spectral and structural analysis of high precision finite difference matrices for elliptic operators abstract in this paper we study the structural properties of matrices coming from high precision finite difference fd formulae when discretizing elliptic or semielliptic differential operators l a u of the form k d k d x k a x d k d x k u x strong relationships with toeplitz structures and linear positive operators lpo are highlighted these results allow one to give a detailed analysis of the eigenvalues localisation distribution of the arising matrices the obtained spectral analysis is then used to define optimal toeplitz preconditioners in a very compact and natural way and in addition to prove szego like and widom like ergodic theorems for the spectra of the related preconditioned matrices a wide numerical experimentation confirming the theoretical results is also reported

a new noncoforming finite element method for the computation of electromagnetic guided waves i mathematical analysis we are concerned with the problem of computing electromagnetic guided waves in a closed inhomogeneous cylindrical wave guide these waves correspond to solutions of an eigenvalue problem and classical methods produce in addition to approximations to the solutions spurious modes which are particularly troublesome because they correspond to nonzero approximations of the 0 eigenvalue a nonconforming finite element method for the calculation of guided waves without spurious modes is introduced and analyzed

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

spectral behavior of matrix sequences and discretized boundary value problems abstract in this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension more specifically we give a unified treatment of notions such as distribution equal distribution localization equal localization clustering and sub clustering as a case study we consider the matrix sequences arising from the finite difference fd discretization of elliptic and semielliptic boundary value problems bvps the spectral analysis is then extended to toeplitz based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular belong to l 1 and to the case of multidimensional problems the related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices

a high order isoparametric finite element method for the computation of waveguide modes we have studied the approximation of optical waveguide eigenvalues by a high order isoparametric vector finite element method isoparametric mappings are used for the approximation of domains with curved boundaries or curved material interfaces eigenvalue convergence for curved elements is investigated numerical results verify the predicted order of convergence and show the remarkable accuracy of the method

equivalence between glt sequences and measurable functions abstract the theory of generalized locally toeplitz glt sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity key concepts in this theory are the notions of approximating classes of sequences a c s and spectral symbols that lead to defining a metric structure on the space of matrix sequences and provide a link with the measurable functions in this paper we prove additional results regarding theoretical aspects such as the completeness of the matrix sequences space with respect to the metric a c s and the identification of the space of glt sequences with the space of measurable functions

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

distribution results on the algebra generated by toeplitz sequences a finite dimensional approach abstract let a be an l 1 symbol defined on q d q with d 1 and let us consider the multi indexed sequence of toeplitz matrices t n a with n n d it is well known that t n a is spectrally distributed as a in the sense of the singular values in this paper we prove that a sequence as t n a is spectrally distributed in the sense of the singular values as the measurable function a for a l 1 and and ranging in any finite set of values

self adaptive hp finite element method with iterative mesh truncation technique accelerated with adaptive cross approximation abstract to alleviate the computational bottleneck of a powerful two dimensional self adaptive h p finite element method fem for the analysis of open region problems which uses an iterative computation of the integral equation over a fictitious boundary for truncating the fem domain we propose the use of adaptive cross approximation aca to effectively accelerate the computation of the integral equation it will be shown that in this context aca exhibits a robust behavior yields good accuracy and compression levels up to 90 and provides a good fair control of the approximants which is a crucial advantage for h p adaptivity theoretical and empirical results of performance computational complexity comparing the accelerated and non accelerated versions of the method are presented several canonical scenarios are addressed to resemble the behavior of aca with h p and h p adaptive strategies and higher order methods in general

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

approximation of eigenvalues in mixed form discrete compactness property and application to hp mixed finite elements abstract in this paper we discuss the discrete compactness property dcp which is a well known tool for the analysis of finite element approximations of maxwell s eigenvalues we restrict our presentation to maxwell s eigenvalues but the theory applies to more general situations and in particular to mixed finite element schemes that can be written in the framework of de rham complex and which enjoy suitable compactness properties we investigate the relationships between dcp and standard mixed conditions for the good approximation of eigenvalues as a consequence of our theory the convergence analysis of the rectangular hp version of raviart thomas finite elements for the approximation of laplace eigenvalues is presented as a corollary of the analogous result for hp edge elements applied to the approximation of maxwell s eigenvalues d boffi m costabel m dauge l demkowicz discrete compactness for the hp version of rectangular edge finite elements siam j numer anal 44 3 2006 979 1004

spectral analysis and structure preserving preconditioners for fractional diffusion equations fractional partial order diffusion equations are a generalization of classical partial differential equations used to model anomalous diffusion phenomena when using the implicit euler formula and the shifted grunwald formula it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a toeplitz like structure in this paper we focus our attention on the case of variable diffusion coefficients under appropriate conditions we show that the sequence of the coefficient matrices belongs to the generalized locally toeplitz class and we compute the symbol describing its asymptotic eigenvalue singular value distribution as the matrix size diverges we employ the spectral information for analyzing known methods of preconditioned krylov and multigrid type with both positive and negative results and with a look forward to the multidimensional setting we also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system with krylov methods such as cgnr and gmres a number of numerical examples show that our proposal is more effective than recently used circulant preconditioners

