supports of derivations free factorizations and ranks of fixed subgroups in free groups for f a free group of finite rank it is shown that the fixed subgroup of any set b of endomorphisms of f has rank rank f generalizing a recent result of dicks and ventura the proof involves the combinatorics of derivations of groups some related questions are examined

treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses finite element approximations of the stokes and navier stokes equations with in homogeneous essential boundary conditions are considered boundary conditions are enforced weakly by introducing lagrange multipliers optimal error estimates including some for the stress vector on the boundary are derived under minimal regularity assumptions on the data particular attention is paid to the analysis of a practical choice of finite element spaces for which the lagrange multiplier calculation uncouples from that for the velocity and pressure the results are also applicable to general second order elliptic systems

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

enumeration of algebras close to absolutely free algebras and binary trees abstract we study three classes of algebras absolutely free algebras free commutative non associative and free anti commutative non associative algebras we study asymptotics of the growth for free algebras of these classes and for their subvarieties as well mainly we study finitely generated algebras also the codimension growth for varieties in theses classes is studied for these purposes we use ordinary generating functions as well as exponential generating functions the following subvarieties are studied in these classes solvable completely solvable right nilpotent and completely right nilpotent subvarieties the obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees we enumerate these trees using generating functions for solvable and right nilpotent algebras the generating functions are algebraic for completely solvable and completely right nilpotent algebras the respective functions are rational it is known that these three varieties of algebras satisfy schreier s property i e subalgebras of free algebras are free for free groups there is schreier s formula for the rank of a subgroup of a free group we find analogues of this formula for these varieties they are written in terms of series as an application we study invariants of finite groups acting on absolutely free algebras

study of coating flow by the finite element method abstract silliman s analysis of slot coating is extended to accommodate film flows with highly bent menisci as in slide and curtain coating by combining polar and cartesian coordinate parametrizations of meniscus shape the nonlinear algebraic equations from the subdomain galerkin weighted residual method and finite element basis functions are solved by newton s method which is perfected for the free boundary problem convergence behavior is examined slot coating results reveal slow flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot comparison of coyne and elrod s approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes

consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenishment we consider several subclasses of the problem of grouping n items indexed 1 2 n into m subsets so as to minimize the function gs1 sm in general these problems are very difficult to solve to optimality even for the case m 2 we provide several sufficient conditions on gi that guarantee that there is an optimum partition in which each subset consists of consecutive integers or else the partition s1 sm satisfies a more general condition called semiconsecutiveness moreover by restricting attention to consecutive or semiconsecutive partitions we can solve the partition problem in polynomial time for small values of m if in addition g is symmetric then the partition problem is solvable in purely polynomial time we apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper we also relate the results to the neyman pearson lemma in statistical hypothesis testing and to a graph partitioning problem of barnes and hoffman

asymptotic analysis of the laminar viscous flow over a porous bed we consider the laminar viscous channel flow over a porous surface the size of the pores is much smaller than the size of the channel and it is important to determine the effective boundary conditions at the porous surface we study the corresponding boundary layers and by a rigorous asymptotic expansion we obtain saffman s modification of the interface condition observed by beavers and joseph the effective coefficient in the law is determined through an auxiliary boundary layer type problem whose computational and modeling aspects are discussed in detail furthermore the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size ep finally we give the interface condition linking the effective pressure fields in the porous medium and in the channel and we determine the jump of the effective pressures explicitly

finite element matrices in congruent subdomains and their effective use for large scale computations the structure of finite element matrices in congruent subdomains is studied when a domain has a form of symmetries and or periodicities it is decomposed into a union of congruent subdomains each of which is an image of a reference subdomain by an affine transformation with an orthogonal matrix whose components consist of 1 0 and 1 stiffness matrices in subdomains are expressed by one in the reference subdomain with renumbering indices and changing signs corresponding to the orthogonal matrices the memory requirements for a finite element solver are reduced by the domain decomposition which is useful in large scale computations reducing rates of memory requirements to store matrices are reported with examples of domains both applicability and limitations of the algorithm are discussed with an application to the earth s mantle convection problem

error estimates of finite element methods for nonstationary thermal convection problems with temperature dependent coefficients general error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature dependent coefficients these variable coefficients turn the diffusion and the buoyancy terms to be nonlinear which increases the nonlinearity of the problems an argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions error estimates are also provided for schemes modified by approximate coefficients which are used conveniently in practical computations

