fixed points of uniformly lipschitzian mappings abstract two fixed point theorems for uniformly lipschitzian mappings in metric spaces due respectively to e lifsic and to t c lim and h k xu are compared within the framework of the so called cat 0 spaces it is shown that both results apply in this setting and that lifsic s theorem gives a sharper result also a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces a class which includes those possessing property p of lim and xu

fixed points of asymptotic contractions abstract let r r be a contractive gauge function in the sense that is continuous s 0 and if f m m satisfies d f x f y d x y for all x y in a complete metric space m d then f always has a unique fixed point it is proved that if t m m satisfies d t n x t n y n d x y x y m where each n is continuous and n uniformly on the range of d then t has a unique fixed point and moreover all of the picard iterates of t converge to this fixed point

pointwise lipschitzian mappings in uniformly convex and uniformly smooth banach spaces abstract let x be a uniformly convex and uniformly smooth banach space let t c c be an asymptotic pointwise nonexpansive mapping where c is a bounded closed and convex subset of x in this paper we investigate conditions sufficient for the weak convergence of the generalized mann and ishikawa processes to a fixed point of t

an upwinding embedded boundary method for maxwell s equations in media with material interfaces 2d case we propose a new upwinding embedded boundary method to solve time dependent maxwell s equations in media with material interfaces a global second order finite difference method is obtained by combining central difference schemes away from the interfaces and upwinding technique with jump conditions near the interfaces the proposed finite difference method allows time step based on a uniform mesh independent of the locations and shapes of the interfaces moreover the scheme is simple to implement in multidimensional cases numerical tests of wave equations with various types of material interfaces and electromagnetic scattering of 2d cylinders confirm the stability uniform accuracy and ease of implementation of the method

study of a 3d ginzburg landau functional with a discontinuous pinning term abstract in a convex domain r 3 we consider the minimization of a 3 d ginzburg landau type energy e e u 1 2 u 2 1 2 e 2 a 2 u 2 2 with a discontinuous pinning term a among h 1 c maps subject to a dirichlet boundary condition g h 1 2 s 1 the pinning term a r 3 r takes a constant value b 0 1 in an inner strictly convex subdomain of and 1 outside we prove energy estimates with various error terms depending on assumptions on and g in some special cases we identify the vorticity defects via the concentration of the energy under hypotheses on the singularities of g the singularities are polarized and quantified by their degrees which are 1 vorticity defects are geodesics computed w r t a geodesic metric d a 2 depending only on a joining two paired singularities of g p i n i where is a minimal connection computed w r t a metric d a 2 of the singularities of g and p 1 p k are the positive resp n 1 n k are the negative singularities

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

a pseudospectral method for time domain computation of electromagnetic scattering by bodies of revolution we present a multidomain pseudospectral method for the accurate and efficient time domain computation of scattering by body of revolution bor perfectly electrically conducting objects in the bor formulation of the maxwell equations the azimuthal dependence of the fields is accounted for analytically through a fourier series the numerical scheme in the r z plane is developed in general curvilinear coordinates and the method of characteristics is applied for patching field values in the individual subdomains to obtain the global solution a modified matched layer method is used for terminating the computational domain and special attention is given to proper treatment of the coordinate singularity in the scattered field formulation and correct time domain boundary conditions along edges numerical results for monochromatic plane wave scattering by smooth and nonsmooth axis symmetric objects including spheres cone spheres and finite cylinders is compared with results from the literature illustrating the accuracy and computational efficiency associated with the use of properly constructed spectral methods to emphasize the versatility of the presented framework we compute plane wave scattering by a missile and find satisfactory agreement with method of moment mom computations

energy stable and high order accurate finite difference methods on staggered grids for wave propagation over distances of many wavelengths high order finite difference methods on staggered grids are widely used due to their excellent dispersion properties however the enforceme

a quantization property for static ginzburg landau vortices for any critical point of the complex ginzburg landau functional in dimension 3 we prove that for large coupling constants if the energy of this critical point on a ball of a given radius r is relatively small compared to r log then the ball of half radius contains no vortex the modulus of the solution is larger than we then show how this property can be applied to describe limiting vortices as e 0 2001 john wiley sons inc

