an error based approach to complementary formulations of static field solutions the constitutive relationship between fields in electromagnetic media is cast in error form for well posed static problems the error splits systematically into complementary functionals error minimization provides a universal variational principle that generates complementary and dual solution formulations the error which is wholly attributed to the numerical constraints in approximate solutions is always defined and computable locally and globally

general constraint preconditioning iteration method for singular saddle point problems for the singular saddle point problems with nonsymmetric positive definite 1 1 block we present a general constraint preconditioning gcp iteration method based on a singular constraint preconditioner using the properties of the moore penrose inverse the convergence properties of the gcp iteration method are studied in particular for each of the two different choices of the 1 1 block of the singular constraint preconditioner a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the gcp iteration method moreover the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual gmres and quasi minimal residual qmr methods are also tested

the hp local discontinuous galerkin method for low frequency time harmonic maxwell equations the local discontinuous galerkin method for the numerical approximation of the time harmonic maxwell equations in a low frequency regime is introduced and analyzed topologically nontrivial domains and heterogeneous media are considered containing both conducting and insulating materials the presented method involves discontinuous galerkin discretizations of the curl curl and grad div operators derived by introducing suitable auxiliary variables and so called numerical fluxes an hp analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained

the set of unattainable points for the rational hermite interpolation problem abstract we describe geometrically and algebraically the set of unattainable points for the rational hermite interpolation problem i e those points where the problem does not have a solution we show that this set is a union of equidimensional complete intersection varieties of odd codimension the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem each of these equidimensional varieties can be further decomposed as a union of as many rational irreducible varieties as input data points we exhibit algorithms and equations defining all these objects

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

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

tree cotree decompositions for first order complete tangential vector finite elements edge based finite elements have important applications in modelling both quasi static and high frequency electromagnetic problems recent work has demonstrated that formulations that exploit the graph structure of the edge based finite element mesh may be extended to higher order elements this paper presents the details of how such an extension to a first order complete finite element is accomplished and illustrates its application in two dimensional magnetostatics 1997 by john wiley sons ltd

multilevel iterative solvers for the edge finite element solution of the 3d maxwell equation in the edge vector finite element solution of the frequency domain maxwell equations the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers we extend the approach of r hiptmair multigrid method for maxwell s equations siam j numer anal 36 1 1999 204 225 and using domain decomposition ideas construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements the new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain maxwell equations the solver can be seen as a variable preconditioner and thus accelerated by krylov subspace techniques e g gcr or fgmres we demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity

computationally efficient applications of the euclidean algorithm to zero location abstract we characterize the inertia of the n n hankel matrix associated with a pair of real polynomials in terms of the quotients generated by the euclidean algorithm applied to these polynomials in this way we solve certain classical zero location problems at the computational cost of o n 2 arithmetic operations which can be reduced to o n log 2 n if fast polynomial arithmetic is used

construction of a finite element basis of the first de rham cohomology group and numerical solution of 3d magnetostatic problems we devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three dimensional magnetostatic problems in particular we construct a finite element basis of the first de rham cohomology group of the computational domain the proposed method works for general topological configurations and does not need the determination of cutting surfaces

error analysis of a finite element integral equation scheme for approximating the time harmonic maxwell system in 1996 hazard and lenoir suggested a variational formulation of maxwell s equations using an overlapping integral equation and volume representation of the solution siam j math anal 27 1996 pp 1597 1630 they suggested a numerical scheme based on this approach but no error analysis was provided in this paper we provide a convergence analysis of an edge finite element scheme for the method the analysis uses the theory of collectively compact operators its novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation

block triangular preconditioners for nonsymmetric saddle point problems field of values analysis a preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed the proposed preconditioner is of block triangular form the aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system as an example a saddle point problem obtained from a mixed finite element discretization of the oseen equations is considered the convergence estimates obtained by using a field of values analysis are independent of the discretization parameter h several computational experiments supplement the theoretical results and illustrate the performance of the method

a variational approach for the vector potential formulation of the stokes and navier stokes problems in three dimensional domains abstract a few well posed variational problems are constructed whose solutions are vector potentials of the velocity field of the three dimensional stokes problem a choice of the adequate boundary conditions to be imposed is performed in a systematic manner even in the case of not simply connected domains the test functions do not have to be divergence free in the formulations given here it is then seen that a similar approach is well suited to the treatment of the non linear navier stokes problem