finite element approximations of the lame system with penalized ideal contact boundary conditions we consider finite element approximations of the lame system of elasticity with ideal contact boundary conditions imposed with the penalty method for a polygonal or polyhedral boundary we prove convergence estimates in terms of both the penalty and discretization parameters in the case of a smooth curved boundary we show through a numerical two dimensional example that convergence may not hold due to a babuska s type paradox we also propose and test numerically several remedies

semi discrete error estimates of the evolutionary stokes equations with inhomogeneous dirichlet boundary data abstract this work concerns finite element analysis of the evolutionary stokes equation with inhomogeneous dirichlet boundary data the dirichlet boundary data are enforced using a lagrange multiplier approach generalized from the work of gunzburger and hou 1992 for the steady stokes equations with a practical choice of finite element spaces the computation of the lagrange multiplier the boundary stress is uncoupled from that of the velocity and the pressure various semi discrete in space error estimates are presented for the velocity pressure and the lagrange multiplier

optimal partitions having disjoint convex and conic hulls leta1 an be distinctk dimensional vectors we consider the problem of partitioning these vectors intom sets so as to maximize an objective which is a quasi convex function of the sum of vectors in each set we show that there exists an optimal partition whose sets have pairwise disjoint conic hulls we also show that if the number of vectors in each of the sets is constrained then a weaker conclusion holds namely there exists an optimal partition whose sets have pairwise disjoint convex hulls the results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope

mixed velocity stress current and potential boundary conditions for stationary mhd flow this paper is concerned with a boundary value problem describing the stationaryflow of a viscous incompressible electrically conducting fluid confined to a bounded region of space under mixed boundary conditions the flow is governed by the navier stokes equations ohm s law and the biot savart law the boundary conditions involve the velocity field stress tensor electric current density and electric potential we derive a mixed variational formulation of the problem which lends itself naturally to finite element discretizations and prove the existence and uniqueness of a small solution under the assumption of sufficiently small data

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

partially dissipative 2d boussinesq equations with navier type boundary conditions abstract this paper concerns itself with two systems of the 2d boussinesq equations with partial dissipation in bounded domains with the navier type boundary conditions we attempt to achieve two main goals first to prove the global existence and uniqueness under minimal regularity assumptions on the initial data and second to provide a direct and transparent approach that explicitly reveals the impacts of the navier boundary conditions the 2d boussinesq equations with partial dissipation have attracted considerable interests in the last few years although most of the results are aimed at sufficiently regular solutions in the whole space or periodic domains larios et al 2013 made serious efforts to minimize the regularity assumptions necessary for the uniqueness of solutions in the spatially periodic setting in contrast to the whole space and the periodic domains the navier boundary conditions generate boundary terms and require compatibility conditions in addition due to the lack of boundary conditions for the pressure we resort to the existence and regularity result on the associated stokes problem with navier boundary conditions the uniqueness relies on the yudovich techniques and the introduction of a lower regularity counterpart of the temperature

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

polynomial retracts and the jacobian conjecture vladimir shpilrain and jie tai yuabstract let k x y be the polynomial algebra in two variables over a eldk of characteristic 0 a subalgebra r of k x y is called a retract if there is anidempotent homomorphism a retraction or projection k x y k x y suchthat k x y r the presence of other equivalent de nitions of retractsprovides several di erent methods of studying them and brings together ideasfrom combinatorial algebra homological algebra and algebraic geometry in thispaper we characterize all the retracts of k x y up to an automorphism andgive several applications of this characterization in particular to the well knownjacobian conjecture notably we prove that if a polynomial mapping of k x y has invertible jacobian matrix and xes a non constant polynomial then is anautomorphism

numerical analysis of a finite element projection based vms turbulence model with wall laws abstract this paper deals with the numerical analysis of a finite element projection based vms turbulence model that includes general non linear wall laws only a single mesh and interpolation operators on a virtual coarser mesh are needed to implement the model we include a projection stabilization of pressure to use the same interpolation for velocity and pressure good accuracy is obtained with benchmark turbulent flow problems on coarse grids that justify the interest of this approach also the model solves smooth flows with optimal accuracy

penalty finite element approximation of stokes equations with slip boundary conditions we consider the finite element approximation of the stationary stokes equations with slip boundary conditions on a domain with a smooth curved boundary the slip boundary condition is imposed weakly with the penalty method on polygonal domains approaching the smooth domain for taylor hood elements we derive error estimates which depend on the penalty parameter varepsilon the disctretization parameter h h and the approximation error of the normal to the boundary in particular if in the penalty term we use the normal to the polygonal boundary the best convergence order is 2 3 2 3 and it is obtained with varepsilon c h 2 3 c h 2 3 this convergence result shows that babuska s paradox associated to stokes equations with slip boundary conditions is circumvented a numerical example illustrates the theoretical results notably that regularized normal approximations give better approximations and convergence orders