difference approximations for the second order wave equation difference approximations are derived for the second order wave equation in one and two space dimensions without first writing it as a first order system both the dirichlet and the neumann problems are treated for the one dimensional case relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary general two dimensional domains are considered for the dirichlet problem where the domain is embedded in a cartesian grid and the boundary conditions are approximated by interpolation a stable conservative scheme is derived where the time step is determined only by the interior discretization formula discretization cells cut by the boundary are treated implicitly but the resulting scheme becomes explicit because the implicit dependence only is pointwise numerical examples are provided to verify the stability and accuracy of the proposed method

high order fdtd methods via derivative matching for maxwell s equations with material interfaces this paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high order finite difference time domain fdtd schemes of computational electromagnetics cem with material interfaces in one 1d and two spatial dimensions 2d by making use of fictitious points systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage to arbitrarily high orders of accuracy in principle while often limited by numerical instability orders up to 16 and 12 are achieved respectively in 1d and 2d detailed stability analyses are presented for the present approach to examine the upper limit in constructing embedded fdtd methods as natural generalizations of the high order fdtd schemes the proposed derivative matching methods automatically reduce to the standard fdtd schemes when the material interfaces are absent an interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code another feature of the present approach is that it can be robustly implemented with other high accuracy time domain approaches such as the multiresolution time domain method and the local spectral time domain method to cope with material interfaces numerical experiments on both 1d and 2d problems are carried out to test the convergence examine the stability access the efficiency and explore the limitation of the proposed methods it is found that operating at their best capacity the proposed high order schemes could be over 2000 times more efficient than their fourth order versions in 2d in conclusion the present work indicates that the proposed hierarchical derivative matching methods might lead to practical high order schemes for numerical solution of time domain maxwell s equations with material interfaces

large solutions for harmonic maps in two dimensions we seek critical points of the functionale u mathop smallint limits omega u 2 where is the unit disk in 2 andu s2 satisfies the boundary conditionu on we prove that if is not a constant thene has a local minimum which is different from the absolute minimum we discuss in more details the case where x y rx ry sqrt 1 r 2 andr 1

a second order dispersive fdtd algorithm for transverse electric maxwell s equations with complex interfaces this work overcomes the difficulty of the finite difference time domain fdtd algorithm in solving the transverse electric te maxwell s equations with inhomogeneous dispersive media for such te problems the electric fields are discontinuous across the dispersive interfaces moreover such discontinuities are time variant a novel matched interface and boundary time domain mibtd method is proposed to solve such problems through new developments in both mathematical formulations and numerical algorithms mathematically instead of handling all zeroth and first order jump conditions in a local coordinate we directly construct the te jump conditions which are needed in the fdtd computations in the cartesian coordinate such cartesian direction conditions depend on the time as well as tangential and normal components of the electric flux driven by the jump condition modeling we adopt the standard maxwell s equations in coupling with the debye auxiliary differential equations for the electric flux as the governing equations computationally the leapfrog scheme is employed for integrating the maxwell system and time dependent jump conditions sophisticated interface treatments are developed in both producing the te jump conditions and enforcing them in the fdtd algorithm based on a staggered yee lattice the numerical accuracy stability and efficiency of the proposed scheme are investigated by considering dispersive interfaces of various shapes the mibtd method achieves a spatially second order of convergence in all tests to the best of our knowledge the present mibtd scheme is the first fdtd method in the literature that can restore second order accuracy in treating curved dispersive interfaces for the te maxwell system

a high order accurate parallel solver for maxwell s equations on overlapping grids a scheme for the solution of the time dependent maxwell s equations on composite overlapping grids is described the method uses high order accurate approximations in space and time for maxwell s equations written as a second order vector wave equation high order accurate symmetric difference approximations to the generalized laplace operator are constructed for curvilinear component grids the modified equation approach is used to develop high order accurate approximations that use only three time levels and have the same time stepping restriction as the second order scheme discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed the implementation is optimized for component grids that are cartesian resulting in a fast and efficient method the solver runs on parallel machines with each component grid distributed across one or more processors numerical results in two and three dimensions are presented for the fourth order accurate version of the method these results demonstrate the accuracy and efficiency of the approach

perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods we derive a correct first order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted most previous perturbative methods give incorrect results for this case even to lowest order because of the complicated discontinuous boundary conditions on the electric field at such an interface our final expression is simply a surface integral over the material interface of the continuous field components from the unperturbed structure the derivation is based on a localized coordinate transformation technique which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking the limit in which the coordinate perturbation is infinitesimally localized around the boundary not only is our result potentially useful in evaluating boundary perturbations e g from fabrication imperfections in highly anisotropic media such as many metamaterials but it also has a direct application in numerical electromagnetism in particular we show how it leads to a subpixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes

conformal finite different time domain fdtd with overlapping grids it is shown how to use many overlapping conformal grids in the neighborhood of the scatterer while using a rectangular grid away from the scatterer for solving maxwell s equations the locally conformal grids allow better approximation of the boundary condition on the scattering object while the rectangular grid preserves the simplicity and accuracy of the regular finite difference time domain fdtd algorithm these grids overlap typically about three zones it is shown how to connect together the calculations done in different overlapping grids

an upwinding boundary condition capturing method for maxwell s equations in media with material interfaces by using ghost points on either side of the interfaces a global second order accurate upwinding boundary condition capturing method for time domain maxwell s equations in media with material interfaces is proposed the equations are discretized on a uniform cartesian grid and the interfaces are allowed to intersect the grid in an arbitrary fashion the method is then obtained by combining central finite difference schemes with applicable nodes being replaced by the ghost points and upwinding technique with jump conditions across the interfaces being captured in a manner that the upwind property is always satisfied the resulting discretization has the desirable property that the allowed time step size is independent of the locations and the shapes of the interfaces numerical examples are then given to demonstrate the second order accuracy as well as the stability of the method where it is used to study wave equations with various types of material interfaces including electromagnetic scattering of a plane incident wave by a dielectric circular cylinder

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

fixed point iteration processes for asymptotic pointwise nonexpansive mappings in banach spaces abstract let x be a uniformly convex banach space with the opial property let t c c be an asymptotic pointwise nonexpansive mapping where c is bounded closed and convex subset of x in this paper we prove that the generalized mann and ishikawa processes converge weakly to a fixed point of t in addition we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex banach spaces both processes converge strongly to a fixed point

a 3 d enlarged cell technique ect for the conformal fdtd method in this paper we present an enlarged cell technique ect to avoid the time step reduction encountered in the conformal finite difference time domain cfdtd method due to small irregular cells truncated by metallic boundaries we focus our efforts on the discussion of the accuracy and stability of the ect and its comparison with other conformal methods especially the one called the uniformly stable conformal usc method we also provide a simplified ect which is much easier to implement in the ect a stability criterion is first constructed to identify instable irregular cells i e those having so small an area to cause instability those instable cells are then enlarged into their adjacent cells to obtain a large stable area careful treatment is performed on the communication between the intruding and intruded cells in terms of electromotive force by keeping the total electromotive force conservative this technique is verified by several 3 d numerical experiments results show that the ect is second order accurate and numerically stable at the regular courant time step limit

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

a general three dimensional tensor fdtd formulation for electrically inhomogeneous lossy media using the z transform a general three dimensional tensor finite difference time domain tfdtd formulation is derived to model electrically inhomogeneous lossy media of arbitrary shapes the time domain representation of electric losses is achieved using z transforms the regular cubical grid structure is maintained everywhere in the calculation domain by defining a 3 d face fraction based 3 x 3 permittivity tensor on the interfaces that describes the relationship between the known average flux density vector and the unknown local electric field vector for electrically lossy media this tensor is complex in the frequency domain however it can be modified for use with the z transform only this modified real form is inverted then transformed from the frequency into the z domain and finally into the time domain furthermore a local interface matrix is used to describe the relationship between the local electric field in the grid node and its counterpart on the other side of the interface this matrix is complex in the frequency domain for lossy media by applying the z transform this matrix can also be transformed into the time domain using only real modified matrix elements the accuracy of the method is confirmed by comparisons with analytical solutions

