric quadratic triangular element for undistorted meshes however the performance is similar to that of the classical element

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

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

a stitching method for the generation of unstructured meshes for use with co volume solution techniques abstract the successful implementation of co volume time domain solution techniques requires the use of high quality smooth dual meshes the generation of delaunay voronoi diagrams with these properties for two dimensional domains of complicated geometrical shape is considered in the adopted approach near boundary regions are discretised according to prescribed criteria and the remainder of the computational domain is discretised using an ideal mesh the two meshes are stitched together to provide a consistent mesh for the complete domain after smoothing the resulting mesh is found to be of a quality which exceeds that of meshes produced by standard automatic unstructured mesh generation methods examples involving electromagnetic wave scattering are included to demonstrate the computational performance that can be achieved with a co volume time domain solution algorithm on these meshes

mesh smoothing using a posteriori error estimates we develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations the algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations in this paper we derive the method present some numerical illustrations and give a brief analysis of the issue of uniqueness

on the adjacencies of triangular meshes based on skeleton regular partitions for any 2d triangulation the 1 skeleton mesh of is the wireframe mesh defined by the edges of while that for any 3d triangulation the 1 skeleton and the 2 skeleton meshes respectively correspond to the wireframe mesh formed by the edges of and the surface mesh defined by the triangular faces of a skeleton regular partition of a triangle or a tetrahedra is a partition that globally applied over each element of a conforming mesh where the intersection of adjacent elements is a vertex or a common face or a common edge produce both a refined conforming mesh and refined and conforming skeleton meshes such a partition divides all the edges and all the faces of an individual element in the same number of edges faces we prove that sequences of meshes constructed by applying a skeleton regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements faces edges and vertices of the nth and n 1 th meshes by using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes number of triangles by node and number of tetrahedra by vertex is constant when n goes to infinity we relate these results with the non degeneracy properties of longest edge based partitions in 2d and include empirical results which support the conjecture that analogous results hold in 3d

nodal based finite element modeling of maxwell s equations weak forms are derived for maxwell s equations which are suitable for implementation on conventional c sup 0 elements with scalar bases the governing equations are expressed in terms of general vector and scalar potentials for the electric field intensity vector gauge theory is invoked to close the system and dictates the continuity requirements for the potentials at material interfaces as well as the blend of boundary conditions at exterior boundaries two specific gauges are presented both of which lead to helmholtz weak forms which are parasite free and enjoy simple physically meaningful boundary conditions a general and numerically efficient procedure for enforcing the jump discontinuities on the normal components of vector fields at dielectric interfaces and boundary conditions on curved surfaces is also given

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

a close look at valences in hexahedral element meshes this paper discusses using valences in objective functions for topological modification of 3d hexahedral meshes for topological optimization of 2d quadrilateral meshes node valence i e number of element edges attached to each node is used to maximize the number of regular nodes i e nodes with four attached edges difficulties in developing 3d hexahedral local topology modifications have limited the success of hexahedral topology optimization although published literature suggests using an object function based on node valence however in this paper we show that node valence is not a consistent measure of good hexahedral element topology and objective functions based on node valence can lead to element topology which will only admit concave element shapes instead we propose that objective functions based on edge valence i e number of quadrilateral faces attached to each element edge will provide a consistent measure of element topology copyright 2010 john wiley sons ltd

automated adaptive time discontinuous finite element method for unsteady compressible airfoil aerodynamics an adaptive finite element methodology for unsteady compressible aerodynamics is presented the automated adaptive environment consists of the finite quadtree automatic mesh generator a time discontinuous galerkin least squares finite element flow solver an error indicator based on interpolation estimates and a mesh enrichment procedure specifically a linear space time wedge element formulation was developed and implemented for two dimensional problems with relative body motion the mesh enrichment procedure is developed to enable the solution process to be spatially adaptive h refinement is edge based and stores no mesh enrichment history