weak imposition of boundary conditions for the navier stokes equations we prove the convergence of a finite element method for the navier stokes equations in which the no slip condition email protected i 0 on c for i 1 2 is imposed by a penalty method and the no penetration condition u n 0 on c is imposed by lagrange multipliers this approach has been studied for the stokes problem in weak imposition of boundary conditions in the stokes problem ph d thesis university of pittsburgh pa 1999 in most flows the reynolds number is not negligible so the email protected u inertial effects are important thus the extension beyond the stokes problem to the navier stokes equations is critical we show existence and uniqueness of the approximate solution and optimal order of convergence can be achieved if the computational mesh follows the real boundary our results for the nonlinear navier stokes equations improve known results for this approach for the stokes problem

numerical approximation of a quasi newtonian stokes flow problem with defective boundary conditions in this article we study the numerical approximation of a quasi newtonian stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries a variational formulation of the problem using lagrange multipliers to enforce the stated flow rates is given the existence and the uniqueness to the continuous and discrete variational formulations of the solution are shown an error analysis for the numerical approximation is also given numerical computations are included which demonstrate the approximation scheme studied

on the regularity of flows with ladyzhenskaya shear dependent viscosity and slip or non slip boundary conditions navier stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by o a ladyzhenskaya and in the case of gradient dependent viscosity by j l lions a particular case is the well known smagorinsky turbulence model this is nowadays a central subject of investigation on the other hand boundary conditions of slip type seems to be more realistic in some situations in particular in numerical applications they are a main research subject the existence of weak solutions u to the above problems with slip or nonslip type boundary conditions is well known in many cases however regularity up to the boundary still presents many open questions in what follows we present some regularity results in the stationary case for weak solutions to this kind of problem see theorem 3 1 and theorem 3 2 the evolution problem is studied in a forthcoming paper 6 a cornerstone in our proof is the classical nirenberg translation method see 38

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

test elements retracts and automorphic orbits abstract let a 2 be a free associative or polynomial algebra of rank two over a field k of characteristic zero based on the degree estimate of makar limanov and j t yu we prove 1 an element p a 2 is a test element if p does not belong to any proper retract of a 2 2 every endomorphism preserving the automorphic orbit of a nonconstant element of a 2 is an automorphism

a stabilized finite element method for the rayleigh benard equations with infinite prandtl number in a spherical shell abstract a finite element scheme is developed and analyzed for a thermal convection problem of boussinesq fluid with infinite prandtl number in a spherical shell this problem is a mathematical model of the earth s mantle movement and has been a topic of interest for geophysicists it is described by the rayleigh benard equations with infinite prandtl number that is a system of the stokes equations and the convection diffusion equation coupled with the buoyancy and the convection terms a stabilized finite element scheme with p1 p1 p1 element is presented and an error estimate is established the obtained theoretical convergence order is also recognized by a numerical result another numerical result is shown as an example of the earth s mantle movement simulation

mixed finite element approximation of the vector potential we consider a mixed finite element approximation of the three dimensional vector potential which plays an important role in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows the central point of our approach is a saddlepoint formulation of the essential boundary conditions in particular this avoids the wellknown babuska paradox when approximating smooth domains by polyhedrons using piecewise linear piecewise constant elements for the vector potential the boundary terms we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

finite element approximation of viscoelastic fluid flow with slip boundary condition we present a theoretical study of a creeping steady state isothermal flow of a viscoelastic fluid obeying an oldroyd type constitutive law with slip boundary condition the slip boundary condition is appropriate for problems that involve free boundaries and other examples where the usual no slip condition u 0 is not valid such as fiber spinning and microfluidics first we study the newtonian problem with slip boundary condition where the viscoelastic stress is added into the list of unknowns in addition the normal viscoelastic stress component associated with the slip boundary condition is introduced in order to balance its effects a second inf sup condition is proven to treat the discrete case we assume that the continuous solution of the non newtonian problem exists and is small and smooth approximating the extra stress velocity pressure and normal viscoelastic stress component via p 1 discontinuous p 2 continuous p 1 continuous and p 0 discontinuous elements respectively yields a stable finite element scheme finally via a fixed point argument we establish the existence of an approximate solution and derive error estimates

the equations of stationary incompressible magnetohydrodynamics with mixed boundary conditions abstract this paper deals with the questions of existence uniqueness and finite element approximation of solutions to the equations of steady state magnetohydrodynamics with mixed boundary conditions posed on a bounded three dimensional domain the boundary conditions for the velocity equations are of dirichlet neumann and mixed type these boundary conditions are important when considering free boundary value problems problems on artificially truncated domains and control problems which are governed by these equations