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

hp vector finite element method for eigenmode analysis of waveguides a new stable variational formulation for full wave analysis of waveguides is presented numerical experiments show that the method remains stable at very low frequencies theoretical results apply to a wide class of vector finite elements including two families of nedelec numer math 35 1980 315 numer math 50 1986 57 and their generalization hp edge elements comput methods appl mech engrg 152 1998 103 comput methods appl mech engrg 169 1999 331

maxwell eigenvalues and discrete compactness in two dimensions abstract we present an elementary proof of the discrete compactness result for a general class of hp finite elements introduced in 1 2 we discuss h convergence of 2d elements only and in this context the results are not new as the analysis of h curl conforming elements for maxwell s equations can be reduced to the long known results for raviart thomas elements 3 the work is based on the result of kikuchi 4 5 for nedelec s edge triangular elements of the lowest order and presents an alternative to techniques presented in 3 6 in particular the present version does not use an inverse inequality argument and therefore is valid for h adaptive meshes we conclude the presentation with a number of 2d computational experiments including nonconvex domains

optimizing a multigrid runge kutta smoother for variable coefficient convection diffusion equations abstract the theory of generally locally toeplitz or glt for short sequences of matrices is proposed in the analysis of a multigrid solver for the linear systems generated by finite volume finite difference approximations of variable coefficients linear convection diffusion equations in 1d proposed by birken in 2012 and extended here to 2d problems the multigrid solver is used with a runge kutta smoother optimal coefficients for the smoother are found by considering the unsteady linear advection equation and using optimization algorithms in particular in order to reduce the issues of having multiple local minima the sequential quadratic programming sqp mixed with genetic and particle swarm optimization algorithms are proposed numerical results show that our proposals are competitive with respect to other multigrid implementations

approximating classes of sequences the hermitian case given a sequence an of matrices an of increasing dimension dn with dk dq for k q k q n we recently introduced the concept of approximating class of sequences a c s in order to define a basic approximation theory for matrix sequences we have shown that such a notion is stable under inversion linear combinations and product whenever natural and mild conditions are satisfied in this note we focus our attention on the hermitian case and we show that f bn m m n is an a c s for f an if bn m m n is an a c s for an an is sparsely unbounded and f is a suitable continuous function defined on r we also discuss the potential impact and future developments of such a result

a two dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities part i energy norm based automatic hp adaptivity abstract the accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering in some cases the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized in other cases the discontinuities are deliberately introduced into the waveguide to perform a certain electric function this is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp adaptive finite element method in this paper a fully automatic energy norm based hp adaptive finite element fe strategy applied to a number of relevant waveguide structures is presented the methodology produces exponential convergence rates in terms of the energy norm error of the solution against the problem size number of degrees of freedom extensive numerical results demonstrate the suitability of the hp method for solving different rectangular waveguide discontinuities these results illustrate the flexibility reliability and high accuracy of the method the self adaptive hp fem provides similar sometimes more accurate results than those obtained with semi analytical techniques such as the mode matching method for problems where semi analytical methods can be applied at the same time the hp fem provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics metallic screws round corners etc which cannot be easily considered when using semi analytical techniques in the second part the work is extended by presenting a fully automatic goal oriented hp adaptive fe in terms of the scattering or s parameters which are widely used in microwave engineering for the characterization of microwave devices

lusin theorem glt sequences and matrix computations an application to the spectral analysis of pde discretization matrices lusin theorem glt sequences and matrix computations an application to the spectral analysis of pde discretization matrices

the conditioning of fd matrix sequences coming from semi elliptic differential equations abstract in this paper we are concerned with the study of spectral properties of the sequence of matrices a n a coming from the discretization using centered finite differences of minimal order of elliptic or semielliptic differential operators l a u of the form 1 d d x a x d d x u x f x on 0 1 dirichlet b c on where the nonnegative bounded coefficient function a x of the differential operator may have some isolated zeros in more precisely we state and prove the explicit form of the inverse of a n a and some formulas concerning the relations between the orders of zeros of a x and the asymptotic behavior of the minimal eigenvalue condition number of the related matrices as a conclusion and in connection with our theoretical findings first we extend the analysis to higher order semi elliptic differential operators and then we present various numerical experiments showing that similar results must hold true in 2d as well

