least squares methods for stokes equations based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for stokes and elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied we provide two methods the first is posed in terms of the velocity pressure pair without the introduction of additional variables the second adds a vorticity variable in both cases we employ least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak 1994

faces of poisson voronoi mosaics for a stationary poisson voronoi tessellation in euclidean d space and for k in 1 dots d we consider the typical k dimensional face with respect to a natural centre function we express the distribution of this typical k face in terms of a certain poisson process of closed halfspaces in a k dimensional space then we show that under the condition of large inradius the relative boundary of the typical k face lies with high probability in a narrow spherical annulus

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

a comparative study of finite element and finite difference methods for cauchy riemann type equations a least squares formulation of the system operatorname div bf u rho operatorname curl bf u zeta is surveyed from the viewpoint of both finite element and finite difference

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

a mixed finite element method for the stokes problem an acceleration pressure formulation the existing finite element methods 6 9 11 for the stokes equation lead to a saddle point problem for removing certain restrictions on the elements and improving the accuracy this paper develops a new method the stokes equation is treated as a first order linear system using a least squares method to minimize the defect in the differential operator we find that the result converges asymptotically to optimal order in sobolev spaces in the 2 d case if the domain is subdivided into regular triangles and the piecewise linear functions in the space h 1 are chosen to be trial functions for both velocity and pressure then we have error estimates p p h 0 u u h 0 ch 2

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

constructive quantum shannon decomposition from cartan involutions the work presented here extends upon the best known universal quantum circuit the quantum shannon decomposition proposed by shende et al 2006 ieee trans comput aided des integr circuits syst 25 1000 we obtain the basis of the circuit s design in a pair of cartan decompositions this insight gives a simple constructive factoring algorithm in terms of the cartan involutions corresponding to these decompositions

quantum conductance fluctuations and maximum entropy ensembles for the transfer matrix abstract as an introduction we present conductance fluctuations which have been observed in a mesoscopic wire at very low temperature those experiments show either time independent quantum fluctuations observed by varying an applied magnetic field of the fermi energy or quantum noises induced by short applied voltage pulses we review and extend a new approach to the theory of those quantum fluctuations based on the study of multiplicative random matrix ensembles defined using a hypothesis of maximum information entropy the distribution of the radial parameters a of the transfer matrix m is given both in the metallic and localized regime for disordered conductors it is shown that this approach gives a distribution of the a of a type similar to those describing the spectral fluctuations of the gaussian ensembles in random matrix theory for disordered insulators the logarithms of the a have quasi independent gaussian fluctuations the conductance g being a linear statistic of the a the universal conductance fluctuations in the metallic regime and lognormal fluctuations in the localized regime are then a consequence of those macroscopic maximum entropy models which can be understood by simple coulomb gas analogies universal results relative to localization lengths and conductance fluctuations of anderson insulators are given the importance of the multiplicative composition law of m in explaining the success of this approach is underlined

cells with many facets in a poisson hyperplane tessellation abstract let z be the typical cell of a stationary poisson hyperplane tessellation in r d the distribution of the number of facets f z of the typical cell is investigated it is shown that under a well spread condition on the directional distribution the quantity n 2 d 1 p f z n n is bounded from above and from below when f z is large the isoperimetric ratio of z is bounded away from zero with high probability these results rely on one hand on the complementary theorem which provides a precise decomposition of the distribution of z and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets from the asymptotics of the distribution of f z tail estimates for the so called content of z are derived as well as results on the conditional distribution of z when its content is large

a least squares finite element method for incompressible flow in stress velocity pressure version abstract in this paper we are concerned with the incompressible flow in 2 d introducing additional variables of derivatives of velocity which are called stresses here the second order dynamic equations are reduced into a first order system with variables of stress velocity and pressure combining the compatability conditions and the divergence ice condition we have a system with six first order equations and six unknowns least squares method performed over this extended system the analysis shows that this method achieves optimal rates of convergence in the h1 norm as the h approaches to zero numerical experiences are also available

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

asymptotic expansions for high contrast linear elasticity we study linear elasticity problems with high contrast in the coefficients using asymptotic limits recently introduced we derive an asymptotic expansion to solve heterogeneous elasticity problems in terms of the contrast in the coefficients we study the convergence of the expansion in the h 1 norm

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

least squares finite element methods for the elasticity problem abstract a new first order system formulation for the linear elasticity problem in displacement stress form is proposed the formulation is derived by introducing additional variables of derivatives of the displacements whose combinations represent the usual stresses standard and weighted least squares finite element methods are then applied to this extended system these methods offer certain advantages such as that they need not satisfy the inf sup condition which is required in the mixed finite element formulation that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns that the resulting algebraic systems are symmetric and positive definite and that accurate approximations of the displacements and the stresses can be obtained simultaneously with displacement boundary conditions it is shown that both methods achieve optimal rates of convergence in the h 1 norm and in the l 2 norm for all the unknowns numerical experiments with various poisson ratios are given to demonstrate the theoretical error estimates