approximation of the thermally coupled mhd problem using a stabilized finite element method a numerical formulation to solve the mhd problem with thermal coupling is presented in full detail the distinctive feature of the method is the design of the stabilization terms which serve several purposes first convective dominated flows in the navier stokes and the heat equation can be dealt with second there is no restriction in the choice of the interpolation spaces of all the variables and finally flows highly coupled with the magnetic field can be accounted for different aspects related to the design of the final fully discrete and linearized algorithm are also discussed

development of a 3d staggered fdtd scheme for solving maxwell s equations in drude medium an explicit finite difference scheme is developed to solve the three dimensional maxwell s equations in drude medium our aim of developing this scheme in time domain is to compute solutions in staggered grids from the faraday s and ampere s equations the electric and magnetic field solutions are sought subject to the discrete zero divergence condition gauss s law the local conservation laws in ideal maxwell s equations are also numerically preserved all the time using the explicit second order accurate symplectic partitioned runge kutta temporal scheme the spatial derivative terms in the faraday s and ampere s equations are discretized to obtain fourth order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error in addition to performing the fundamental analysis on the proposed scheme the computational efficiency and long term accurate properties embedded in the proposed symplectic dispersion error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and drude media

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

two remarks on frequent hypercyclicity abstract we show that if t x x is a continuous linear operator on an f space x 0 then the set of frequently hypercyclic vectors of t is of first category in x and this answers a question of a bonilla and k g grosse erdmann we also show that if t x x is a bounded linear operator on a banach space x 0 and if t is frequently hypercyclic or more generally syndetically transitive then the t orbit of every non zero element of x is bounded away from 0 and in particular t is not hypercyclic

application of reproducing kernel particle methods in electromagnetics abstract motivated by the success of the reproducing kernal particle methods rkpm for solving the helmholtz equation rkpm is used as the interpolating basis functions for solving three dimensional electromagnetics problems the electromagnetics problem is the vector form of the wave equation treatment of the divergenceless condition is discussed as well as accurate treatment of the boundary conditions a simple shorted cylindrical waveguide is solved under te 11 excitation the accuracy of the rkpm is demonstrated for the three dimensional problem

planar mesh refinement cannot be both local and regular we show that two desirable properties for planar mesh refinement techniques are incompatible mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data local refinements are modifications of the mesh that involve a fixed maximum amount of computation independent of the number of triangles in the mesh regular meshes are meshes for which every interior vertex has degree 6 at least for some simple model meshing problems optimal meshes are known to be regular hence it would be desirable to have a refinement technique that if applied to a regular mesh produced a larger regular mesh we call such a technique a regular refinement in this paper we prove that no refinement technique can be both local and regular our results also have implications for non local refinement techniques such as delaunay insertion or rivara s refinement

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

analysis of a least squares finite element method for the circular arch problem1this work was supported in part by nsc grant 88 2119 m 214 001 taiwan 1 the stability and convergence of a least squares finite element method for the circular arch problem with shear deformation in a first order system formulation are investigated it is shown that the least squares finite element approximations are stable and convergent in a natural energy norm associated with the least squares functional for the shallow arch case the optimal order of convergence in the h 1 norm for all the unknowns can be achieved uniformly with respect to the small thickness parameter and thus the locking phenomenon does not occur in this case a simple and sharp a posteriori error estimator is also addressed

mesh relaxation a new technique for improving triangulations summary given a list of points defining a domain boundary a three stage process is often used to triangulate a domain first an appropriate distribution of interior points is generated next the points are connected to form triangles and finally the connectivity data are used to reposition the interior points using the laplacian smoothing technique thereby usually improving the shapes of the triangles this paper describes a new technique for mesh improvement adjusting the connection structure during the second stage of this process the new scheme which we call mesh relaxation consists of a procedure for iteratively making the mesh topology more regular by edge swapping for each interior edge a relaxation index is computed that depends on the degrees of its end points and adjacent points any edge for which this index exceeds a prescribed threshold will be swapped i e replaced by a new edge connecting the adjacent points of the original edge after all edge swaps are completed laplacian smoothing is applied to the mesh examples show that when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes

