the quantum jeffreys prior bures metric volume element for squeezed thermal states and a universal coding conjecture clarke and barron have recently established that the classical jeffreys prior yields universal codes ones which do relatively well in the sense of relative entropy no matter what the true state twamley has recently computed the bures metric proportional to the quantum fisher information statistical distinguishability metric as shown by braunstein and caves for squeezed thermal states the volume elements of these metrics that is the quantum jeffreys prior are found to be simply the product of a function of the squeeze factor and a function of the inverse temperature the phase being irrelevant in this regard a computational strategy to find a universal coding using the quantum jeffreys prior previously implemented for the two level quantum systems is then discussed

a multigrid method for the pseudostress formulation of stokes problems the purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator mathcal a gamma mathbf grad mathbf div over 2 times 2 matrix valued functions this system is derived from the pseudostress velocity formulation z cai b lee and p wang siam j numer anal 42 2004 pp 843 859 of two dimensional stokes problems through the penalty method or natural time discretization for the respective steady or unsteady state problems here gamma 0 is a constant associated with either the penalty parameter or the time step size and mathcal a is a singular map in this paper we develop a multigrid solver for the discrete problem using both the raviart thomas rt and the brezzi douglas marini bdm finite elements we show that the multigrid convergence rate is o frac 1 gamma 2 1 gamma 2 m where m is the number of smoothings this convergence rate is independent of the mesh size and the number of levels used in multigrid moreover numerical results presented in this paper show that the multigrid convergence rate for the bdm elements does not depend on the parameter gamma

a posteriori error estimation for an augmented mixed primal method applied to sedimentation consolidation systems abstract in this paper we develop the a posteriori error analysis of an augmented mixed primal finite element method for the 2d and 3d versions of a stationary flow and transport coupled system typically encountered in sedimentation consolidation processes the governing equations consist in the brinkman problem with concentration dependent viscosity written in terms of cauchy pseudo stresses and bulk velocity of the mixture coupled with a nonlinear advection nonlinear diffusion equation describing the transport of the solids volume fraction we derive two efficient and reliable residual based a posteriori error estimators for a finite element scheme using raviart thomas spaces of order k for the stress approximation and continuous piecewise polynomials of degree k 1 for both velocity and concentration for the first estimator we make use of suitable ellipticity and inf sup conditions together with a helmholtz decomposition and the local approximation properties of the clement interpolant and raviart thomas operator to show its reliability whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle bubble and edge bubble functions next we analyse an alternative error estimator whose reliability can be proved without resorting to helmholtz decompositions finally we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed primal finite element method

monotone riemannian metrics on density matrices with non monotone scalar curvature the theory of monotone riemannian metrics on the state space of a quantum system was established by denes petz in 1996 in a recent paper he argued that the scalar curvature of a statistically relevant monotone metric can be interpreted as an average statistical uncertainty the present paper contributes to this subject it is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed the manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state we show that not every monotone metric fulfils this expectation some of them behave in a very different way a mathematical condition is given for monotone riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics

a dual mixed approximation method for a three field model of a nonlinear generalized stokes problem in this work a dual mixed approximation of a nonlinear generalized stokes problem is studied the problem is analyzed in sobolev spaces which arise naturally in the problem formulation existence and uniqueness results are given and error estimates are derived it is shown that both lowest order and higher order mixed finite elements are suitable for the approximation method numerical experiments that support the theoretical results are presented

analysis of a pseudostress based mixed finite element method for the brinkman model of porous media flow in this paper we introduce and analyze a new mixed finite element method for the two dimensional brinkman model of porous media flow with mixed boundary conditions we use a dual mixed formulation in which the main unknown is given by the pseudostress the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply the babuska brezzi theory to establish sufficient conditions for the well posedness of the resulting continuous and discrete formulations in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k ge 0 k 0 for the pseudostress and continuous piecewise polynomials of degree k 1 k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem suitable auxiliary problems the continuous inf sup conditions satisfied by the bilinear forms involved a discrete helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators are the main tools for proving the reliability then helmholtz s decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are employed to show the efficiency finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithm are provided

