preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

dissipative terms and local time stepping improvements in a spatial high order discontinuous galerkin scheme for the time domain maxwell s equations in this paper we present some improvements in terms of accuracy and speed up for a particular well adapted discontinuous galerkin method devoted to the time domain maxwell equations first to reduce spurious modes on very distorted meshes the addition of dissipative terms as penalization in the numerical scheme is studied and compared on examples second in order to increase the efficiency of the method a multi class local time stepping strategy is presented and its validation and advantages are highlighted on different examples

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

a discontinuous galerkin augmented electric field integral equation for multiscale electromagnetic scattering problems a discontinuous galerkin dg augmented electric field integral equation method based on the domain decomposition is proposed in this paper for full wave solution of multiscale targets the conventional surface integral equation based dg method allowing both conformal and nonconformal discretizations for multiscale structures suffers from low frequency breakdown by augmenting the dg efie with current continuity equation the proposed scheme can alleviate the low frequency breakdown in the augmented system the electric field integral equation and the current continuity equation are discretized by using hybrid basis functions including rao wilton glisson rwg and half rwg basis functions since the half rwg basis is not divergence conforming line charge degrees of freedom on the adjoining edges are introduced in this paper it is observed that the resulting linear system is well conditioned at low frequencies which leads to a rapid convergence over wide frequency band numerical examples demonstrate the accuracy and efficiency of the augmented system

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 hybrid time domain discontinuous galerkin boundary integral method for electromagnetic scattering analysis a scheme hybridizing discontinuous galerkin time domain dgtd and time domain boundary integral tdbi methods for accurately analyzing transient electromagnetic scattering is proposed radiation condition is enforced using the numerical flux on the truncation boundary the fields required by the flux are computed using the tdbi from equivalent currents introduced on a huygens surface enclosing the scatterer the hybrid dgtdbi ensures that the radiation condition is mathematically exact and the resulting computation domain is as small as possible since the truncation boundary conforms to scatterer s shape and is located very close to its surface locally truncated domains can also be defined around each disconnected scatterer additionally reducing the size of the overall computation domain numerical examples demonstrating the accuracy and versatility of the proposed method are presented

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

a new family of exponential based high order dgtd methods for modeling 3 d transient multiscale electromagnetic problems a new family of exponential based time integration methods are proposed for the time domain maxwell s equations discretized by a high order discontinuous galerkin dg scheme formulated on locally refined unstructured meshes these methods which are developed from the lawson method remove the stiffness on the time explicit integration of the semidiscrete operator associated with the fine part of the mesh and allow for the use of high order time explicit scheme for the coarse part operator they combine excellent stability properties with the ability to obtain very accurate solutions even for very large time step sizes here the explicit time integration of the lawson transformed semidiscrete system relies on a low storage runge kutta lsrk scheme leading to a combined lawson lsrk scheme in addition efficient techniques are presented to further improve the efficiency of this exponential based time integration for the efficient calculation of matrix exponential we employ the krylov subspace method numerical experiments are presented to assess the stability verify the accuracy and numerical convergence of the lawson lsrk scheme they also demonstrate that the dg time domain methods based on the proposed time integration scheme can be much faster than those based on classical fully explicit time stepping schemes with the same accuracy and moderate memory usage increase on locally refined unstructured meshes and are thus very promising for modeling 3 d multiscale electromagnetic problems

interior penalty discontinuous galerkin finite element method for the time dependent first order maxwell s equations an interior penalty discontinuous galerkin method is described with a conformal perfectly matched layer pml for solving the two first order maxwell s equations in the time domain both central and upwind fluxes are studied in this work in both cases the proposed method is explicit and conditionally stable additionally a local time stepping strategy is applied to increase efficiency and reduce the computational time finally numerical examples are presented to validate the method

asymptotics of the mean field heisenberg model we consider the mean field classical heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity in particular we obtain cramer and sanov type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second order phase transition in the gibbs measures we also study the asymptotics of the total spin throughout the phase transition using stein s method proving central limit theorems in the sub and supercritical phases and a nonnormal limit theorem at the critical temperature

interconnect and lumped elements modeling in interior penalty discontinuous galerkin time domain methods in this paper we present an approach on how to incorporate passive lumped elements such as resistors capacitors and inductors in dgtd methods and their application to interconnect modeling starting from the voltage and current relationships we derive the equivalent relationships that describe each of the r l c in terms of the electric and magnetic fields next these field expressions are weakly enforced through the interior penalty ip dg formulation the proposed method is explicit and conditionally stable additionally a local time stepping strategy is applied to increase efficiency and reduce the computational time finally a numerical example is presented to validate the proposed approach