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

generalized shift splitting preconditioners for nonsingular and singular generalized saddle point problems based on the shift splitting technique a class of generalized shift splitting preconditioners are proposed for both nonsingular and singular generalized saddle point problems the generalized shift splitting preconditioner is induced by a generalized shift splitting of the generalized saddle point matrix resulting in a generalized shift splitting fixed point iteration theoretical analyses show that the generalized shift splitting iteration method is convergent and semi convergent unconditionally for solving the nonsingular and the singular generalized saddle point problems respectively numerical experiments of a model navier stokes problem are implemented to demonstrate the feasibility and effectiveness of the proposed preconditioners

analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain this paper considers finite element approximation and three coupled type iterations of stationary magnetohydrodynamics mhd equations on a general lipschitz domain an additional lagrange multiplier r is introduced related to divergence free of magnetic field b and the b is analyzed in h curl space which is proposed in schotzau 2004 in finite element discretization the hydrodynamic unknowns are approximated by stable finite element pairs and the magnetic unknown is discretized by curl conforming nedelec element spaces the well posedness of this formula and the optimal error estimate are provided based on this for numerical implementation of this scheme we propose and discuss three coupled type iterative methods which are stable and convergent under different conditions specifically iteration i is stable and convergent under strong condition iteration ii is stable and convergent under weaker condition iteration iii is unconditionally stable and convergent under the weakest condition finally some numerical tests confirm our theoretical analysis

convergence analysis of modified pgss methods for singular saddle point problems abstract recently variants of shift splitting iteration method have been proposed for solving singular saddle point problems however these methods can only be proved to converge to one of the solutions of the consistent singular linear system not knowing any further information about this solution in this work we consider a modified preconditioned generalized shift splitting mpgss iteration method for solving both consistent and inconsistent singular saddle point problems this method is proved to converge to the best least squares solution moreover based on the iteration form a preconditioner is obtained to accelerate krylov subspace methods theoretical analysis shows that the preconditioned gmres method also converges to the best least squares solution of the consistent singular saddle point problem in addition numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner

multigrid computation of vector potentials in this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity we show how the mixed finite element discrete problem may be reformulated as a symmetric positive definite system this involves the proof of a discrete analogue of the second kind friedrichs inequality then we show how the multigrid method may be applied to compute the magnetic vector potential the difficulty here is that the bilinear forms on different meshes are not related in the standard way two different intergrid transfer operators are defined in the paper for the multigrid method the convergence theory and the numerical tests include both algorithms

the bgs uzawa and bj uzawa iterative methods for solving the saddle point problem in this paper we propose two new iterative methods for solving the nonsingular saddle point problem based on partitioning the coefficient matrix one is combining the block gauss seidel iterative method with the uzawa iterative method and the other one is combining the block jacobi iterative method with the uzawa iterative method then we study the convergence of the two novel methods under suitable restrictions on the iteration parameters respectively numerical experiments are also presented to illustrate the behavior of the considered algorithms

transport equation with boundary conditions for free surface localization during the filling stage of an injection moulding process which consists in casting a melt polymer in order to manufacture plastic pieces the free interface between polymer and air has to be precisely described we set this interface as a zero level set of an unknown function this function satisfies a transport equation with boundary conditions where the velocity field has few regularity properties

analysis and computation of compatible least squares methods for div curl equations we develop and analyze least squares finite element methods for two complementary div curl elliptic boundary value problems the first one prescribes the tangential component of the vector field on the boundary and is solved using curl conforming elements the second problem specifies the normal component of the vector field and is handled by div conforming elements we prove that both least squares formulations are norm equivalent with respect to suitable discrete norms yield optimal asymptotic error estimates and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