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

sur une conjecture de d g kendall concernant la cellule de crofton du plan et sur sa contrepartie brownienne in connection with a conjecture stated by d g kendall in the forties we describe the asymptotic behavior of the distribution function of the area of the planar crofton cell we deduce from this in support of his conjecture that expressed in terms of eigenvalues the large crofton cells are nearly circular we obtain also the asymptotic behavior of the laplace transform of the law of the perimeter of the convex hull of planar brownian motion run until time 1 this last result implies that the small convex hulls of brownian motion are nearly circular

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

scattering theory and conductance fluctuations in mesoscopic systems abstract a random matrix model developed originally in the framework of the statistical theory of nuclear reactions is via the many channel approximation of landauer s formula applied to universal conductance fluctuations ucf in a disordered sample connected to two or three leads the leads are modelled as ideal conductors for large sample lengths the results agree with those obtained by other approaches this shows that ucf can be modelled successfully in terms of random matrices for sample lengths of several tens of elastic mean free paths the conductance fluctuations are seen to depend sensitively on the coupling to the leads this is due to the appearance of a new energy scale the width for emission of an electron from the disordered sample into the leads this quantity also influences the autocorrelation function of the conductance results are presented for the three canonical random matrix ensembles the gaussian orthogonal unitary and symplectic ensembles

an adaptively refined least squares finite element method for generalized newtonian fluid flows using the carreau model we implemented an adaptively refined least squares finite element approach for the navier stokes equations that govern generalized newtonian fluid flows using the carreau model to capture the flow region we developed an adaptive mesh refinement approach based on the least squares method the generated refined grids agree well with the physical attributes of the flows we also proved that the least squares approximation converges to the linearized versions solutions of the carreau model at the best possible rate model problems considered in the study are the flow past a planar channel and 4 to 1 contraction problems we presented the numerical results of the model problems revealing the efficiency of the proposed scheme and investigated the physical parameter effects

finite element approximation for grad div type systems in the plane this paper studies a 3 times 3 linear system of differential equations of grad div type with appropriate boundary conditions in constructing numerical approximate solutions to this boundary val

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

a weighted least squares method for the backward forward heat equation a weighted least squares method is given for the numerical solution of parabolic partial differential equations where the diffusion coefficient changes sign the second order equation is transformed into a first order system of symmetric positive differential equations in the sense of friedrichs and the system is solved using least squares techniques error estimates and some numerical examples are presented

a family of higher order mixed finite element methods for plane elasticity the dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the raviart thomas mixed finite elements for a scalar second order elliptic equation estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inl 2 and optimal order negative norm estimates are obtained inh s for a range ofs depending on the index of the finite element space an optimal order estimate inl for the displacement error is given also a quasioptimal estimate is derived in an appropriate space all estimates are valid uniformly with respect to the compressibility and apply in the incompressible case the formulation of the elements is presented in detail

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

the anderson transition new numerical results for the critical exponents abstract the results of our recent evaluation of the critical exponents of the localization length v and the conductivity s at the anderson transition for box and gaussian distributions of the random potential are reported data are evaluated not only in the centre of the band but also near the band edge near the band centre we obtain s v 1 4 0 2 and s v 0 9 0 3 for the box and the gaussian distribution respectively near the band edge we found it impossible to determine accurately the exponents

the least squares finite element method in elasticity part i plane stress or strain with drilling degrees of freedom a new least squares finite element method lsfem for plane elasticity problems is developed based on the first order displacement stress rotation formulation which includes two new first order compatibility constraints among the stresses and the drilling rotation this lsfem can accommodate all kinds of equal order interpolations numerical experiments on various examples including incompressible materials show that the method achieves an optimal rate of convergence for all variables copyright 2001 john wiley sons ltd

on the construction of optimal mixed finite element methods for the linear elasticity problem the mixed finite element method for the linear elasticity problem is considered we propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement the ideas are applied in some examples

large faces in poisson hyperplane mosaics a generalized version of a well known problem of d g kendall states that the zero cell of a stationary poisson hyperplane tessellation in d under the condition that it has large volume approximates with high probability a certain definite shape which is determined by the directional distribution of the underlying hyperplane process this result is extended here to typical k faces of the tessellation for k 2 d 1 this requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction

discretization by finite elements of a model parameter dependent problem the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness which appears as a parameter in the problem on the accuracy it is shown that the approximation achieved by a standard finite element method degenerates for thin beams in contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter the most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by gauss quadratures

analysis of some mixed finite element methods for plane elasticity equations we analyze some mixed finite element methods based on rectangular elements for solving the two dimensional elasticity equations we prove error estimates for a method proposed by taylor and zienkiewicz and for some new variants of the known equilibrium methods a numerical example is given demonstrating the performance of the various algorithms considered 1 introduction in the numerical solution of problems of continuum mechanics the stresses are normally of primary interest in the elastic region it is therefore natural to design the numerical algorithms so that the stresses can be obtained directly without first computing the displacements such methods can be derived from the dual variational formulation of the elasticity problem the corresponding finite element algorithms are usually formulated as mixed methods where both the displacements and the stresses are first approximated and the displacements are then eliminated from the discrete equations in many cases the elimination can be rather effectively done using penalty perturbation techniques or their iterative variants cf 3 11 12 the best known finite element methods of the above type are the so called equilibrium methods first proposed by fraejis de veubeke 17 cf also 14 16 18 and analyzed theoretically by johnson and mercier 9 cf also 8 in these methods one uses specific composite elements which allow the equilibrium condi tion between the stresses and the volume load to be satisfied exactly in the case where the volume load is zero

