a new conservative high order modified fdtd 2 4 scheme in this paper we propose a new 3 d conservative fdtd 2 4 scheme exhibiting properties close to the 2 d m24 scheme this scheme is designed to retain a 2 d stencil in the field update equations and has high phase accuracy at low grid resolutions moreover an accurate courant friedrichs lewy condition has been validated for this scheme some experiments are conducted to assess the higher accuracy of the proposed scheme

a high order compact fdtd algorithm for electrically large waveguide analysis to model electrically large waveguiding structures the compact finite difference time domain fdtd algorithm needs to use severely scaled down time steps to properly contain the rapidly growing numerical dispersion errors with increased operating frequency in this work a high order compact fdtd algorithm based on fourth order spatial central finite differencing and fourth order temporal backward finite differencing is developed the accuracy and efficiency of this proposed algorithm are verified through its dispersion relation analysis and validated by modeling high frequency wave propagation through an earth tunnel the obtained computational efficiency allows this high order algorithm to model wireless propagation through longitudinally invariant road and railway tunnels using several hundred compact fdtd cells as opposed to the several million fdtd cells required by three dimensional fdtd algorithms

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

a finite volumes based 3 d low dispersion fdtd algorithm an algorithm extension to three dimensions is developed and presented for the highly phase coherent modified second order in time fourth order in space or m24 finite difference time domain fdtd algorithm a finite volumes approach in conjunction with yee s standard fdtd lattice is used for algorithm development the corresponding dispersion relation is also developed analyzed and compared to both the standard second order and fourth order fdtd algorithms as well as to two closely related high order phase coherent algorithms wideband algorithm attributes are also presented as well as sets of ready to use optimized algorithm coefficients

a note on the convergence of linear finite elements let u be the solution of the dirichlet problem for laplace s equation in a smooth domain omega and let u h be its piecewise linear finite element approximation if the family of partitions is quasi uniform it is known that left u u h right l p leqq ch 2 u w 2 p quad text for 1 p infty and left u u h right l infty leqq ch 2 log frac 1 h u w 2 infty and moreover the log 1 h factor cannot be removed from the last estimate the objective of this paper is to show that if u in w 2 infty the error is of optimal order in the bounded mean oscillation bmo norm that is left u u h right text bmo leqq ch 2 u w 2 infty this norm is very close to the uniform norm and moreover the space bmo plays the role of l infty in many applications in real analysis the result is obtained by a duality argument as in the l p case making use of an appropriate regularity result

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a low dispersion 3 d second order in time fourth order in space fdtd scheme m3d sub 24 a second order in time fourth order in space modified finite difference time domain fdtd scheme for three dimensional electromagnetic problems m3d sub 24 is presented the algorithm enables the numerical phase error to be minimized so that it leads to high accuracy with low resolution grids the advantage of this method is demonstrated by considering the long distance propagation of the wave radiated from a time harmonic elementary dipole using a low resolution grid and comparing the results with other fdtd schemes

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

a versatile split field 1 d propagator for perfect fdtd plane wave injection a new staggered field design and formulation for the one dimensional propagator of the total field scattered field source implementation in finite difference time domain fdtd scattering simulations are presented the new equations are based on split field maxwell s equations and the resulting technique extends the functionality of the multipoint auxiliary propagator to sourcing fdtd lattices hosting extended stencil high order algorithms this technique virtually eliminates numerical dispersion field location and polarization mismatches between propagator and main grid the resulting machine accuracy level leakage error from implementing this technique is confirmed for the standard low and high order fdtd schemes as well as the m24 high order algorithm normalized field leakage for all three algorithm implementations outside the total field region was measured at below 295 db