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

a priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems abstract in this paper we introduce and analyze new augmented mixed finite element methods allowing much more flexibility in the choice of the corresponding finite dimensional subspaces for a class of nonlinear elasticity problems arising in hyperelasticity the starting mixed method is based on the incorporation of the strain tensor as an auxiliary unknown which together with the usual stress displacement rotation approach employed in linear elasticity yields a nonlinear twofold saddle point operator equation as the resulting weak formulation we first extend known results on the well posedness of the associated galerkin scheme with peers of order k 0 to the case k 1 then the augmented schemes are obtained by adding consistent galerkin type terms arising first from the constitutive equation and then from the equilibrium equation and the relations defining the rotation in terms of the displacement and the strain tensor as independent unknown all of them multiplied by suitably chosen stabilization parameters we apply classical results on the solvability analysis of nonlinear saddle point and strongly monotone operator equations to prove that the corresponding continuous and discrete augmented schemes are well posed more precisely and as the main advantages of our present approach we show that the well posedness of a partially augmented galerkin scheme is ensured by any finite element subspace for the strain tensor together with the peers space of order k 0 for the remaining unknowns whereas any finite element subspace of the whole continuous space will do in the case of a fully augmented scheme then we derive reliable and efficient residual based a posteriori error estimators for all the schemes finally we provide several numerical results illustrating the good performance of the mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

volume of the quantum mechanical state space the volume of the quantum mechanical state space over n dimensional real complex and quaternionic hilbert spaces with respect to the canonical euclidean measure is computed and explicit formulae are presented for the expected value of the determinant also in the general setting the case when the state space is endowed with a monotone metric or a pull back metric is also considered we give formulae for the volume of the state space with respect to the given riemannian metric we present the volume of the space of qubits with respect to various monotone metrics it turns out that the volume of the space of qubits can also be infinite we characterize those monotone metrics which generate infinite volume

a residual based a posteriori error estimator for a fully mixed formulation of the stokes darcy coupled problem abstract in this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2d flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we consider dual mixed formulations in both media which yields the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium and the traces of the porous media pressure and the fluid velocity on the interface as the resulting unknowns the set of feasible finite element subspaces includes raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the traces we derive a reliable and efficient residual based a posteriori error estimator for the coupled problem the proof of reliability makes use of the global inf sup condition helmholtz decompositions in both media and local approximation properties of the clement interpolant and raviart thomas operator on the other hand inverse inequalities the localization technique based on element bubble and edge bubble functions and known results from previous works are the main tools for proving the efficiency of the estimator finally some numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution are reported

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

analysis of an augmented fully mixed approach for the coupling of quasi newtonian fluids and porous media abstract in this paper we introduce and analyze an augmented mixed finite element method for the coupling of quasi newtonian fluids and porous media the flows are governed by a class of nonlinear stokes and linear darcy equations respectively and the transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law we apply dual mixed formulations in both domains and in order to handle the nonlinearity involved in the stokes region we set the strain and vorticity tensors as auxiliary unknowns in turn since the transmission conditions become essential they are imposed weakly which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated lagrange multipliers moreover in order to facilitate the analysis we augment the formulation in the fluid by incorporating a redundant galerkin type term arising from the quasi newtonian constitutive law multiplied by a suitable stabilization parameter in this way under a suitable and explicit choice of this parameter a generalization of the babuska brezzi theory is utilized to show the well posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate in particular the feasible finite element subspaces include peers and arnold falk winther elements for the stress velocity and vorticity in the fluid raviart thomas elements and piecewise constants for the velocity and pressure in the porous medium together with piecewise constant stokes strain tensor and continuous piecewise linear elements for the traces next we employ classical approaches which include linearization techniques clement s interpolator and helmholtz s decomposition together with known efficiency estimates to derive a reliable and efficient residual based a posteriori error estimator for the coupled problem finally several numerical results confirming the good performance of the method and the properties of the a posteriori error estimator and illustrating the capability of the corresponding adaptive algorithm to identify the singular regions of the solution are reported