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

an embedded boundary method for the wave equation with discontinuous coefficients a second order accurate embedded boundary method for the two dimensional wave equation with discontinuous wave propagation speed is described the wave equation is discretized on a cartesian grid with constant grid size and the interface across which the wave speed is discontinuous is allowed to intersect the mesh in an arbitrary fashion by using ghost points on either side of the interface previous embedded boundary techniques for the neumann and dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy the resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points resulting in a fully explicit time integration method we prove that the one dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid for the two dimensional case the previously developed fourth order a t a dissipation is generalized to handle jump conditions we demonstrate that this operator provides sufficient stabilization to enable long time simulations while being weak enough to preserve the accuracy of the solution numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder the numerical solutions are evaluated against the analytical solution due to mie and pointwise second order accuracy is confirmed

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically

a note on the admissibility of modular function spaces abstract in this paper we prove the admissibility of modular function spaces e considered and defined by kozlowski in 17 as an application we get that any compact and continuous mapping t e e has a fixed point moreover we prove that the same holds true for any retract of e

a boundary difference method for electromagnetic scattering problems with perfect conductors and corners boundary integral equations bie are widely used in applied electromagnetics with the boundary element method bem typically employed for numerical solution the paper extends the recently proposed boundary difference method bdm an alternative to bem to electromagnetic scattering problems with perfect conductors and with corners bdm shares several key features and advantages of bem but avoids the singular kernels inherent in bie convergence rates of the method established via numerical experiments are in agreement with the theoretical predictions

spectral methods based on prolate spheroidal wave functions for hyperbolic pdes we examine the merits of using prolate spheroidal wave functions pswfs as basis functions when solving hyperbolic pdes using pseudospectral methods the relevant approximation theory is reviewed and some new approximation results in sobolev spaces are established an optimal choice of the band limit parameter for pswfs is derived for single mode functions our conclusion is that one might gain from using the pswfs over the traditional chebyshev or legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions

accuracy of staircase approximations in finite difference methods for wave propagation while a number of increasingly sophisticated numerical methods have been developed for time dependent problems in electromagnetics the yee scheme is still widely used in the applied fields mainly due to its simplicity and computational efficiency a fundamental drawback of the method is the use of staircase boundary approximations giving inconsistent results usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result in this paper we derive exact discrete solutions to the yee scheme close to the staircase approximated boundary enabling a detailed theoretical study of the amplitude phase and frequency errors created furthermore we show how evanescent waves of amplitude o 1 o 1 occur along the boundary these characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why in practice the yee scheme works as well as it does the analysis is supported by detailed proofs and numerical examples

finite difference time domain modeling of curved surfaces em scattering the finite difference time domain fdtd method is generalized to include the accurate modeling of curved surfaces this generalization the contour path cp method accurately models the illumination of bodies with curved surfaces yet retains the ability to model corners and edges cp modeling of two dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

asymptotic pointwise contractions a simple and elementary without using the ultrapower technique proof for the existence of a fixed point of an asymptotic pointwise contraction is provided the existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex banach space is also investigated

fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in generalized coordinates in two dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in cylindrical coordinates it is shown that for a perfect conductor such a formulation leads to a stable well posed algorithm and that in regions where the curvature of the coordinate lines is not great the dispersion and anisotropy effects are negligible such effects become more pronounced in regions of high curvature leading to unphysical phase shifts the magnitude of such shifts and the amount of wavefront distortion is studied via numerical experiments using a cylindrical mesh near field results are given for two canonical shapes for each polarization the circular cylinder and cylinders of square and rectangular cross sections these results are compared with those obtained by exact eigenfunction expansion techniques with method of moments mm solutions and with solutions obtained from an alternate fdtd approach in each case agreement is excellent the propagation of a plane wave through a polar space in the absence of a scatterer is also examined and it is shown that the fdtd algorithm is capable of tracking the incident wave closely

