a saddle point least squares approach to mixed methods we investigate new pde discretization approaches for solving variational formulations with different types of trial and test spaces the general mixed formulation we consider assumes a stability lbb condition and a data compatibility condition at the continuous level we expand on the bramble pasciak s least square formulation for solving such problems by providing new ways to choose approximation spaces and new iterative processes to solve the discrete formulations our proposed method has the advantage that a discrete inf sup condition is automatically satisfied by natural choices of test spaces first and corresponding trial spaces second in addition for the proposed iterative solver a nodal basis for the trial space is not required applications of the new approach include discretization of first order systems of pdes such as div curl systems and time harmonic maxwell equations

well posedness and numerical performances of the strain gap method a mixed method of approximation is discussed starting from a suitably modified expression of the hu washizu variational principle in which the independent fields are displacements stresses and strain gaps defined as the difference between compatible strains and strain fields the well posedness of the discrete problem is discussed and necessary and sufficient conditions are provided the analysis of the mixed method reveals that the discrete problem can be split into a reduced problem and in a stress recovery accordingly the discrete stress solution is univocally determined once an interpolating stress subspace is chosen the enhanced assumed strain method by simo and rifai is based on an orthogonality condition between stresses and enhanced strains and coincides with the reduced problem it is shown that the mixed method is stable and converges computational issues in the context of the finite element method are discussed in detail and numerical performances and comparisons are carried out copyright 2001 john wiley sons ltd

the constrained shortest path problem with stochastic correlated link travel times abstract this paper investigates the constrained shortest path problem in a transportation network where the link travel times are assumed to be random variables defined on the basis of joint probability mass functions a 0 1 integer programming model is formulated to find the least expected travel time paths besides the flow balance and side constraints the unique link selection constraint is particularly introduced to guarantee that only an optimal path can be finally generated then a lagrangian relaxation solution approach is provided to relax the hard constraints and decompose the relaxed model into two parts an algorithmic framework integrating the sub gradient algorithm label correcting algorithm and k shortest path algorithm is designed to minimize the gap between the upper and lower bounds to find near optimal solutions an extension that considers the joint probability mass functions of link travel times varying with time is discussed we decompose the time dependent problem into three subproblems and solve it by the modified algorithmic framework computational tests are conducted on different scales of transportation networks experimental results in a large scale network indicate that the proposed algorithm can quickly find the high quality solutions with small relative gaps demonstrating the effectiveness of the proposed approaches

mixed displacement rotation pressure formulations for linear elasticity abstract we propose a new locking free family of mixed finite element and finite volume element methods for the approximation of linear elastostatics formulated in terms of displacement rotation vector and pressure the unique solvability of the three field continuous formulation as well as the well definiteness and stability of the proposed galerkin and petrov galerkin methods is established thanks to the babuska brezzi theory optimal a priori error estimates are derived using norms robust with respect to the lame constants turning these numerical methods particularly appealing for nearly incompressible materials we exemplify the accuracy in a suitably weighted norm as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2d and 3d also including cases not covered by our theoretical analysis

a novel alpha finite element method αfem for exact solution to mechanics problems using triangular and tetrahedral elements abstract the paper presents an alpha finite element method fem for computing nearly exact solution in energy norm for mechanics problems using meshes that can be generated automatically for arbitrarily complicated domains three node triangular fem t3 and four node tetrahedral fem t4 elements with a scale factor are formulated for two dimensional 2d and three dimensional 3d problems respectively the essential idea of the method is the use of a scale factor 0 1 to obtain a combined model of the standard fully compatible model of the fem and a quasi equilibrium model of the node based smoothed fem n sfem this novel combination of the fem and n sfem makes the best use of the upper bound property of the n sfem and the lower bound property of the standard fem using meshes with the same aspect ratio a unified approach has been proposed to obtain a nearly exact solution in strain energy for linear problems the proposed elements are also applied to improve the accuracy of the solution of nonlinear problems of large deformation numerical results for 2d using fem t3 and 3d using fem t4 problems confirm that the present method gives the much more accurate solution comparing to both the standard fem and the n sfem with the same number of degrees of freedom and similar computational efforts for both linear and nonlinear problems

rotation based mixed formulations for an elasticity poroelasticity interface problem in this paper we introduce a new formulation for the stationary poroelasticity equations written using the rotation vector and the total fluid solid pressure as additional unknowns and we also wri