elements finis mixtes incompressibles pour l equation de stokes dans ℝ3 we introduce some new families of finite element approximation for the stationary stokes and navier stokes equations in a bounded domain in 3 these elements can used tetahedrons or cubes the approximation satisfie exactly the incompressibility condition

linear algebra methods in a mixed approximation of magnetostatic problems in the solution of magnetostatic problems the use of a mixed formulation based on both the magnetic and magnetic displacement fields is particularly appropriate as it allows us to impose the physical conditions exactly and to maintain the continuity properties of the two fields together with an efficient treatment of boundary conditions the discretization by means of a proper finite element method yields a strongly structured algebraic linear system this paper is concerned with the solution of this large very sparse indefinite linear system in particular we present the implementation of some known direct and preconditioned iterative methods and discuss their performance on two dimensional 2 d and three dimensional 3 d specific models we show that the 2 d system can be efficiently handled by appropriate variants of these schemes while preliminary tests on the 3 d system give some insight in the understanding of the analysis that needs to be done

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

modified edge finite elements for photonic crystals we consider maxwell s equations with periodic coefficients as it is usually done for the modeling of photonic crystals using bloch floquet theory the problem reduces in a standard way to a modification of the maxwell cavity eigenproblem with periodic boundary conditions following 8 a modification of edge finite elements is considered for the approximation of the band gap the method can be used with meshes of tetrahedrons or parallelepipeds a rigorous analysis of convergence is presented together with some preliminary numerical results in 2d which fully confirm the robustness of the method the analysis uses well established results on the discrete compactness for edge elements together with new sharper interpolation estimates

mixed finite element methods for stationary incompressible magneto hydrodynamics a new mixed variational formulation of the equations of stationary incompressible magneto hydrodynamics is introduced and analyzed the formulation is based on curl conforming sobolev spaces for the magnetic variables and is shown to be well posed in possibly non convex lipschitz polyhedra a finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf sup stable velocity pressure space pairs and the magnetic ones by a mixed approach using nedelec s elements of the first kind an error analysis is carried out that shows that the proposed finite element approximation leads to quasi optimal error bounds in the mesh size

supervised parametric and non parametric classification of chromosome images this paper describes a fully automatic chromosome classification algorithm for multiplex fluorescence in situ hybridization m fish images using supervised parametric and non parametric techniques m fish is a recently developed chromosome imaging method in which each chromosome is labelled with 5 fluors dyes and a dna stain the classification problem is modelled as a 25 class 6 feature pixel by pixel classification task the 25 classes are the 24 types of human chromosomes and the background while the six features correspond to the brightness of the dyes at each pixel maximum likelihood estimation nearest neighbor and k nearest neighbor methods are implemented for the classification the highest classification accuracy is achieved with the k nearest neighbor method and k 7 is an optimal value for this classification task

a modified generalized shift splitting method for nonsymmetric saddle point problems abstract in this paper we propose a modified generalized shift splitting denoted by mgssp preconditioned method for solving large sparse saddle point problems by theoretical analyses we verify the mgssp iteration method unconditionally converges to the unique solution of the saddle point problems estimate the sharp eigenvalue bounds of the related iteration matrix and point out the corresponding preconditioned matrix is positive real finally we perform some numerical computations to show the efficiency and the feasibility of the mgssp preconditioner

preconditioners for generalized saddle point problems noindent we propose and examine block diagonal preconditioners and variants of indefinite preconditioners for block two by two generalized saddle point problems that is we consider the nonsymmetric nonsingular case where the 2 2 block is small in norm and we are particularly concerned with the case where the 1 2 block is different from the transposed 2 1 block we provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices we also extend the results of de sturler and liesen siam j sci comput 26 2005 pp 1598 1619 to matrices with nonzero 2 2 block and to the use of approximate schur complements to demonstrate the effectiveness of these preconditioners we show convergence results spectra and eigenvalue bounds for two model navier stokes problems