a hybrid hermite discontinuous galerkin method for hyperbolic systems with application to maxwell s equations a high order discretization strategy for solving hyperbolic initial boundary value problems on hybrid structured unstructured grids is proposed the method leverages the capabilities of two distinct families of polynomial elements discontinuous galerkin discretizations which can be applied on elements of arbitrary shape and hermite discretizations which allow highly efficient implementations on staircased cartesian grids we demonstrate through numerical experiments in 1 1 and 2 1 dimensions that the hybridized method is stable and efficient

high order accurate fdtd schemes for dispersive maxwell s equations in second order form using recursive convolutions abstract we propose a novel finite difference time domain fdtd scheme for the solution of maxwell s equations in which linear dispersive effects are present the method uses high order accurate approximations in space and time for the dispersive maxwell s equations written as a second order vector wave equation with a time history convolution term the modified equation approach is combined with the recursive convolution rc method to develop high order approximations accurate to any desired order in space and time high order accurate centered approximations of the physical maxwell interface conditions are derived for the dispersive setting in order to fully restore accuracy at discontinuous material interfaces second and fourth order accurate versions of the scheme are presented and implemented in two spatial dimensions for the case of the drude linear dispersion model the stability of these schemes is analyzed finally our approach is also amenable to curvilinear numerical grids if used with an appropriate generalized laplace operator

consistent boundary conditions for the yee scheme a new set of consistent boundary conditions for yee scheme approximations of wave equations in two space dimensions are developed and analyzed we show how the classical staircase boundary conditions for hard reflections or in the electromagnetic case conducting surfaces in certain cases give o 1 errors the proposed conditions keep the structure of the yee scheme and are thus well suited for high performance computing the higher accuracy is achieved by modifying the coefficients in the difference stencils near the boundary this generalizes our earlier results with gustafsson and wahlund in one space dimension we study stability and convergence and we present numerical examples

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

a second order 3d electromagnetics algorithm for curved interfaces between anisotropic dielectrics on a yee mesh a new frequency domain electromagnetics algorithm is developed for simulating curved interfaces between anisotropic dielectrics embedded in a yee mesh with second order error in resonant frequencies the algorithm is systematically derived using the finite integration formulation of maxwell s equations on the yee mesh second order convergence of the error in resonant frequencies is achieved by guaranteeing first order error on dielectric boundaries and second order error in bulk possibly anisotropic regions convergence studies conducted for an analytically solvable problem and for a photonic crystal of ellipsoids with anisotropic dielectric constant both show second order convergence of frequency error the convergence is sufficiently smooth that richardson extrapolation yields roughly third order convergence the convergence of electric fields near the dielectric interface for the analytic problem is also presented

pinning with a variable magnetic field of the two dimensional ginzburg landau model abstract we study the ginzburg landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain supposing that the ginzburg landau parameter and the intensity of the magnetic field are large and of the same order we determine an accurate asymptotic formula for the minimizing energy this asymptotic formula displays the influence of the pinning term also we discuss the existence of non trivial solutions and prove some asymptotics of the third critical field

the finite difference time domain fdtd and the finite volume time domain fvtd methods in solving maxwell s equations the finite difference time domain fdtd and its current generalizations have been demonstrated to be useful and powerful tools for the calculation of the radar cross section rcs of complicated objects the radiation of antennas in the presence of other structures and other applications the mathematical techniques for conformal fdtd have matured the primary impediments to its implementation are the complex geometries and material properties associated with the problem even under these circumstances fdtd is more flexible to implement because it is based on first principles instead of a clever mathematical trick this paper gives an account of some new results on conformal fdtd obtained by the authors and their associates at lockheed martin space company since 1988 the emphasis is on nonsmooth boundary condition simulation