a bridge between u frequent hypercyclicity and frequent hypercyclicity abstract given a the family of weights a a n n decreasing to 0 such that the series n 0 a n diverges we show that the supremum on a of lower weighted densities coincides with the unweighted upper density and that the infimum on a of upper weighted densities coincides with the unweighted lower density we then investigate the notions of u frequent hypercyclicity and frequent hypercyclicity associated to these weighted densities we show that there exists an operator which is u frequently hypercyclic for each weight in a but not frequently hypercyclic although the set of frequently hypercyclic vectors always coincides with the intersection of sets of u frequently hypercyclic vectors for each weight in a

least squares finite element approximations to the timoshenko beam problem in this paper a least squares finite element method for the timoshenko beam problem is proposed and analyzed the method is shown to be convergent and stable without requiring extra smoothness of the exact solutions for sufficiently regular exact solutions the method achieves optimal order of convergence in the h 1 norm for all the unknowns displacement rotation shear moment uniformly in the small parameter which is generally proportional to the ratio of thickness to length thus the locking phenomenon disappears as the parameter tends to zero a sharp a posteriori error estimator which is exact in the energy norm and equivalent in the h 1 norm is also briefly discussed

improved vector fem solutions of maxwell s equations using grid pre conditioning the time domain vector finite element method is a promising new approach for solving maxwell s equations on unstructured triangular grids this method is sensitive to the quality or condition of the grid in this study grid pre conditioning techniques such as edge swapping laplacian smoothing and energy minimization are shown to improve the accuracy of the solution and also reduce the overall computational effort 1997 john wiley sons ltd

an adaptive method for computing invariant manifolds in non autonomous three dimensional dynamical systems abstract we present a computational method for determining the geometry of a class of three dimensional invariant manifolds in non autonomous aperiodically time dependent dynamical systems the presented approach can be also applied to analyse the geometry of 3d invariant manifolds in three dimensional time dependent fluid flows the invariance property of such manifolds requires that at any fixed time they are given by surfaces in r 3 we focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in r 3 the presented technique can be employed in particular to compute codimension one invariant stable and unstable manifolds of hyperbolic trajectories in 3d non autonomous dynamical systems which are crucial in the lagrangian transport analysis the same approach can also be used to determine evolution of an orientable material surface in a fluid flow these developments represent the first step towards a non trivial 3d extension of the so called lobe dynamics a geometric invariant manifold based framework which has been very successful in the analysis of lagrangian transport in unsteady two dimensional fluid flows in the developed algorithm the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise c 2 interpolating functions the method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution a variant of the advancing front technique is used for remeshing whenever necessary such an approach allows for computationally efficient determination of highly convoluted evolving geometry of codimension one invariant manifolds in unsteady three dimensional flows we show that the developed method is capable of providing detailed information on the evolving lagrangian flow structure in three dimensions over long periods of time which is crucial for a meaningful 3d transport analysis

interior penalty discontinuous galerkin method for maxwell s equations energy norm error estimates we develop the symmetric interior penalty discontinuous galerkin dg method for the time dependent maxwell equations in second order form we derive optimal a priori error estimates in the energy norm for smooth solutions we also consider the case of low regularity solutions that have singularities in space

an approach to the generation of triangular grids possessing few obtuse triangles a triangular mesh generator is presented which makes extensive use of side swapping and mesh smoothing to create a grid with few obtuse triangles to perform the above task a data structure is presented which holds full adjacency information both for nodes and elements it is shown how this data structure is employed in the process mesh generation and how the methods of computational geometry such as region decomposition are used not only to decompose a complex region but also to reflect the boundary grading into the interior the region algorithms are provided to show the mechanism of these processes and practical examples are given to support the approach