a class of preconditioned generalized local pss iteration methods for non hermitian saddle point problems in this paper a class of new methods based on the positive definite and skew hermitian splitting scheme called preconditioned generalized local positive definite and skew hermitian splitting pglpss methods are considered to solve non hermitian saddle point problems the convergence properties of the proposed methods are analyzed which prove that the pglpss methods are convergent if the iteration parameters and parameter matrices satisfy appropriate conditions some numerical experiments are provided to verify the efficiency of the proposed method showing the competitiveness and efficiency of this novel method over other testing methods whether it served as a preconditioned iteration method or as a preconditioner to the krylov subspace method

the generalized hss method for solving singular linear systems for the singular non hermitian and positive semidefinite linear systems we propose an alternating direction iterative method with two parameters based on the hermitian and skew hermitian splitting the semi convergence analysis and the quasi optimal parameters of the proposed method are discussed moreover the corresponding preconditioner based on the splitting is given to improve the semi convergence rate of the gmres method numerical examples are given to illustrate the theoretical results and the efficiency of the generalized hss method either as a solver or a preconditioner for gmres

feature normalization via expectation maximization and unsupervised nonparametric classification for m fish chromosome images multicolor fluorescence in situ hybridization m fish techniques provide color karyotyping that allows simultaneous analysis of numerical and structural abnormalities of whole human chromosomes chromosomes are stained combinatorially in m fish by analyzing the intensity combinations of each pixel all chromosome pixels in an image are classified often the intensity distributions between different images are found to be considerably different and the difference becomes the source of misclassifications of the pixels improved pixel classification accuracy is the most important task to ensure the success of the m fish technique in this paper we introduce a new feature normalization method for m fish images that reduces the difference in the feature distributions among different images using the expectation maximization em algorithm we also introduce a new unsupervised nonparametric classification method for m fish images the performance of the classifier is as accurate as the maximum likelihood classifier whose accuracy also significantly improved after the em normalization we would expect that any classifier will likely produce an improved classification accuracy following the em normalization since the developed classification method does not require training data it is highly convenient when ground truth does not exist a significant improvement was achieved on the pixel classification accuracy after the new feature normalization indeed the overall pixel classification accuracy improved by 20 after em normalization

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

two modified block triangular splitting preconditioners for generalized saddle point problems abstract for the generalized saddle point problems firstly we introduce a modified generalized relaxed splitting mgrs preconditioner to accelerate the convergence rate of the krylov subspace methods based on a block triangular splitting of the saddle point matrix secondly we propose a modified block triangular splitting mbts this new preconditioner is easily implemented since it has simple block structure the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively moreover we apply the mgrs and the mbts preconditioners to three dimensional linearized navier stokes equations then we derive the quasi optimal parameters of the mgrs and the mbts preconditioners for two and three dimensional navier stokes equations respectively finally numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

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

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

a field based mixed formulation for the two dimensional magnetostatic problem a mixed variational formulation for the two dimensional magnetostatic problem involving both the magnetic field and the magnetic displacement field is presented and analyzed such a formulation is obtained by minimizing the error in the constitutive equation and imposing as constraints the canonical equations a numerical discretization by means of edge and face elements is proposed together with a stability analysis and error estimates

on block preconditioners for nonsymmetric saddle point problems we discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations in particular the navier stokes equation we prove that block diagonal and block triangular preconditioners based on symmetric positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h

a domain decomposition approach for heterogeneous time harmonic maxwell equations the time harmonic maxwell equations for a composite medium which behaves like a conductor in one part and like a perfect insulator in the other one are considered an existence and uniqueness theorem is proven for this degenerate problem in the case of dirichlet boundary conditions a finite element domain decomposition approach is then proposed for the numerical approximation of the exact solution this leads to an iteration by subdomain procedure where at each step a non degenerate boundary value problem has to be solved in each subdomain the convergence of these iterations is proven and the rate of convergence turns out to be independent of the mesh size h showing that the preconditioner implicitly defined by the iteration procedure is optimal