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

error bounded schemes for time dependent hyperbolic problems in this paper we address the error growth in time for hyperbolic problems in first order form in bounded domains the energy method is used to study when an error growth or a fixed error bound is obtained it is shown that the choice of boundary procedure is a crucial point numerical experiments corroborate the theoretical findings

on asymptotic pointwise contractions in metric spaces abstract we discuss the existence of fixed points of asymptotic pointwise mappings in metric spaces this is the nonlinear version of some known results proved in banach spaces we also discuss the case of multivalued mappings

a new compact difference scheme for second derivative in non uniform grid expressed in self adjoint form a single parameter family of self adjoint compact difference sacd schemes is developed for discretizing the laplacian operator in self adjoint form developed implicit scheme is formally second order accurate with respect to truncation error with a free parameter a which helps control the numerical properties in the spectral plane the sacd scheme is analyzed in the spectral plane for its resolution properties for both periodic and non periodic problems using the matrix spectral analysis t k sengupta g ganeriwal s de analysis of central and upwind schemes j comput phys 192 2 2003 677 694 the major objective here is to identify the advantages of the new scheme over the traditional explicit second order cd2 scheme in discretizing the laplacian operator in self adjoint form this appears in navier stokes equation and in other transport equations and boundary value problems bvp expressed in orthogonal co ordinate systems either in physical or in transformed plane we also compare the developed method with the higher order compact schemes for non uniform grids to demonstrate the accuracy of sacd scheme we have tested it for i bi directional wave propagation problem given by the second order wave equation and ii an elliptic bvp as in the stommel ocean model for the stream function these examples help infer the properties of sacd scheme when solving different types of partial differential equations most importantly the effects of grid stretching and choice of value of the free parameter a are investigated here we also compare the present implicit compact method with explicit compact method known as the higher order compact hoc method also the practical applications of the sacd scheme are explored by solving the navier stokes equation for the cases of a a flow inside a lid driven cavity and b the receptivity and instability of an external adverse pressure gradient flow over a flat plate in the former unsteadiness of the flow is captured and in the latter the receptivity of the flow is studied in causing flow instability by triggering tollmien schlichting waves the new scheme shows a marked improvement over the explicit scheme for low reynolds number steady flow in lid driven cavity whereas for the flow in the same geometry at higher reynolds numbers efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer

regular article comparison of kinetic theory and discrete element schemes for modelling granular couette flows discrete element based simulations of granular flow in a 2d velocity space are compared with a particle code that solves kinetic granular flow equations in two and three dimensions the binary collisions of the latter are governed by the same forces as those governing the discrete elements both methods are applied to a granular shear flow of equally sized discs and spheres the two dimensional implementation of the kinetic approach shows excellent agreement with the results of the discrete element simulations during change to a three dimensional velocity space the qualitative features of the flow are maintained however some flow properties change quantitatively

conformal fdtd methods to avoid time step reduction with and without cell enlargement during the last decades there have been considerable efforts to develop accurate and yet simple conformal methods for modelling curved boundaries within the finite difference time domain fdtd algorithm in an earlier publication we proposed the uniformly stable conformal usc approach as a general three dimensional extension of fdtd without the need to reduce the maximum stable time step the main idea of usc is the usage of virtually enlarged cells near to the boundary leading to an increased implementation effort in this paper we review the usc method and introduce a new simple and accurate conformal scheme which does not use such enlarged cells this simplified conformal sc scheme has the same number of operations and algorithmic logic as the standard staircase method and thus is easily realizable in existing fdtd codes like usc it leads to accurate results without time step reduction showing a nearly second order convergence in practice the method is verified and compared to other approaches by means of several numerical 2d and 3d examples

on asymptotic pointwise nonexpansive mappings in modular function spaces we prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

a high order accurate embedded boundary method for first order hyperbolic equations abstract a stable and high order accurate embedded boundary method for first order hyperbolic equations is derived where the grid boundaries and the physical boundaries do not coincide high order interpolation is used the boundary stencils are based on a summation by parts framework and the boundary conditions are imposed by the sat penalty method which guarantees linear stability for one dimensional problems second fourth and sixth order finite difference schemes are considered the resulting schemes are fully explicit accuracy and numerical stability of the proposed schemes are demonstrated for both linear and nonlinear hyperbolic systems in one and two spatial dimensions

pseudospectral vs finite difference methods for initial value problems with discontinuous coefficient an initial value problem with piecewise constant coefficients is considered the accuracies for both finite difference methods and the pseudospectral method are analyzed and a modification of the initial value problem is suggested the modified problem is shown to have the same temporal period as the original problem does and a second order accuracy is obtained for the pseudospectral method at integral multiples of the temporal period