applications of quantum and classical fisher information to two level complex and quaternionic and three level complex systems in the bayesian theory of statistical inference as first suggested by harold jeffreys in highly influential work one can employ the square root of the determinant of an n n fisher information matrix as a reparametrization invariant prior generally unnormalized measure over an n dimensional family riemannian manifold of probability distributions jeffreys ansatz is adopted here to the quantum context that is with regard to density matrices rather than probability distributions by computing the quantum fisher information matrices associated with helstrom and holevo for the three five and eight dimensional convex sets of two level complex two level quaternionic and three level complex systems respectively in both the two level cases the priors have been normalized to probability distributions over the 2 2 density matrices while in the much more computationally demanding three level situation no such normalization has been accomplished an argument is made for the general form in term

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

analysis of an augmented pseudostress based mixed formulation for a nonlinear brinkman model of porous media flow abstract in this paper we introduce and analyze an augmented mixed finite element method for the two dimensional nonlinear brinkman model of porous media flow with mixed boundary conditions more precisely we extend a previous approach for the respective linear model to the present nonlinear case and employ a dual mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress in this way and similarly as before the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k 0 for the pseudostress piecewise polynomials of degree k for the gradient of the velocity and continuous piecewise polynomials of degree k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behavior of the associated adaptive algorithm are provided

a flux free a posteriori error estimator for the incompressible stokes problem using a mixed fe formulation in this contribution we present an a posteriori error estimator for the incompressible stokes problem valid for a conventional mixed fe formulation due to the saddle point property of the problem conventional error estimators developed for pure minimization problems cannot be utilized straight forwardly the new estimator is built up by two key ingredients at first a computed error approximation exactly fulfilling the continuity equation for the error is obtained via local dirichlet problems secondly we adopt the approach of solving local equilibrated flux free problems in order to bound the remaining incompressible error in this manner guaranteed upper and lower bounds of the velocity energy norm of the error as well as goal oriented linear output functionals with respect to a reference overkill mesh are obtained in particular it should be noted that this approach requires no computation of hybrid fluxes furthermore the estimator is applicable to mixed fe formulations using continuous pressure approximations such as the mini and taylor hood class of elements in conclusion a few simple numerical examples are presented illustrating the accuracy of the error bounds c 2010 elsevier b v all rights reserved

a posteriori error analysis of a mixed virtual element method for a nonlinear brinkman model of porous media flow abstract in this paper we present an a posteriori error analysis of a mixed vem discretization for a nonlinear brinkman model of porous media flow which has been proposed by the authors in a previous work therein the system is formulated in terms of a pseudostress tensor and the velocity gradient whereas the velocity and the pressure of the fluid are computed via postprocessing formulas furthermore the well posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established we now propose reliable and efficient residual based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem the resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes for the analysis we make use of a global inf sup condition helmholtz decomposition local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions finally we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm

augmented mixed finite element methods for the stationary stokes equations in this paper we introduce and analyze two augmented mixed finite element methods for a velocity pressure stress formulation of the stationary stokes equations our approach which extends analog results for linear elasticity problems is based on the introduction of the galerkin least squares type terms arising from the constitutive and equilibrium equations and the dirichlet boundary condition for the velocity all of them multiplied by suitable stabilization parameters we show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms whence the associated galerkin schemes become well posed for any choice of finite element subspaces in particular we can use continuous piecewise linear velocities piecewise constant pressures and raviart thomas elements for the stresses thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation alternatively the above factor reduces to 4 when a second augmented variational formulation involving only the velocity and the stress as unknowns is employed next we derive a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element schemes finally extensive numerical experiments illustrating the performance of the augmented mixed finite element methods confirming the properties of the a posteriori estimators and showing the behavior of the associated adaptive algorithms are reported