on positive solutions of a nonlinear equilibrium boundary value problem abstract we give necessary and sufficient conditions for the existence of positive solutions to a nonlinear elliptic system arising from mathematical ecology with the dirichlet boundary data assigned by the equilibria of the corresponding ode system the main result is theorem 1 the tools are dancer fixed point index and degree theory

preconditioned hss methods for the solution of non hermitian positive definite linear systems and applications to the discrete convection diffusion equation we study the role of preconditioning strategies recently developed for coercive problems in connection with a two step iterative method based on the hermitian skew hermitian splitting hss of the coefficient matrix proposed by bai golub and ng for the solution of nonsymmetric linear systems whose real part is coercive as a model problem we consider finite differences fd matrix sequences an a p n discretizing the elliptic convection diffusion problem with being a plurirectangle of rd with a x being a uniformly positive function and p x denoting the reynolds function here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes more precisely in connection with preconditioned hss gmres like methods we consider the preconditioning sequence pn a n pn a dn1 2 a an 1 0 dn1 2 a where dn a is the suitably scaled main diagonal of an a 0 if a x is positive and regular enough then the preconditioned sequence shows a strong clustering at unity so that the sequence pn a n turns out to be a superlinear preconditioning sequence for an a 0 n where an a 0 represents a good approximation of re an a p namely the real part of an a p the computational interest is due to the fact that the preconditioned hss method has a convergence behavior depending on the spectral properties of pn 1 a re an a p n pn 1 a an a 0 n therefore the solution of a linear system with coefficient matrix an a p is reduced to computations involving diagonals and to the use of fast poisson solvers for an 1 0 n some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

analysis of preconditioning strategies for collocation linear systems in the past few years several authors have studied the preconditioning of collocation matrices by finite differences fds matrices arising from the associated collocation points here we discuss how to solve in an efficient way nonuniform grid fd linear systems including those related to a generic fd collocation preconditioner the main idea is based on a further step of preconditioning defined in terms of diagonal and toeplitz matrices first we identify the limit spectral distributions of the involved fd collocation matrix sequences and then we prove that the proposed toeplitz based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1d case in the 2d case the situation is different so that more appropriate strategies are discussed a wide numerical experimentation emphasizing the correctness of the theoretical results is also reported

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

stability of the notion of approximating class of sequences and applications given an approximating class of sequences b n m n m for a n n we prove that b n m n m x being the pseudo inverse of moore penrose is an approximating class of sequences for a n n where a n n is a sparsely vanishing sequence of matrices a n of size d n with d k d q for k q k email protected n as a consequence we extend distributional spectral results on the algebra generated by toeplitz sequences by including the pseudo inversion operation in the case where the sequences that are pseudo inverted are distributed as sparsely vanishing symbols applications to preconditioning and a potential use in image signal restoration problems are presented

bifurcation for an elliptic system coupled in the linear part we study linearly coupled elliptic systems without any assumption on the signs of the coefficients of the linear part we derive first the leray schauder degree of the associated linear system then we obtain sufficient conditions on the nonlinear terms for having bifurcation to coexistence states our results include the sublinear case keys words linearly coupled elliptic systems bifurcation to positive solutions coexistence states leray schauder degree email jfleck cict fr rpardo sunma4 mat ucm es

least squares preconditioners for stabilized discretizations of the navier stokes equations this paper introduces two stabilization schemes for the least squares commutator lsc preconditioner developed by elman howle shadid shuttleworth and tuminaro siam j sci comput 27 2006 pp 1651 1668 for the incompressible navier stokes equations this preconditioning methodology is one of several choices that are effective for navier stokes equations and it has the advantage of being defined from strictly algebraic considerations it has previously been limited in its applicability to div stable discretizations of the navier stokes equations this paper shows how to extend the same methodology to stabilized low order mixed finite element approximation methods

symbol based multigrid methods for galerkin b spline isogeometric analysis we consider the stiffness matrices arising from the galerkin b spline isogeometric analysis discretization of classical elliptic problems by exploiting their specific spectral properties compactly described by a symbol we design an efficient multigrid method for the fast solution of the related linear systems the convergence rate of general purpose multigrid methods based on classical stationary smoothers is optimal i e bounded independently of the matrix size but it also worsens exponentially with respect to the spline degree the symbol allows us to give a detailed theoretical explanation of this exponential worsening in the case of the two grid scheme in addition thanks to a specific factorization of the symbol we are able to design an ad hoc multigrid method with an effective preconditioned cg or gmres smoother at the finest level in the spirit of the multi iterative idea the convergence rate of this multi iterative multigrid method is not only optimal but also robust i e bounded su