the uzawa pps iteration methods for nonsingular and singular non hermitian saddle point problems abstract in this paper based on the positive definite and positive semidefinite splitting pps iteration scheme we establish a class of uzawa pps iteration methods for solving nonsingular and singular non hermitian saddle point problems with the 1 1 part of the coefficient matrix being non hermitian positive definite theoretical analyses show that the convergence and semi convergence properties of the proposed methods can be guaranteed under suitable conditions furthermore we consider acceleration of the uzawa pps methods by krylov subspace like gmres methods and discuss the spectral properties of the corresponding preconditioned matrix numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners

on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

a semi circulant preconditioner for the convection diffusion equation in this paper we define and analyze a semi circulant preconditioner for the convection diffusion equation we derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations we show that for mesh peclet numbers less than 2 the rate of convergence depends only on the mesh peclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

stability of hodge decompositions in finite element spaces of differential forms in arbitrary dimension we elaborate on the interpretation of some mixed finite element spaces in terms of differential forms in particular we define regularization operators which combined with the standard interpolators enable us to prove discrete poincare friedrichs inequalities and discrete rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

convergence conditions for splitting iteration methods for non hermitian linear systems abstract necessary and sufficient convergence conditions are studied for splitting iteration methods for non hermitian system of linear equations when the coefficient matrix is nonsingular when this theory is specialized to the generalized saddle point problem we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods which include the uzawa and the inexact uzawa methods as special cases moreover we apply this theory to the two stage iteration methods for non hermitian positive definite linear systems and obtain sufficient conditions for guaranteeing the convergence of these methods

corrected uzawa methods for solving large nonsymmetric saddle point problems abstract in this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods which are similar in some respects to the uzawa type methods and are called corrected uzawa methods their convergence rates are analyzed the results of numerical experiments are presented when we apply them to navier stokes equations discretized by mixed finite elements

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

a pseudospectral matrix element method for solution of three dimensional incompressible flows and its parallel implementation abstract a new pseudospectral matrix element psme method employing the primitive variable formulation of the navier stokes equations was used to simulate 3 dimensional time dependent driven cavity flow at a reynolds number of 3200 with an aspect ratio of 3 in the spanwise direction as well as 3 dimensional flow over a backward step the new method only requires functions which are c 0 continuous across the interface between two adjacent elements it also ensures that the continuity equation is satisfied everywhere in the interior including the inter element points and on the boundary the resulting complex geometry for flow over a backward step can be divided into a number of overlapping subdomains by a domain decomposition of simpler geometry with patched grid points in which the solution is more easily obtained with an iterative procedure between subdomains the complete solution is found by the schwarz alternating procedure sap with an eigenfunction expansion for the pressure storage requirements for the 3d inversion step o n 6 are reduced to o n 3 if the inverse of pressure equation is not stored the parallel implementation of the three most time consuming procedures i computing the partial derivatives of scalar fields in terms of dotproduct ii transforming between eigenfunction space and physical space for the pressure in terms of matrix multiplications and iii performing the forward and backward sweeps of an lu decomposition to solve for the pressure have been efficiently performed on a parallel computer the numerical results have reproduced dynamic longitudinal taylor gortler like tgl vortices in qualitative agreement with the experimental results of koseff et al and also indicated other 3 dimensional effects on the flow development computational results for both 2 and 3 dimensional flow over a backward facing step at different reynolds numbers are also presented in this paper no pronounced 3 dimensional effects are observed for reynolds numbers up to 450 except in the boundary layer along the spanwise direction

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

a mixed formulation for 3d magnetostatic problems theoretical analysis and face edge finite element approximation a mixed field based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed this method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship while constraints represented by maxwell s equations are imposed by means of lagrange multipliers in this way both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements and boundary conditions can be imposed in a natural way moreover this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability as is usually the case a finite element discretization involving face and edge elements is also proposed performing stability analysis and giving error estimates

on semi convergence of a class of uzawa methods for singular saddle point problems in this paper a class of uzawa methods are presented for singular saddle point problems these methods contain the recently proposed uzawa aor and uzawa saor methods as special cases the 1 1 block of the corresponding uzawa preconditioner is positive definite both nonsingular and singular preconditioning matrices are considered the semi convergence of these methods is analyzed by using the techniques of singular value decomposition and moore penrose inverse numerical results show that they need less workload per iteration step comparing with the gsor and piu methods in some situations so they are feasible and effective for singular saddle point problems in some situations