the limit shape of the zero cell in a stationary poisson hyperplane tessellation in the early 1940s d g kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic poisson line process in the plane tends to circularity given that the area of the zero cell tends to a proof was given by i n kovalenko in 1997 this paper generalizes kovalenko s result in two directions to higher dimensions and to not necessarily isotropic stationary poisson hyperplane processes in the anisotropic case the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of minkowski s existence theorem for convex bodies with given area measures

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

analysis of a least squares finite element method for the circular arch problem1this work was supported in part by nsc grant 88 2119 m 214 001 taiwan 1 the stability and convergence of a least squares finite element method for the circular arch problem with shear deformation in a first order system formulation are investigated it is shown that the least squares finite element approximations are stable and convergent in a natural energy norm associated with the least squares functional for the shallow arch case the optimal order of convergence in the h 1 norm for all the unknowns can be achieved uniformly with respect to the small thickness parameter and thus the locking phenomenon does not occur in this case a simple and sharp a posteriori error estimator is also addressed

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

least squares finite element approximations to the timoshenko beam problem in this paper a least squares finite element method for the timoshenko beam problem is proposed and analyzed the method is shown to be convergent and stable without requiring extra smoothness of the exact solutions for sufficiently regular exact solutions the method achieves optimal order of convergence in the h 1 norm for all the unknowns displacement rotation shear moment uniformly in the small parameter which is generally proportional to the ratio of thickness to length thus the locking phenomenon disappears as the parameter tends to zero a sharp a posteriori error estimator which is exact in the energy norm and equivalent in the h 1 norm is also briefly discussed

least squares finite element for second order boundary value problem with optimal rates of convergence abstract during the past twenty years engineers and mathematicians have successfully developed the mixed methods and the hybrid methods to solve problems 1 1 1 2 and 1 3 numerically such as 1 6 the discrete inf sup or its equivalent condition for the subspaces of the approximate functions of u and p has been applied for it ensures the numerical results converge to the real world solution steadily in this paper we try to put an extra equation which is the compatibility condition of this system we then apply the least squares method to the so called enriched system analysis shows that this method achieves the optimal rate of convergence in the h1and l2 norms without using the above condition numerical examples are still available

algebraic relations between the total least squares and least squares problems with more than one solution this paper completes our previous discussion on the total least squares tls and the least squares ls problems for the linear systemax b which may contain more than one solution 12 13 generalizes the work of golub and van loan 1 2 van huffel 8 van huffel and vandewalle 11 the tls problem is extended to the more general case the sets of the solutions and the squared residuals for the tls and ls problems are compared the concept of the weighted squares residuals is extended and the difference between the tls and the ls approaches is derived the connection between the approximate subspaces and the perturbation theories are studied it is proved that under moderate conditions all the corresponding quantities for the solution sets of the tls and the modified ls problems are close to each other while the quantities for the solution set of the ls problem are close to the corresponding ones of a subset of that of the tls problem

first order system least squares fosls for planar linear elasticity pure traction problem this paper develops two first order system least squares fosls approaches for the solution of the pure traction problem in planar linear elasticity both are two stage algorithms that first solve for the gradients of displacement which immediately yield deformation and stress then for the displacement itself if desired one approach which uses l 2 norms to define the fosls functional is shown under certain h 2 regularity assumptions to admit optimal h 1 like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the poisson ratio for all variables the second approach which is based on h 1 norms is shown under general assumptions to admit optimal uniform performance for displacement flux in an l 2 norm and for displacement in an h 1 norm these methods do not degrade as other methods generally do when the material properties approach the incompressible limit

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

intersection and proximity of processes of flats abstract weakly stationary random processes of k dimensional affine subspaces flats in r n are considered if 2 k n then intersection processes are investigated while in the complementary case 2 k n a proximity process is introduced the intensity measures of these processes are described in terms of parameters of the underlying k flat process by a translation into geometric parameters of associated zonoids and by means of integral transformations several new uniqueness and stability results for these processes of flats are derived they rely on a combination of known and novel estimates for area measures of zonoids which are also developed in the paper finally an asymptotic second order analysis as well as central and non central limit theorems for length power direction functionals of proximity processes derived from stationary poisson k flat process complement earlier works for intersection processes

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

adaptive refinement for least squares finite elements with element by element conjugate gradient solution an error analysis and an adaptive refinement scheme are developed for use with a class of least squares finite element methods the error indicators for the refinement procedure are based on element residuals which are calculated as part of the least squares method the solution scheme used in the supporting numerical studies employs an elment by element conjugate gradient algorithm consequently storage and computation are not influenced by nodal numbering