a new class of higher order mixed finite volume methods for elliptic problems we introduce a new class of higher order mixed finite volume methods for elliptic problems we start from the usual way of changing the given equation into a mixed system using the darcy s law bf u cal k nabla p by integrating the system of equations with some judiciously chosen test spaces on each element we define new mixed finite volume methods of higher order we show that these new schemes are equivalent to the nonconforming finite element spaces used to define them the darcy velocity can be locally recovered from the solution of nonconforming finite element method hence our work opens a way to make higher order mixed method more practicable three dimensional extensions to parallelepiped elements are also presented this work can be viewed as a generalization of earlier works which were devoted to reducing the lowest order mixed finite element method to a corresponding nonconforming finite element method an optimal error analysis is carried out and numerical results are presented w

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

guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity we obtain fully computable a posteriori error estimators for the energy norm of the error in second order conforming and nonconforming finite element approximations in planar elasticity these estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error the estimators are shown to also provide local lower bounds up to a constant and higher order data oscillation terms numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator

constructing both lower and upper bounds for the eigenvalues of elliptic operators by nonconforming finite element methods this paper introduces a method of constructing nonconforming finite elements which can produce lower bounds for the eigenvalues of elliptic operators based on such nonconforming discrete eigenfunctions we propose a simple method to produce upper bounds of eigenvalues more precisely we construct conforming approximations of exact eigenfunctions by a projection average interpolation operator of nonconforming discrete eigenfunctions after showing the approximation property of the projection average interpolation operator we prove that the rayleigh quotients of the aforementioned conforming approximations are convergent to the exact eigenvalues from above finally we combine lower and upper bounds of eigenvalues to obtain high accuracy approximations of eigenvalues numerical examples verify our theoretical results

a non conforming piecewise quadratic finite element on triangles it is shown in this paper that non conforming finite elements on the triangle using second degree polynomials can be easily built and used indeed they appear as an enriched version of the standard piecewise quadratic six node element this work is divided into two parts in the first we present the basic properties of the element namely how it can be built and basic error estimates we also show that this element exhibits a very peculiar regularity property in the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

inf sup stable non conforming finite elements of arbitrary order on triangles we introduce a family of scalar non conforming finite elements of arbitrary order k 1 with respect to the h1 norm on triangles their vector valued version generates together with a discontinuous pressure approximation of order k 1 an inf sup stable finite element pair of order k for the stokes problem in the energy norm for k 1 the well known crouzeix raviart element is recovered

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

a three dimensional quadratic nonconforming element we define a second degree nonconforming element on tetrahedra we build a basis for the opproximation space derived from this element we prove a discrete regularity property similar to the one that holds for the corresponding two dimensional element

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

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

intrinsic finite element methods for the computation of fluxes for poisson s equation in this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory we develop a general method for the derivation of intrinsic conforming and non conforming finite element spaces and appropriate lifting operators for the evaluation of the right hand side from abstract theoretical principles related to the second strang lemma this intrinsic finite element method is analyzed and convergence with optimal order is proved

first order system least squares for the signorini contact problem in linear elasticity a first order system least squares formulation for the signorini problem modeling frictionless contact in linear elasticity is studied in addition to the displacement field the stress tensor is used as an independent process variable a contact boundary term is added to the usual least squares functional in order to achieve coercivity and continuity in appropriate norms the discrete functional is shown to constitute an a posteriori error estimator on which an adaptive refinement strategy may be based as finite element spaces standard conforming piecewise polynomials for the displacement approximation are combined with raviart thomas elements for the rows in the stress tensor computational results for a test problem of hertzian contact illustrate the effectiveness of our least squares approach

a posteriori error estimators for nonconforming approximation of some quasi newtonian flows this paper deals with a posteriori error estimators for nonconforming approximations of quasi newtonian flows we consider the crouzeix raviart piecewise linear approximations of scalar elliptic problems and define an error estimator when omega is a simply connected domain the error is dominated by the estimator this estimator can be generalized to higher order elements we define a posteriori error estimators for the fortin soulie piecewise quadratic approximations of quasi newtonian flows and prove that the error is dominated by the estimator this estimator can be computed locally in terms of the approximate solution and is therefore suitable for adaptive refinement

the linear dynamic complementarity problem is a special case of the extended linear complementarity problem in this paper we consider some extensions of the linear complementarity problem which is one of the fundamental problems in mathematical programming more specifically we consider the linear dynamic complementarity problem ldcp the generalized linear complementarity problem glcp and the extended linear complementarity problem elcp in this note we show that the ldcp is a special case of the elcp and the glcp

from compressible to incompressible materials via an asymptotic expansion in linear elasticity problems the pressure is usually introduced for computing the incompressible state in this paper is presented a technique which is based on a power series expansion of the displacement with respect to the inverse of lame s coefficient lambda it does not require to introduce the pressure as an auxiliary unknown moreover low degree finite elements can be used the same technique can be applied to stokes or navier stokes equations and can be extended to more general parameterized partial differential equations discretization error and convergence are analyzed and illustrated by some numerical results