block lu preconditioners for symmetric and nonsymmetric saddle point problems in this paper a general purpose block lu preconditioner for saddle point problems is presented the main difference between the approach presented here and that of other studies is that an explicit accurate approximation of the schur complement matrix is efficiently computed this is used to compute a preconditioner to the schur complement matrix which in turn defines a preconditioner for the global system a number of different variants are developed and results are reported for a few linear systems arising from cfd applications

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

hermitian and normal splitting methods for non hermitian positive definite linear systems in this paper a hermitian and normal splitting hns iterative method is proposed for solving non hermitian positive definite linear systems arising from convection diffusion reaction equations theoretical analysis shows that the hns method converges unconditionally to the exact solution of the linear system choices of the parameters are discussed the corresponding inexact hns ihns iterative method which is developed by employing some krylov subspace methods as its inner process is studied finally some numerical experiments are presented to demonstrate the effectiveness of the new method

on constraint preconditioners for generalized saddle point matrices for the iterative solution of large sparse generalized saddle point problems a class of new constraint preconditioners is presented and the spectral properties and parameter choices are discussed numerical experiments are used to demonstrate the feasibility and effectiveness of the new preconditioners as well as their advantages over the modified product type skew hermitian triangular splitting mpsts preconditioners

a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

an analysis of ne de lec s method for the spatial discretization of maxwell s equations abstract in 1980 nedelec developed a family of curl and divergence conforming finite elements in r 3 he proposed the use of these elements to discretize the time dependent maxwell equations noting that the elements have the advantage that the discrete magnetic displacement can be made exactly divergence free in this paper we shall analyze a slight generalization of nedelec s scheme and prove essentially optimal order convergence estimates in a variety of situations we also demonstrate that the nedelec method can be superconvergent at certain special points and we relate the method to yee s finite difference scheme a by product of our analysis will be a convergence proof for yee s method

segmentation of m fish images for improved classification of chromosomes with an adaptive fuzzy c means clustering algorithm an adaptive fuzzy c means algorithm was developed and applied to the segmentation and classification of multicolor fluorescence in situ hybridization m fish images which can be used to detect chromosomal abnormalities for cancer and genetic disease diagnosis the algorithm improves the classical fuzzy c means algorithm fcm by the use of a gain field which models and corrects intensity inhomogeneities caused by a microscope imaging system flairs of targets chromosomes and uneven hybridization of dna other than directly simulating the inhomogeneousely distributed intensities over the image the gain field regulates centers of each intensity cluster the algorithm has been tested on an m fish database that we have established which demonstrates improved performance in both segmentation and classification when compared with other fcm clustering based algorithms and a recently reported region based segmentation and classification algorithm our method gave the lowest segmentation and classification error which will contribute to improved diagnosis of genetic diseases and cancers

preconditioning strategies for hermitian indefinite toeplitz linear systems in this paper we propose and analyze preconditioning strategies for hermitian indefinite linear systems by using indefinite preconditioners under very elementary assumptions we show that the eigenvalues are real moreover in the case of multilevel toeplitz structures we prove distributional and localization results these techniques used in connection with the cg gmres bicgstab and qmr algorithms allow us to solve in an optimal way the corresponding linear systems a wide numerical experimentation confirms the efficiency of the proposed procedures

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

the preconditioned iterative methods with variable parameters for saddle point problem abstract in this paper by transforming the original problem equivalently we propose a new preconditioned iterative method for solving saddle point problem we call the new method as ptu preconditioned transformative uzawa method and we study the convergence of the ptu method under suitable restrictions on the iteration parameters moreover we show the choices of the optimal parameters and the spectrum of the preconditioned matrix deriving from the ptu method based on the ptu iterative method we propose another iterative method nonlinear inexact ptu method for solving saddle point problem we also prove its convergence and study the choices of the optimal parameters in addition we present some numerical results to illustrate the behavior of the considered algorithms

a one grid overlapped spectral multidomain method for the pdes abstract in this paper a novel chebyshev pseudospectral multidomain technique is introduced for the numerical solution of the partial differential equations careful consideration is given to the proper interface condition which decomposes the solution domain into subdomains by overlapping one grid point the effectiveness of the technique is illustrated for the solution of the partial differential equations exhibiting weak or discontinuous solutions