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

slip with friction and penetration with resistance boundary conditions for the navier stokes equations numerical tests and aspects of the implementation we consider slip with friction and penetration with resistance boundary conditions in the steady state navier stokes equations this paper describes some aspects of the implementation of these boundary conditions for finite element discretizations numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line of the recirculating vortex respectively

consistent lattice boltzmann modeling of low speed isothermal flows at finite knudsen numbers in slip flow regime ii application to curved boundaries gaseous flows inside microfluidic devices often fall in the slip flow regime according to this theoretical description the navier stokes model remains applicable in bulk while at solid walls a slip velocity boundary model shall be considered physically it is well established that to properly account for the wall curvature the wall slip velocity must be determined by the shear stress rather than the normal component of the velocity derivative alone as commonly applied to planar surfaces it follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics cfd techniques this paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice boltzmann method lbm to substantiate this conclusion this work undertakes the following three studies first we examine the conditions under which the generic reflection type boundary rules used by lbm become consistent models for the slip velocity boundary condition this effort makes use of the second order chapman enskog expansion method where we address both planar and curved boundaries the analysis also clarifies the capabilities and limitations behind the considered reflection type slip schemes second we revisit the family of parabolic accurate lbm slip boundary schemes originally formulated in phys rev e 96 013311 2017 2470 004510 1103 physreve 96 013311 on the basis of the multireflection framework and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate lbm slip boundary formulations such as the linearly accurate slip schemes and the widely popular kinetic based boundary schemes in addition we also discuss the numerical stability of the parabolic slip schemes previously developed providing an heuristic strategy to improve their stable range of operation third we evaluate the performance of the several slip boundary schemes debated in this paper the numerical tests correspond to two classical 2d benchmark flow problems of slip over non planar solid surfaces namely i the velocity profile of the cylindrical couette flow and ii the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles the obtained numerical results confirm the competitiveness of the lbm when equipped with slip boundary schemes of parabolic accuracy as cfd tool to simulate slippage phenomena over arbitrarily non planar surfaces indeed although operating on a simple uniform mesh discretization the lbm yields a similar or even superior level of accuracy compared to state of the art fem simulations conducted on hardworking body fitted meshes this conclusion establishes the lbm as a very appealing cfd technique for simulating microfluidic flows in the slip flow regime a result that deserves further exploration in future studies

a nitsche cut finite element method for the oseen problem with general navier boundary conditions abstract in this work a nitsche based imposition of generalized navier conditions on cut meshes for the oseen problem is presented other methods from literature dealing with the generalized navier condition impose this condition by means of substituting the tangential robin condition in a classical galerkin way these methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero we introduce a novel method for the weak imposition of the generalized navier condition which remains well posed and stable for arbitrary choice of slip length including zero the method proposed here builds on the formulation done by juntunen and stenberg 2009 they impose a robin condition for the poisson problem by means of nitsche s method for an arbitrary combination of the dirichlet and neumann parts of the condition the analysis conducted for the proposed method is done in a similar fashion as in massing et al 2018 but is done here for a more general type of boundary condition the analysis proves stability for all flow regimes and all choices of slip lengths also an l 2 optimal estimate for the velocity error is shown which was not conducted in the previously mentioned work a numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length even though proofs and formulations are presented for the more general case of an unfitted grid method they can easily be reduced to the simpler case of a boundary fitted grid with the removal of the ghost penalty stabilization terms

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

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

majorization and schur convexity with respect to partial orders majorization and schur convexity constitute an important tool for establishing inequalities in particular due to the schur ostrowski theorem which provides a simple characterization of schur convex functions through local two coordinate conditions however the usefulness of the approach is limited by two stringent requirements first every schur convex function must be symmetric second the necessary and sufficient conditions for schur convexity involve every pair of coordinates on its symmetric domain several attempts have been made to relax these two requirements in particular a more general concept of majorization with respect to partial orders was introduced to prepare for such tasks but it did not capture the classic theory for symmetric functions in the current paper we obtain the desired generalization by developing a theory of majorization and schur convexity with respect to partial orders over subsets of euclidean spaces our results are used in hwang rothblum and shepp 1993 to address some optimal assembly problems