a parallel non conforming multi element dgtd method for the simulation of electromagnetic wave interaction with metallic nanoparticles the present work is about the development of a parallel non conforming multi element discontinuous galerkin time domain dgtd method for the simulation of the scattering of electromagnetic waves by metallic nanoparticles such nanoparticles most often have curvilinear shapes therefore we propose a numerical modeling strategy which combines the use of an unstructured tetrahedral mesh for the discretization of the scattering structures with a structured uniform cartesian mesh for treating efficiently the rest of the domain the overall goal is to increase the flexibility in the meshing process while decreasing the needs in computational resources for the target applications the latter are here modeled by the system of 3d time domain maxwell equations coupled to a drude dispersion model for taking into account the material properties of nanoparticles at optical frequencies we propose an auxiliary differential equation ade based dgtd method for solving the resulting system and present numerical results demonstrating the benefits of using non conforming multi element meshes in this particular application context

a comparative study of three finite element based explicit numerical schemes for solving maxwell s equations three finite element based explicit numerical algorithms named the dual field domain decomposition at the element level dfdd eld the discontinuous galerkin time domain method with upwind fluxes dgtd upwind and the discontinuous galerkin time domain method with central fluxes dgtd central are investigated and compared in terms of accuracy and efficiency all three algorithms can perform an efficient domain decomposition and avoid the inversion of a global system matrix yet the study shows that they differ from each other in terms of accuracy and efficiency hybrid implicit explicit schemes which can relax the restriction on the time step size imposed by the smallest elements in the computational domain are also investigated for both dfdd and dgtd and compared in terms of efficiency

preconditioning nonconforming finite element methods for treating dirichlet boundary conditions ii this work deals with theh 1 condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are finite element discretizations of second order elliptic operators a andb respectively b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval goldstein also has derived results on the spectral distribution ofb h 1 a h using a different approach as a generalization of the results of parter and wong the current work includes nonconforming finite element methods which deal with dirichlet boundary conditions it will be shown that in this more general setting the singular values also cluster in a positive finite interval in particular if the leading part ofb is the same as the leading part ofa then the singular values cluster about the point 1 two specific methods are given as applications of this theory they are the penalty method of babuska and the method of nearly zero boundary conditions of nitsche finally it will be shown that the same results can be proven by an approach generalized from the work of goldstein

an implicit hybridized discontinuous galerkin method for the 3d time domain maxwell equations we present a time implicit hybridizable discontinuous galerkin hdg method for numerically solving the system of three dimensional 3d time domain maxwell equations this method can be seen as a fully implicit variant of classical so called dgtd discontinuous galerkin time domain methods that have been extensively studied during the last 10 years for the simulation of time domain electromagnetic wave propagation the proposed method has been implemented for dealing with general 3d problems discretized using unstructured tetrahedral meshes we provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand and its performance when applied to more realistic problems we also include some performance comparisons with a centered flux time implicit dgtd method

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

time domain finite element methods various time domain finite element methods for the simulation of transient electromagnetic wave phenomena are discussed detailed descriptions of test trial spaces explicit and implicit formulations nodal and edge facet element basis functions are given along with the numerical stability properties of the different methods the advantages and disadvantages of mass lumping are examined finally the various formulations are compared on the basis of their numerical dispersion performance

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

preconditioning isoparametric finite element methods taking into account numerical integration this paper is concerned with the h 1 condition numbers and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h and b h are isoparametric finite element discretizations of second order elliptic operators a and b by taking into account numerical integration respectively where a is nonselfad joint and possibly indefinite and b is selfadjoint and positive definite it is proved that the h 1 condition numbers of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally the implementation of b h 1 is discussed

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

a domain decomposition method with nonconformal meshes for finite periodic and semi periodic structures we present a domain decomposition method as a preconditioner for krylov type solvers to model complex electromagnetic problems containing periodicities the method reduces memory requirements by decomposing the original problem into several nonoverlapping sub domains the 1st order transmission condition is employed on interfaces between adjacent sub domains to enforce continuity of electromagnetic fields and to ensure the sub domain problems are well posed by following the spirit of duality paring a symmetric system is obtained to reduce the computational burdens of the present method the finite element tearing and interconnecting like algorithm is adopted this algorithm results in the computation of the so called numerical green s function which can be compressed efficiently via a rank revealing matrix factorization algorithm the final system matrix is solved by krylov solvers instead of classical stationary solvers to improve the convergence of iterative solvers several robust implementation details are discussed and the choice of some popular krylov subspace solvers is studied through numerical examples

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

optimal equivalent preconditioners preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem the technique considered here makes use of a selfadjoint positive definite operator b as the preconditioner for the nonselfadjoint convection diffusion operator a preconditioning by a s leading term plus a positive zeroth order term sigma i and optimizing over sigma is proposed the analysis of a model problem indicates that when a has large first order terms both the spectral and norm condition of b 1 a are minimized by sigma gg 0