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

a least squares mixed finite element method for biot s consolidation problem in porous media a least squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper for the analysis we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process this is known in the literature as biot s consolidation problem for simplicity the analysis is presented for the problem in two space dimensions our least squares approach is motivated by the fact that all process variables i e fluid pressure and flux as well as displacement field and stress tensor are approximated directly by suitable finite element spaces ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time dependent case standard h1 conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for each component of the displacement respectively for the flux and stress components the h div conforming raviart thomas spaces of lowest order are used computational results are presented for some two dimensional test problems

a discontinuous subgrid eddy viscosity method for the time dependent navier stokes equations in this paper we provide an error analysis of a subgrid scale eddy viscosity method using discontinuous polynomial approximations for the numerical solution of the incompressible navier stokes equations optimal continuous in time error estimates of the velocity are derived the analysis is completed with some error estimates for two fully discrete schemes which are first and second order in time respectively

discontinuous finite element methods for incompressible flows on subdomains with non matching interfaces abstract in this paper an improved inf sup condition is derived for a class of discontinuous galerkin methods for solving the steady state incompressible stokes and navier stokes equations the computational domain is subdivided into subdomains with non matching meshes at the interfaces optimal error estimates are obtained numerical experiments including two benchmark problems are presented

nonconforming finite element methods for the equations of linear elasticity in the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions the main difficulty in the analysis is to prove that an appropriate discrete version of korn s second inequality is valid such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears optimal order error estimates uniform for poisson ratio v e 0 1 2 are then derived for the corresponding p2 and p3 methods this contrasts with the use of c finite elements where there is a deterioration in the convergence rate as v 1 2 for piecewise polynomials of degree 3 modifications of the continuous methods and the nonconforming linear method which also give uniform optimal order error estimates are discussed

crouzeix velte decompositions for higher order finite elements we consider polynomial finite elements of order k 1 for the approximation of stokes and linear elasticity problems which are continuous in the gauss legendre points of the elements sides i e generalize the crouzeix raviart k 1 fortin soulie k 2 and crouzeix falk k 3 elements we show that for odd orders these gauss legendre elements do not possess a crouzeix velte decomposition for even orders not only a crouzeix velte decomposition can be shown to exist which is advantageous when solving the corresponding linear equations and eigenvalue problems but also the grid singularity of the well known scott vogelius elements is avoided by these elements which are shown to differ from the former ones by nonconforming bubbles we also consider quadrilateral elements of order k 1 where the requirement of a crouzeix velte decomposition is shown to exclude most commonly used elements

fully computable a posteriori error estimates for the stokes equation without the global inf sup constant in this paper we consider the a posteriori error estimates for the stokes equation which provide computable upper bounds on the actual errors it is known that such error estimates typically involve the global inf sup constant which can be quite tricky to find and lead to extra difficulty in numerical computations to resolve this difficulty we propose a new error estimate which relies only upon the inf sup constants local to the subdomains forming a partition of the original domain hence our new error estimate is fully computable whenever the subdomains are simple enough to make the local inf sup constants readily available and moreover provides a sharper upper bound than the previous estimate when these local constants are bigger than the global constant application to the crouzeix raviart and fortin soulie nonconforming finite elements is presented along with some effective minimization technique for further improvement of the upper bound on the error finally numerical experiments are carried out to investigate the performance of the new error estimate

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

a discontinuous galerkin method with nonoverlapping domain decomposition for the stokes and navier stokes problems a family of discontinuous galerkin finite element methods is formulated and analyzed for stokes and navier stokes problems an inf sup condition is established as well as optimal energy estimates for the velocity and l 2 estimates for the pressure in addition it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces

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

nonconforming finite elements for the stokes problem on obtient un nouveau resultat de stabilite pour l approximation du probleme de stokes stationnaire avec des elements finis non conformes

robust a posteriori error estimation for the nonconforming fortin soulie finite element approximation we obtain a computable a posteriori error bound on the broken energy norm of the error in the fortin soulie finite element approximation of a linear second order elliptic problem with variable permeability this bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms the estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error

first order system least squares for the stress displacement formulation linear elasticity this paper develops a least squares finite element method for linear elasticity in both two and three dimensions the least squares functional is based on the stress displacement formulation with the symmetry condition of the stress tensor imposed in the first order system for the respective displacement and stress using the crouzeix raviart and raviart thomas finite element spaces our least squares finite element method is shown to be optimal in the broken h1 and h div norms uniform in the incompressible limit

new nonconforming finite elements on arbitrary convex quadrilateral meshes in this paper we construct new nonconforming finite elements on the meshes consisting of arbitrary convex quadrilaterals especially for the quadratic and cubic cases for each case we first define a quadrilateral element that adopts edge moments as the degrees of freedom dofs and then enforce a linear constraint on this element we have for the quadratic case eight degrees of freedom per element and for the cubic case eleven dofs per element respectively the dimensions and the bases of different types for the global finite element spaces are provided we consider the approximations of two dimensional second order elliptic problems for both of these elements error estimates with optimal convergence order in both broken h 1 norm and l 2 norm are given moreover we consider the discretization of the stokes equations adopting our quadratic element to approximate each component of the velocity along with piecewise discontinuous p 1 element for the pressure this mixed scheme is stable and optimal error estimates both for the velocity and the pressure are also achieved numerical examples verify our theoretical analysis