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

a stabilized finite volume element formulation for sedimentation consolidation processes a model of sedimentation consolidation processes in so called clarifier thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the stokes system for an incompressible fluid describing the motion of the mixture in cylindrical coordinates and if an axially symmetric solution is assumed the original problem reduces to two space dimensions this poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted sobolev spaces a novel finite volume element method is introduced for the spatial discretization where the velocity field and the solids concentration are discretized on two different dual meshes the method is based on a stabilized discontinuous galerkin formulation for the concentration field and a multiscale stabilized pair of mathbb p 1 mathbb p 1 elements for velocity and pressure respectively numerical experiments illustrate properties of the model and the satisfactory p

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

weak imposition of the slip boundary condition on curved boundaries for stokes flow we study the finite element approximation of two methods to weakly impose a slip boundary condition for incompressible fluid flows the lagrange multiplier method and nitsche s method for each method we can distinguish several formulations depending on the values of some real parameters in the case of a spatial domain with a polygonal or polyhedral boundary we prove convergence results of their finite element approximations extending previous results of verfurth 33 and we show numerical results confirming them in the case of a spatial domain with a smooth curved boundary numerical results show that approximations computed on polygonal domains approximating the original domain may not converge to the exact solution depending on the values of the aforementioned parameters and on the finite element discretization these negative results seem to highlight babuska s like paradox due to the approximation of the boundary by polygonal ones in particular they seem to contradict some of verfurth s theoretical convergence results

penalty method with p1 p1 finite element approximation for the stokes equations under the slip boundary condition we consider the p1 p1 or p1b p1 finite element approximations to the stokes equations in a bounded smooth domain subject to the slip boundary condition a penalty method is applied to address the essential boundary condition u cdot n g on partial omega which avoids a variational crime and simultaneously facilitates the numerical implementation we give o h 1 2 epsilon 1 2 h epsilon 1 2 error estimate for velocity and pressure in the energy norm where h and epsilon denote the discretization parameter and the penalty parameter respectively in the two dimensional case it is improved to o h epsilon 1 2 h 2 epsilon 1 2 by applying reduced order numerical integration to the penalty term the theoretical results are confirmed by numerical experiments

polynomial automorphisms and grobner reductions abstract letpn k x1 xn be the polynomial algebra over a fieldkof characteristic 0 we show that applying an automorphism to a given polynomialp pnis mimicked by grobner transformations of a basis of the ideal ofpngenerated by partial derivatives of this polynomial in the case ofp2 this yields a miraculously simple algorithm for deciding whether or not a given polynomial fromp2is part of a basis another application is an algorithm which given a polynomialp p2that is part of a basis finds a sequence of elementary automorphisms that reducesptox1 we also speculate on how our method may be used for constructing a possible counterexample to the jacobian conjecture in higher dimensions

optimality of monotone assemblies for coherent systems composed of series modules we consider a system with m modules as components these modules are composed of parts of finitely many types and the number of parts of each type that is needed in each of the modules is given e g module i requires nui parts of type u parts of the same type may have different reliabilities but they are functionally interchangeable a module works if and only if all of its parts work i e the internal composition of the modules has series structure an assembly of the modules consists of an assignment of each of the u i nui parts to the modules such that each module meets its specification by getting the required number of parts of each type such an assembly is called monotone if the best parts of each type go to one module the next best parts of each type go to a second module and so on until finally the last module gets the worst parts of each type we prove that for coherent systems there always exists a monotone assembly which maximizes the reliability of the system furthermore we obt

weak imposition of no slip conditions in finite element methods abstract we begin the analysis of two schemes for imposing weakly the no slip boundary conditions in the navier stokes equations the first is via a lagrange multiplier constraining no tangential velocity on the boundary and the second is via penalization of tangential velocities along boundaries our analysis in this first step is for the stokes problem although the motivation for relaxing this essential boundary condition comes from higher reynolds number flow