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

on the asymptotic spectrum of finite element matrix sequences we derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying p 1 finite elements with standard mesh refinement to the semielliptic pde of second order in divergence form nabla k nabla t u f on o u g on partial omega here omega subset r 2 and k is supposed to be piecewise continuous and pointwise symmetric semipositive definite the symbol describing this asymptotic eigenvalue distribution depends on the pde but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain our work is motivated by recent results on the superlinear convergence behavior of the conjugate gradient method which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h to 0 we compare our findings with similar results for the finite difference method which were published in recent years in particular we observe that our sequence of stiffness matrices is part of the class of generalized locally toeplitz sequences for which many theoretical tools are available this enables us to derive some results on the conditioning and preconditioning of such stiffness matrices

spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

an hp adaptive finite element method for electromagnetics part 3 a three dimensional infinite element for maxwell s equations this paper is a continuation of reference 26 cecot demkowicz and rachowicz computer methods in applied mechanics and engineering 2000 188 625 643 and describes an implementation of the infinite element for three dimensional time harmonic maxwell s equations proposed in reference 15 demkowicz and pal computer methods in applied mechanics and engineering 1998 164 77 94 the element is compatible with the hp finite element discretizations for maxwell s equations in bounded domains reported in references 16 18 computer methods in applied mechanics and engineering 1998 152 103 124 1999 169 331 344 2000 187 307 337 copyright 2003 john wiley sons ltd

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

spectra of multilevel toeplitz matrices advanced theory via simple matrix relationships we consider the eigenvalue and singular value distributions for m level toeplitz matrices generated by a complex valued periodic function of m real variables we show that familiar formulations for l due to szeg and others can be preserved so long as f l1 and what is more with g weyl s definitions just a bit changed in contrast to other approaches the one we follow is based on simple matrix relationships

mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides in this paper we propose a finite element method to compute dielectric waveguides which does not present spurious solutions in addition to a theoretical analysis showing the good stability properties of this method we give numerical results for some classical test problems

full wave analysis of dielectric waveguides at a given frequency new variational formulation to compute propagation constants is proposed based on it vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property convergence analysis is conducted in the framework of collectively compact operators reported theoretical results apply to a wide class of vector finite elements including two families of nedelec and their generalization the hp edge elements numerical experiments fully support theoretical estimates for convergence rates

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

inhomogeneous lossy waveguide mode analysis abstract this paper discusses electromagnetic numerical mode analysis in waveguides with materially inhomogeneous cross sections and material dissipation a full wave formulation of maxwell s homogeneous equations including gauss electric law stable at vanishing propagation constant is implemented and verified in terms of the hp adaptive version of the finite element method it provides the possibility to use high order polynomial enrichments combined with strongly graded meshes it is considered most efficient in resolving the loss of solution regularity at material interfaces with large contrast numerical examples including materially lossless homogeneous and inhomogeneous cross sections with and without losses are analysed to corroborate the implementation the efficiency of using higher order polynomial enrichments is shown the approach is anticipated to have a broad application from modern on chip interconnect and antenna technologies to the design of low observable aerial vehicles

an optimal domain decomposition preconditioner for low frequency time harmonic maxwell equations the time harmonic maxwell equations are considered in the low frequency case a finite element domain decomposition approach is proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure which is proven to converge the rate of convergence turns out to be independent of the mesh size showing that the preconditioner implicitly defined by the iterative procedure is optimal for obtaining this convergence result it has been necessary to prove a regularity theorem for dirichlet and neumann harmonic fields

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

a v cycle multigrid for multilevel matrix algebras proof of optimality we analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant tau hartley or are of toeplitz type in the case of matrix algebra linear systems we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill conditioning this happens iff f takes the zero value more precisely if the d level coefficient matrix has partial dimension nr at level r with r 1 ldots d then the size of the system is n varvec n prod r 1 d n r varvec n n 1 ldots n d and o n n operations are required by the considered v cycle multigrid in order to compute the solution within a fixed accuracy since the total arithmetic cost is asymptotically equivalent to the one of a matrix vector product the proposed method is optimal some numerical experiments concerning linear systems arising in 2d and 3d applications are considered and discussed

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au