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

convergence rates of the dpg method with reduced test space degree this paper presents a duality theorem of the aubin nitsche type for discontinuous petrov galerkin dpg methods this explains the numerically observed higher convergence rates in weaker norms considering the specific example of the mild weak or primal dpg method for the laplace equation two further results are obtained first the dpg method continues to be solvable even when the test space degree is reduced provided it is odd second a non conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

supg stabilization for the nonconforming virtual element method for advection diffusion reaction equations abstract we present the design convergence analysis and numerical investigations of the nonconforming virtual element method with streamline upwind petrov galerkin vem supg stabilization for the numerical resolution of convection diffusion reaction problems in the convective dominated regime according to the virtual discretization approach the bilinear form is split as the sum of a consistency and a stability term the consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form including the supg stabilization term polynomial projections can be computed exactly from the degrees of freedom the stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients the nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface optimal convergence properties of the method are proved in a suitable norm which includes contribution from the advective stabilization terms experimental results confirm the theoretical convergence rates

stable finite element pair for stokes problem and discrete stokes complex on quadrilateral grids in this paper we construct a stable finite element pair for incompressible stokes problem and a discrete stokes complex associated with that pair on general quadrilateral grids the finite element spaces involved consist of piecewise polynomials only and the divergence free condition is imposed in a primal formulation these constructions can be generated onto grids that consist of both triangular and quadrilateral cells

a w cycle algorithm for a weakly over penalized interior penalty method we study a w cycle multigrid algorithm for a weakly over penalized interior penalty method for the two dimensional poisson problem on convex polygonal domains

a first order system least squares method for hyperelasticity a least squares mixed finite element method for deformations of hyperelastic materials using stress and displacement as process variables is presented and studied the method is investigated in detail for the specific case of a neo hookean material law and is based on the representation of the strain stress relation a formulation is derived for compressible materials and shown to remain valid in the incompressible limit automatically enforcing the incompressibility constraint the mapping properties of the first order system operator are studied in appropriate sobolev spaces under the assumption of a locally unique solution with sufficient regularity it is proved that the first order least squares residual constitutes an upper bound for the error measured in a suitable norm provided that the finite element approximation is sufficiently close the method is tested numerically in a plane strain situation using next to lowest order raviart thomas elements for the stress tensor and conforming quadratic

a new mixed formulation for elasticity we propose a new mixed variational formulation for the equations of linear elasticity it does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations

a stable nonconforming finite element on hexahedra summary a new nonconforming brick element is introduced which only has 13 dofs locally and takes p2 span x13 x23 x33 as its shape functions space the vector valued version generates together with a discontinuous p1 approximation an inf sup stable finite element pair of order 2 for the stokes problem in the energy norm copyright 2016 john wiley sons ltd

the generalized linear complementarity problem and an algorithm to find all its solutions motivated by a number of typical applications a generalization of the classicallinear complementarity problem is presented together with an algorithm to determine the complete solution set the algorithm is based on the double description method for solving linear inequalities and succeeds in describing continuous as well as unbounded solution sets

analyse d un element mixte pour le probleme de stokes we build a mixed finite element method for the stokes problem this method is nonconforming we apply the results of part i to prove convergence and obtain optimal error estimates we also consider the treatment of interface conditions by multipliers and show that a superconvergent approximation can be built from those multipliers nous construisons un element fini mixte pour le probleme de stokes cet element est non conforme nous appliquons les resultats de la partie i pour obtenir des estimations d erreur optimales nous considerons aussi le traitement des conditions de raccords par des multiplicateurs de lagrange et nous montrons que ces multiplicateurs peuvent etre utilises pour construire une approximation superconvergente

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

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

discontinuous galerkin and mimetic finite difference methods for coupled stokes darcy flows on polygonal and polyhedral grids we study locally mass conservative approximations of coupled darcy and stokes flows on polygonal and polyhedral meshes the discontinuous galerkin dg finite element method is used in the stokes region and the mimetic finite difference method is used in the darcy region dg finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces optimal convergence estimates for the numerical scheme are derived results from computational experiments supporting the theory are presented

stable nonconforming methods or the stokes problem1this work was supported in part by kosef 1 it is the purpose of this paper to show that the mixed finite element scheme with pressure stabilization for the stokes problem converges with an optimal order for some higher order nonconforming triangular elements specifically an optimal order convergence is given for the ncp 4 p 3 element using nonconforming piecewise quartic velocities paired with discontinuous piecewise cubic pressures the ncp 6 p 5 element is analyzed similarly to verify the stability condition and the error estimates for nonconforming elements we combine the ideas of macroelement technique of stenberg and the arguments for galerkin least squares methods

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

minimal realization in the max algebra is an extended linear complementarity problem we demonstrate that the minimal state space realization problem in the max algebra can be transformed into an extended linear complementarity problem elcp we use an algorithm that finds all solutions of an elcp to find all equivalent minimal state space realizations of a single input single output siso discrete event system we also give a geometrical description of the set of all minimal realizations of a siso max linear discrete event system