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

the derivation of hybridizable discontinuous galerkin methods for stokes flow in this paper we introduce a new class of discontinuous galerkin methods for the stokes equations the main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries we present four ways of hybridizing the methods which differ by the choice of the globally coupled unknowns classical methods for the stokes equations can be thought of as limiting cases of these new methods

analysis of finite element approximation for time dependent maxwell problems we provide an error analysis of finite element methods for solving time dependent maxwell problem using nedelec and thomas raviart elements we study the regularity of the solution and develop some new error estimates of nedelec finite elements as a result the optimal l 2 error bound for the semidiscrete scheme is obtained

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

a finite element method for the magnetostatic problem in terms of scalar potentials the aim of this paper is to analyze a numerical method for solving the magnetostatic problem in a three dimensional bounded domain containing prescribed currents and magnetic materials the method discretizes a well known formulation of this problem based on two scalar potentials the total potential defined in magnetic materials and the reduced potential defined in dielectric media and in nonmagnetic conductors carrying currents the topology of the domain of each material is not assumed to be trivial the resulting variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements transmission conditions are imposed by means of a piecewise linear lagrange multiplier on the surface separating the domains of both potentials error estimates for the numerical method are proved and the results of some numerical tests are reported to assess the performance of the method

on semi convergence of the uzawa hss method for singular saddle point problems based on the hermitian and skew hermitian splitting hss iteration scheme an efficient uzawa hss iteration method has been proposed to solve the nonsingular saddle point problems in this paper we discuss the feasibility of the uzawa hss method used for solving singular saddle point problems the semi convergence properties of the uzawa hss iteration method are carefully analyzed which show that the iterative sequence generated by the uzawa hss method converges to a solution of the singular saddle point problem if the iteration parameters satisfy suitable restrictions numerical results verify the robustness and efficiency of the uzawa hss method

splitting based block preconditioning methods for block two by two matrices of real square blocks recently bai proposed rotated block preconditioners for block two by two matrices of real square blocks these rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two by two triangular matrix theoretical and numerical results have shown the superiority of these rotated block preconditioners see bai 2013 10 in this paper inspired by the efficiency and the special structure of the rotated block preconditioners we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises we construct block jacobi and block gauss seidel splitting iteration methods based on the coefficient matrix of the new linear system the convergence of these splitting iterations is also demonstrated then by utilizing the proposed block jacobi and block gauss seidel splittings we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two by two diagonal or a block two by two triangular matrix spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate krylov subspace iteration methods such as gmres for solving the block two by two liner systems

generalized finite element systems for smooth differential forms and stokes problem we provide both a general framework for discretizing de rham sequences of differential forms of high regularity and some examples of finite element spaces that fit in the framework the general framework is an extension of the previously introduced notion of finite element systems and the examples include conforming mixed finite elements for stokes equation in dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes in dimension 3 we define one low order complex which may be branched into whitney forms at a chosen index stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension the finite element spaces all consist of composite polynomials the framework guarantees some nice properties of the spaces in particular the existence of commuting interpolators it also shows that some of the examples are minimal spaces

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method

the nonlinear inexact uzawa hybrid algorithms based on one step newton method for solving nonlinear saddle point problems this paper mainly consider the construction of efficient uzawa type algorithms for solving the nonlinear saddle point problem nspp two nonlinear inexact uzawa hybrid algorithms based on one step newton method are proposed for solving the nspps by using the so called energy norm the convergence of the proposed algorithms is verified under some practical conditions in addition some numerical results are reported to illustrate the behavior of the considered algorithm