comparison of the dispersion properties of several low dispersion finite difference time domain algorithms a comparison of the accuracy of several low dispersion finite difference time domain fdtd schemes in two dimensions is presented each algorithm is reviewed and its fdtd update equations presented the dispersion relation of each fdtd algorithm is also given the accuracy of each fdtd scheme is compared via direct evaluation of the dispersion relation results are presented showing the dispersion errors of each algorithm as a function of propagation angle and cell size tables are shown that present for each algorithm the optimal courant number at a specified discretization as well as the number of floating point operations needed to update each cell three fields at each time step the advantages and disadvantages of each algorithm are briefly discussed while some schemes are more wideband than others almost all provide substantial improvement in the dispersion errors compared with the classical yee 1966 fdtd algorithm

a time domain method with isotropic dispersion and increased stability on an overlapped lattice a time domain method on an overlapped lattice is presented for the accurate and efficient simulation of electromagnetic wave propagation through inhomogeneous media the method comprises a superposition of complementary approximations to electromagnetic theory on a lattice the discrete space time dst method is set on a pair of dual lattices whose field components are collocated on their respective lattice sites the other the time domain element tde method is set on overlapping dual lattices whose field components are noncollocated the tde method is shown to be a generalization and reinterpretation of the yee algorithm the benefits of the combined algorithm over comparable methods include 1 increased accuracy over larger bandwidths 2 increased stability allowing larger time steps 3 local stencil satisfying boundary conditions on interfaces 4 self contained mathematical framework 5 it is physically intuitive

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

optimal convergence for the finite element method in campanato spaces we prove a priori estimates and optimal error estimates for linear finite element approximations of elliptic systems in divergence form with continuous coefficients in campanato spaces the proofs rely on discrete analogues of the campanato inequalities for the solution of the system which locally measure the decay of the energy as an application of our results we derive w 1 p estimates and give a new proof of the well known w 1 results of rannacher and scott

near field pml optimization for low and high order fdtd algorithms using closed form predictive equations the convolutional perfectly matched layer cpml absorbing boundary condition is fully capable of handling near field wave absorption that usually combines near grazing wave incidence with wave evanescence the appropriate choice of the various cpml parameters to realize this potential for any given simulation problem is a challenging task that is typically achieved through exhaustive and time consuming searches that involve large numbers of full scale simulations the presented work here uses a previously developed predictive system of equations that accurately determines numerical reflections off the pml interface and embeds it into a global optimization routine that reliably computes the required optimum cpml parameters this predictive system of equations has also been extended and validated for the m24 and fv24 integral based high order fdtd algorithms with this approach the task of selecting optimum cpml parameters that would usually take several days of intense computations can now be accomplished within a few minutes on an average personal computer

quasi optimal estimates for finite element approximations using orlicz norms on considere l approximation par des elements finis lineaires de la solution du probleme de dirichlet u f on etablit une relation entre l erreur dans la norme infinie et l erreur dans quelques espaces d orlicz on en deduit des estimations uniformes quasi optimales lorsque la solution ne possede des derivees secondes dans l espace d orlicz associe a la fonction exponentielle

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

a modified fdtd 2 4 scheme for modeling electrically large structures with high phase accuracy a new fourth order finite difference time domain fdtd scheme has been developed that exhibits extremely low phase errors at low grid resolutions compared to the conventional fdtd scheme moreover this new scheme is capable of combining with the standard yee 1966 scheme to produce a stable hybrid algorithm the problem of wave propagation through a building is simulated using this new hybrid algorithm to demonstrate the large savings in computing resources it could afford with this new development the fdtd method can now be used to successfully model structures that are thousands of wavelengths large using the present day computer technology

an analysis of a defect correction method for a model convection diffusion equation the error in a defect correction method for a model one dimensional convection diffusion problem without turning points is analyzed although the analysis is limited to the one dimensional problem the scheme is a computationally attractive way to increase the accuracy in smooth regions of first order schemes for general multidimensional problems it is shown that in one dimension the kth approximation converges uniformly in varepsilon in regions bounded away from the layer with rate o varepsilon 0 varepsilon k h 2 varepsilon 0 o h the estimates degrade to o 1 near layers numerical experiments confirm that this is precisely the behavior of the method