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

high order accurate modeling of electromagnetic wave propagation across media grid conforming bodies the maxwell equations contain a dielectric permittivity e that describes the particular media for homogeneous materials at low temperatures this coefficient is constant within a material however it jumps at the interface between different media this discontinuity can significantly reduce the order of accuracy of the numerical scheme we solve the maxwell equations with an interface between two media using a fourth order accurate algorithm we regularize the discontinuous dielectric permittivity by a continuous function either locally near the interface or globally in the entire domain we study the effect of this regularization on the order of accuracy for a one dimensional time dependent problem we then implement this for the three dimensional maxwell equations in spherical coordinates with appropriate physical and artificial absorbing boundary conditions we use fourier filtering of the high frequency modes near the poles to increase the time step

optimal envy free pricing with metric substitutability we study the unit demand envy free pricing problem faced by a profit maximizing seller with unlimited supply when there is metric substitutability among the items consumer i s value for item j is v i c i j and the substitution costs c i j form a metric our model is motivated by the observation that sellers often sell the same product at different prices in different locations and rational consumers optimize the tradeoff between prices and substitution costs while the general envy free pricing problem is hard to approximate we show that the problem of maximizing revenue with metric substitutability among items can be solved exactly in polynomial time we do this by first showing that in any optimal price vector the set of nodes that pay exactly their value uniquely determines which nodes buy an item and what price they pay and therefore the revenue we transform the problem of finding an optimal set of such nodes to an instance of weighted independent set on a perfect graph which can be solved in polynomial time by the strong perfect graph theorem proving the result we then analyze the computational tractability of various extensions to our model we begin with relaxing the metric substitutability requirement and show that when the substitution costs do not form a metric even if a 1 epsilon approximate triangle inequality holds the problem becomes np hard thus the triangle inequality characterizes the threshold at which the problem goes from tractable to hard we then relax assumptions on the supply and demand we consider restricting supplies to a subset of locations or the amount of supplies or allowing buyers to demand more than one unit in all cases the problem becomes np hard in addition the multiunit demand case illustrates an interesting paradoxical nonmonotonicity the optimal revenue the seller can extract can actually decrease when consumers demands increase we show the revenue maximization problem with multiunit demand is apx hard even for the simplest valuations with equal marginal values for all items up to the demand constraint and demands of at most 3

on the influence of numerical integration on mixed finite element approximations of a maxwell eigenvalue problem the behaviour of electromagnetic resonances in cavities is modelled by a maxwell eigenvalue problem evp in the present work we rewrite the corresponding variational problem as it arises with a view to the application of a finite element method in a mixed formulation for the modelling of realistic problems the integrals occurring in this mixed formulation often cannot be evaluated exactly we take into account the error arising from numerical quadrature and show convergence to the approximations using exact integration finally some numerical results are presented

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

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

robust boundary triangulation and delaunay triangulation of arbitrary planar domains a simple and robust boundary triangulation algorithm is proposed and based on it completely automatic delaunay mesh generation procedures are developed the algorithm is equally applicable to convex non convex and multiply connected planar domains in this approach given the nodes the number of triangles formed is precisely known and any desired control over mesh generation is possible

the origin of spurious solutions in computational electromagnetics it is commonly believed that the divergence equations in the maxwell equations are redundant for transient and timeharmonic problems therefore most of the numerical methods in computational electromagnetics solve only two first order curl equations or the second order curl curl equations this misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics by studying the div curl system this paper clarifies that the first order maxwell equations are not overdetermined and the divergence equations must always be included to maintain the ellipticity of the system in the space domain to guarantee the uniqueness of the solution and the accuracy of the numerical methods and to eliminate the infinitely degenerate eigenvalue this paper shows that the common derivation and usage of the second order curl curl equations are incorrect and that the solution of helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary the div curl method and the least squares method introduced in this paper provide rigorous derivation of the equivalent second order maxwell equations and their boundary conditions the node based least squares finite element method lsfem is recommended for solving the first order full maxwell equations directly examples of the numerical solutions by lsfem are given to demonstrate that the lsfem is free of spurious solutions