discretization of the navier stokes equationswith slip boundary condition ii multilevel methods are a common tool for solving nonlinear systems arising from discretizations of elliptic boundary value problems see e g 1 2 multilevel methods consist of solving the nonlinear problem on a coarse mesh and then performing one or two newton correction steps on each subsequent mesh thus solving only one or two larger linear systems in a previous paper by the author 3 a two level method was proposed for the stationary navier stokes equations with slip boundary condition uniqueness of solution to the nonlinear problem step 1 of the two level method is guaranteed provided that the data of the problem i e the reynolds number and the forcing term is bounded in practice however the aforementioned bound on the data is rarely satisfied consequently a priori error estimates should not rely upon this bound such estimates will be established in this report for the two level method the scalings of the meshwidths that guarantee optimal accuracy in the h 1 norm are equally favorable to those in the uniqueness case

primitive almost primitive test and δ primitive elements of free algebras with the nielsen schreier property abstract we study generalized primitive elements of free algebras of finite ranks with the nielsen schreier property and their automorphic orbits a primitive element of a free algebra is an element of some free generating set of this algebra almost primitive elements are not primitive elements which are primitive in any proper subalgebra primitive elements are elements whose partial derivatives generate the same one sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate we prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism an algorithm to determine test elements of free algebras of rank two is described a series of almost primitive elements is constructed and new examples of test elements are given we prove that if the rank n of the free lie algebra l is even n 2m then any primitive element of l is an automorphic image of the element w x1 x2 x2m 1 x2m there are no primitive elements of l if n is odd and the group of automorphisms of the algebra l acts transitively on the set of all primitive elements

test polynomials for automorphisms of polynomial and free associative algebras abstract in this paper we consider test polynomials in the polynomial algebra and the free associative algebra a test polynomial is defined by the following property every endomorphism which fixes the polynomial is an automorphism we construct families of test polynomials for the polynomial algebra and the free associative algebra and show how different techniques may be used in the investigation of test polynomials we also introduce the notion of a test vector space and determine all test vector spaces of the free associative algebra

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

finite element methods for the stokes problem on complicated domains abstract it is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process in case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck the resolution condition links the computational complexity to the number and size of geometric details although the accuracy requirements possibly are moderate and would allow a locally coarse mesh width therefore even the coarsest available discretization can lead to a huge number of unknowns the composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details the basic concept for the stokes problem with uniform no slip boundary conditions has been introduced and analyzed in d peterseim s sauter the composite mini element coarse mesh computation of stokes flows on complicated domains sinum 46 6 2008 3181 3206 here we generalize the composite mini element to slip leak and neumann boundary conditions so that it becomes applicable to this much larger and more important problem class the main results are a the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule b the stability and convergence can be proved under very mild assumption on the domain geometries c the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no slip boundary conditions

shape optimization of peristaltic pumping transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this it is also important to many industrial processes we present a variational method for optimizing the wave shape of a peristaltic pump specifically we optimize the wave profile of a two dimensional channel containing a navier stokes fluid with no assumption on the wave profile other than it is a traveling wave e g we do not assume it is the graph of a function hence this is an infinite dimensional optimization problem the optimization criteria consists of minimizing the input fluid power due to the peristaltic wave subject to constraints on the average flux of fluid and area of the channel sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming sqp method to find a solution of the first order kkt conditions we also use a merit function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape our numerical implementation uses a finite element method for computing a solution of the navier stokes equations adjoint equations as well as for the sqp method when computing perturbations of the channel shape the walls of the channel are deformed by an explicit front tracking approach in computing functional sensitivities with respect to shape we use l 2 type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature we show error estimates for the boundary stress and tangent field approximations as a result we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature specifically we see highly asymmetric wave shapes that are far from being sine waves many examples are shown for a range of fluxes and reynolds numbers up to re 500 which illustrate the capabilities of our method

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

directional quasi convexity asymmetric schur convexity and optimality of consecutive partitions the current paper has the following distinct goals 1 to generalize standard quasi convexity to obtain a weaker property of functions that suffices for optimality of extreme points specifically it is required that intersections of the level sets of the function with line segments having direction in a prescribed set are convex 2 to characterize schur convexity for symmetric functions through the above generalization of quasi convexity thereby obtaining a previously unknown relationship of schur convexity and standard convexity the new characterization is used to extend the definition of schur convexity to functions which are not symmetric 3 to obtain new sufficient conditions for the optimality of consecutive partitions by using the new definition of schur convexity for functions that are not necessarily symmetric the conclusion that schur convexity is an instance of a convexity property that implies the optimality of extreme points unifies two approaches that were used in the literature to prove optimality of subsets of the domain of real valued functions the two tools quasi convexity and schur convexity were previously considered as distinct techniques

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100