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

a second order accurate embedded boundary method for the wave equation with dirichlet data the accuracy of cartesian embedded boundary methods for the second order wave equation in general two dimensional domains subject to dirichlet boundary conditions is analyzed based on the analysis we develop a numerical method where both the solution and its gradient are second order accurate we avoid the small cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the dirichlet boundary condition long time stability of the method is obtained by adding a small fourth order dissipative term several numerical examples are provided to demonstrate the accuracy and stability of the method the method is also used to solve the two dimensional tm z problem for maxwell s equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field

regular article spectral collocation time domain modeling of diffractive optical elements a spectral collocation multi domain scheme is developed for the accurate and efficient time domain solution of maxwell s equations within multi layered diffractive optical elements special attention is being paid to the modeling of out of plane waveguide couplers emphasis is given to the proper construction of high order schemes with the ability to handle very general problems of considerable geometric and material complexity central questions regarding efficient absorbing boundary conditions and time stepping issues are also addressed the efficacy of the overall scheme for the time domain modeling of electrically large and computationally challenging problems is illustrated by solving a number of plane as well as non plane waveguide problems

density results relative to the dirichlet energy of mappings into a manifold let y be a smooth compact oriented riemannian manifold without boundary weak limits of graphs of smooth maps u k b n y with an equibounded dirichlet integral give rise to elements of the space cart 2 1 b n y assume that y is 1 connected and that its 2 homology group has no torsion in any dimension n we prove that every element t in cart 2 1 b n x y with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u k b n y with dirichlet energies converging to the energy of t

a fast higher order solver for scattering by penetrable bodies in three dimensions in this paper we introduce a new fast higher order solver for scattering by inhomogeneous media in three dimensions as in previously existing methods the low complexity of our integral equation method o nlogn operations for an n point discretization is obtained through extensive use of the fast fourier transform fft for the evaluation of convolutions however the present approach obtains significantly higher order accuracy than these previous approaches yielding at worst third order far field accuracy or substantially better for smooth scatterers even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the increased order of convergence of our method results from i a partition of unity decomposition of the green s function into a smooth part with unbounded support and a singular part with compact support and ii replacement of the possibly discontinuous scatterer by an appropriate fourier smoothed scatterer the resulting convolutions can then be computed with higher order accuracy by means of o nlogn ffts we present a parallel implementation of our approach and demonstrate the method s efficiency and accuracy through a variety of computational examples for a very large scatterer considered earlier in the literature with a volume of 3648 l 3 where l is the wavelength using the same number of points per wavelength and in computing times comparable to those required by the previous approach the present algorithm produces far field values whose errors are two orders of magnitude smaller than those reported previously

a curved boundary treatment for discontinuous galerkin schemes solving time dependent problems for problems defined in a two dimensional domain with boundary conditions specified on a curve we consider discontinuous galerkin dg schemes with high order polynomial basis functions on a geometry fitting triangular mesh it is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary h will render any finite element method to be at most second order accurate unless the boundary conditions can be accurately transferred from to h in general curvilinear element method should be used to obtain high order accuracy we discuss a simple boundary treatment which can be implemented as a modified dg scheme defined on triangles adjacent to h even though integration along the curve is still necessary integrals over any curved element are avoided if the domain is convex or if is nonconvex and the true solutions can be smoothly extended to the exterior of the modified dg scheme is high order accurate in these cases numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear dg scheme

high order finite difference methods for wave propagation in discontinuous media high order finite difference approximations are derived for the second order wave equation with discontinuous coefficients on rectangular geometries the discontinuity is treated by splitting the domain at the discontinuities in a multi block fashion each sub domain is discretized with compact second derivative summation by parts operators and the blocks are patched together to a global domain using the projection method this guarantees a conservative strictly stable and high order accurate scheme the analysis is verified by numerical simulations in one and two spatial dimensions