a low order mixed finite element method for a class of quasi newtonian stokes flows part i a priori error analysis abstract this is the second part of a work dealing with a low order mixed finite element method for a class of nonlinear stokes models arising in quasi newtonian fluids in the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation and that the corresponding galerkin scheme becomes well posed with piecewise constant functions and raviart thomas spaces of lowest order as the associated finite element subspaces in this paper we develop a bank weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions several numerical results illustrating the efficiency of the method are also provided

dual mixed finite element approximation of stokes and nonlinear stokes problems using trace free velocity gradients in this work a finite element method for a dual mixed approximation of stokes and nonlinear stokes problems is studied the dual mixed structure which yields a twofold saddle point problem arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning the method approximates the velocity its gradient and the total stress tensor but avoids the explicit computation of the pressure which can be recovered through a simple postprocessing technique this method improves an existing approach for these problems and uses raviart thomas elements and discontinuous piecewise polynomials for approximating the unknowns existence uniqueness and error results for the method are given and numerical experiments that exhibit the reduced computational cost of this approach 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

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

a dual dual mixed formulation for nonlinear exterior transmission problems we combine a dual mixed finite element method with a dirichletto neumann mapping derived by the boundary integral equation method to study the solvability and galerkin approximations of a class of exterior nonlinear transmission problems in the plane as a model problem we consider a nonlinear elliptic equation in divergence form coupled with the laplace equation in an unbounded region of the plane our combined approach leads to what we call a dual dual mixed variational formulation since the main operator involved has itself a dual type structure we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements the main tool of our analysis is given by a generalization of the usual babuska brezzi theory to a class of nonlinear variational problems with constraints

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

a priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows abstract in this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem to the case of incompressible fluid flows with symmetric stress tensor similarly as before the present approach is based on the introduction of the galerkin least squares type terms arising from the constitutive and equilibrium equations and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement all of them multiplied by stabilization parameters we show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form whence the associated galerkin scheme becomes well posed for any choice of finite element subspaces next we present a reliable and efficient residual based a posteriori error estimator for the augmented mixed finite element scheme finally several numerical results confirming the theoretical properties of this estimator and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution are reported

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

a priori and a posteriori error analyses of a velocity pseudostress formulation for a class of quasi newtonian stokes flows abstract in this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear stokes models arising in quasi newtonian fluids the methods are based on a non standard mixed approach in which the velocity the pressure and the pseudostress are the original unknowns however we use the incompressibility condition to eliminate the pressure and set the velocity gradient as an auxiliary unknown which yields a twofold saddle point operator equation as the resulting dual mixed variational formulation in addition a suitable augmented version of the latter showing a single saddle point structure is also considered we apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well posed in particular we show that raviart thomas elements of order k 0 for the pseudostress and piecewise polynomials of degree k for the velocity and its gradient ensure the well posedness of the associated galerkin schemes moreover we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation then we derive a reliable and efficient residual based a posteriori error estimator for each scheme finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithms

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

an augmented mixed finite element method for 3d linear elasticity problems in this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3d our approach is an extension of a technique developed recently for plane elasticity which is based on the introduction of consistent terms of galerkin least squares type we consider non homogeneous and homogeneous dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms in this way the associated galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces in particular raviart thomas spaces of order 0 for the stress tensor continuous piecewise linear elements for the displacement and piecewise constants for the rotation can be utilized moreover we show that in this case the number of unknowns behaves approximately as 9 5 times the number of elements tetrahedrons of the triangulation which is cheaper by a factor of 3 than the classical peers in 3d several numerical results illustrating the good performance of the augmented schemes are provided