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

dynamics of multiple degree ginzburg landau vortices for the two dimensional complex parabolic ginzburg landau equation we prove that asymptotically vortices evolve according to a simple ordinary differential equation which is a gradient flow of the kirchhoff onsager functional this convergence holds except for a finite number of times corresponding to vortex collisions and splittings which we describe carefully the only assumption is a natural energy bound on the initial data

a staggered upwind embedded boundary sueb method to eliminate the fdtd staircasing error in spite of its flexibility and second order accuracy in a homogeneous medium yee s finite difference time domain fdtd method suffers from serious degradation when treating material interfaces greatly reducing its accuracy in the presence of inhomogeneous media and perfect conductors indeed such so called staircasing approximation may lead to local zeroth order and global first order errors in this work an embedded fdtd scheme the staggered upwind embedded boundary sueb method is developed for the solution of oneand two dimensional maxwell s equations this simple embedded technique uses upwind conditions in the fdtd method to correctly represent the location and physical conditions of material and metallic boundaries hence eliminating problems caused by the staircasing approximation accuracy analysis has been made to show that the sueb method maintains a second order accuracy globally since the entire problem has been embedded into the simple staggered grid similar to that employed by the yee s scheme extra effort is only needed when treating the grid points close to the interfaces therefore little additional computational cost is needed over yee s scheme the sueb method has been validated by analytical solutions for plane wave normally incident to a planar boundary and for the tm wave propagation in the presence of a dielectric cylinder and a perfectly electrically conducting cylinder

higher order fourier approximation in scattering by two dimensional inhomogeneous media this paper provides a theoretical analysis of a higher order fft based integral equation method introduced recently ieee trans antennas and propagation 48 2000 pp 1862 1864 for the evaluation of transverse electric polarized electromagnetic scattering from a bounded penetrable inhomogeneity in two dimensional space roughly speaking this method is based on fourier smoothing of the integral operator and the refractive index n x here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher order accuracy even when n x is a discontinuous function as suggested by the numerical experiments contained in the paper mentioned above in detail we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer and we demonstrate with a few numerical examples that the predicted convergence rates are achieved in practice

difference approximations of the neumann problem for the second order wave equation stability theory and numerical experiments are presented for a finite difference method that directly discretizes the neumann problem for the second order wave equation complex geometries are discretized using a cartesian embedded boundary technique both second and third order accurate approximations of the boundary conditions are presented away from the boundary the basic second order method can be corrected to achieve fourth order spatial accuracy to integrate in time we present both a second order and a fourth order accurate explicit method the stability of the method is ensured by adding a small fourth order dissipation operator locally modified near the boundary to allow its application at all grid points inside the computational domain numerical experiments demonstrate the accuracy and long time stability of the proposed method

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

calculation of the sar induced in head tissues using a high order dgtd method and triangulated geometrical models the great majority of numerical calculations of the specific absorption rate sar induced in human tissues exposed to microwaves are performed using the finite difference time domain fdtd method and voxel based geometrical models the straightforward implementation of the method and its computational efficiency are among the main reasons for fdtd being currently the leading method for numerical assessment of human exposure to electromagnetic waves however the rather difficult departure from the commonly used cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context in particular interfaces between tissues where sharp gradients of the electromagnetic field may occur are hardly modeled rigorously in these studies we present here an alternative numerical dosimetry methodology which is based on a high order discontinuous galerkin time domain dgtd method and adapted geometrical models constructed from unstructured triangulations of tissue interfaces and discuss its application to the calculation of the sar induced in head tissues

some geometrical properties and fixed point theorems in orlicz spaces abstract let g be a finite atomless measure space and let l be an orlicz space of measurable functions on g we consider some geometrical properties of the functional g t d t called the orlicz modular these properties like strict convexity uniform convexity or uniform convexity in every direction can be equivalently expressed in terms of the properties of the corresponding orlicz function we use these properties in order to prove some fixed point results for mappings t b b b l that are nonexpansive with respect to the orlicz modular i e t tg g for all and g in b we prove also existence and uniqueness in l of the best approximant with respect to and some convex subsets of l our results are valid also in the case when the orlicz function does not satisfy the 2 condition this demonstrates the advantage of our method because in the latter case both luxemburg s and orlicz s norms cannot possess suitable convexity properties

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we