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

a posteriori error analysis of an augmented mixed method for the navier stokes equations with nonlinear viscosity abstract in this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2d and 3d versions of the navier stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient two different reliable and efficient residual based a posteriori error estimators for this problem on arbitrary convex or non convex polygonal and polyhedral regions are derived our analysis of reliability of the proposed estimators draws mainly upon the global inf sup condition satisfied by a suitable linearisation of the continuous formulation an application of helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators in addition differently from previous approaches for augmented mixed formulations the boundedness of the clement operator plays now an interesting role in the reliability estimate on the other hand inverse and discrete inequalities and the localisation technique based on triangle bubble and edge bubble functions are utilised to show their efficiency finally several numerical results are provided to illustrate the good performance of the augmented mixed method to confirm the aforementioned properties of the a posteriori error estimators and to show the behaviour of the associated adaptive algorithm

a posteriori error analysis of an augmented fully mixed formulation for the stationary boussinesq model abstract in this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the boussinesq approximation setting our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations respectively which possibilitates the elimination of the pressure this unknown however can be easily recovered by a postprocessing formula in turn redundant galerkin terms are included into the weak formulation to ensure well posedness in this way the resulting variational formulation is a four field augmented scheme whose galerkin discretization allows a raviart thomas approximation for the auxiliary unknowns and a lagrange approximation for the velocity and the temperature in the present work we propose a reliable and efficient fully local and computable residual based a posteriori error estimator in two and three dimensions for the aforementioned method standard arguments based on duality techniques stable helmholtz decompositions and well known results from previous works are the main underlying tools used in our methodology several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm

numerical analysis of a three fields model for a quasi newtonian flow abstract we study a finite element approximation of a non linear three fields version of stokes system for incompressible fluids such a problem arises when one considers a white metzner type model for a viscoelastic fluid flow and sets the relaxation time to zero the non linear aspect comes from the fact that a fraction of the viscosity of the fluid depends on the second invariant of the rate of deformation tensor we give existence and unicity results for the continuous problem and its approximation and we prove the error bound

least squares methods for the velocity pressure stress formulation of the stokes equations we formulate and study finite element methods for the solution of the incompressible stokes equations based on the application of least squares minimization principle to an equivalent first order velocity pressure stress system our least squares functional involves the l norms of the residuals of each equation multiplied by a mesh dependent weight each weight is determined according to the agmon douglis nirenberg index of the corresponding equation as a result the approximating spaces are not subject to the lbb condition and conforming discretizations are possible with merely continuous finite element spaces moreover the resulting discrete problems involve only symmetric positive definite systems of linear equations i e assembly of the discretization matrix is not required even at the element level we prove that the least squares approximadons converge to the solutions of the stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results among other things these numerical examples indicate that the method is not optimal without the weights in the least squares functional

analysis of a velocity pressure pseudostress formulation for the stationary stokes equations abstract we consider a non standard mixed approach for the stokes problem in which the velocity the pressure and the pseudostress are the main unknowns alternatively the pressure can be eliminated from the original equations thus yielding an equivalent formulation with only two unknowns in this paper we develop a priori and a posteriori error analyses of both approaches we first apply the babuska brezzi theory to prove the well posedness of the continuous and discrete formulations in particular we show that raviart thomas elements of order k 0 for the pseudostresses and piecewise polynomials of degree k for the velocities and the pressures ensure unique solvability and stability of the associated galerkin schemes then we derive reliable and efficient residual based a posteriori error estimators for both schemes without and with pressure unknown finally we provide several numerical results illustrating the good performance of the resulting mixed finite element methods confirming the theoretical properties of the estimators and showing the behaviour of the associated adaptive algorithms

geometry of canonical correlation on the state space of a quantum system a riemannian metric is defined on the state space of a finite quantum system by the canonical correlation or kubo mori bogoliubov scalar product this metric is infinitesimally induced by the nonsymmetric relative entropy functional or the von neumann entropy of density matrices hence its geometry expresses maximal uncertainty it is proven that the metric is monotone under stochastic mappings however an example shows that it is not the only such riemannian metric this fact is remarkable because in the probabilistic case the markovian monotonicity property characterizes the fisher information metric the essential difference appears in the curvatures of a classical state space and a quantum one a conjecture is made that the scalar curvature is monotone with respect to the more mixed statistical partial order of density matrices furthermore an information inequality resembling the cramer rao inequality of classical statistics is established the inequality provides a lower bound for the c