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

a vorticity based fully mixed formulation for the 3d brinkman darcy problem abstract we propose and analyze a fully mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium an array of low concentration fixed particles described by brinkman equations and its interaction with non viscous flow within classical porous media governed by darcy s law the system is formulated in terms of velocity and pressure in the porous medium together with vorticity velocity and pressure of the viscous fluid in addition and for sake of the analysis the tangential component of the vorticity is supposed to vanish on the whole boundary of the brinkman domain whereas null normal components of both velocities are assumed on the respective boundaries except on the interface where suitable transmission conditions are considered in this way the derivation of the corresponding mixed variational formulation leads to a lagrange multiplier enforcing the pressure continuity across the interface whereas mass balance results from essential boundary conditions on each domain as a consequence a typical saddle point operator equation is obtained and hence the classical babuska brezzi theory is applied to establish the well posedness of the continuous and discrete schemes in particular we remark that the continuous and discrete inf sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema which constitutes a previously known technique recently denominated t coercivity in turn and consistently with the above the stability of the galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives which yields raviart thomas and nedelec finite element subspaces as feasible choices then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the brinkman momentum equation finally several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported

convergence in the incompressible limit of finite element approximations based on the hu washizu formulation the classical hu washizu mixed formulation for plane problems in elasticity is examined afresh with the emphasis on behavior in the incompressible limit the classical continuous problem is embedded in a family of hu washizu problems parametrized by a scalar for which alpha lambda big mu corresponds to the classical formulation with and being the lame parameters uniform well posedness in the incompressible limit of the continuous problem is established for 1 finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes conditions for uniform convergence are made explicit these conditions are shown to be met by particular choices of bases for stresses and strains and include bases that are well known as well as newly constructed bases though a discrete version of the spherical part of the stress exhibits checkerboard modes it is shown that a independent a priori error estimate for the displacement can be established furthermore a independent estimate is established for the post processed stress the theoretical results are explored further through selected numerical examples

generalized difference methods for a nonlinear dirichlet problem the paper is devoted to the construction of generalized difference methods on irregular meshes for second order nonlinear elliptic equations based on the generalized galerkin method the error estimate is proved

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

conditions for equivalence between the hu washizu and related formulations and computational behavior in the incompressible limit the relationship of the hu washizu mixed formulation to other mixed and enhanced formulations is examined in detail in the context of linear elasticity with a view to presenting a unified framework for such formulations the hu washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well posedness in the incompressible limit recently established theoretical results on uniform well posedness are studied computationally by considering a range of numerical examples which illustrate where appropriate the good behavior predicted by the theory as well as the locking behavior that is evident in those problems in which the conditions for stability and convergence are not satisfied 2005 elsevier b v all rights reserved

fast and accurate 4 node quadrilateral an accurate and variationally consistent 4 node quadrilateral element is introduced where high coarse mesh accuracy and low mesh distortion sensitivity are characteristic qualities even when incompressibility is approached for plane strain one point quadrature integration procedure is adopted and a new improved stabilization technique is developed orthogonality conditions are utilized so that the patch test is satisfied for arbitrary quadrilaterals several numerical examples including a convergence rate study are presented which confirm the excellent performance of this element copyright c 2004 john wiley sons ltd

stochastic shortest path problems with piecewise linear concave utility functions this paper considers a stochastic shortest path problem where the arc lengths are independent random variables following a normal distribution in this problem the optimal path is one that maximizes the expected utility with the utility function being piecewise linear and concave such a utility function can be used to approximate nonlinear utility functions that capture risk averse behaviour for a wide class of problems the principal contribution of this paper is the development of exact algorithms to solve large problem instances two algorithms are developed and incorporated in labelling procedures computational testing is done to evaluate the performance of the algorithms overall both algorithms are very effective in solving large problems quickly the relative performance of the two algorithms is found to depend on the curvature of the piecewise linear utility function

vorticity velocity pressure formulation for stokes problem we propose a three field formulation for efficiently solving a twodimensional stokes problem in the case of nonstandard boundary conditions more specifically we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary the proposed computational methodology consists in reformulating the considered boundary value problem via a mixed type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the lagrange multiplier the obtained formulation is then discretized and a convergence analysis is performed a priori error estimates are established and some numerical results are presented to highlight the perfomance of the proposed computational methodology

a dynamic traveling salesman problem with stochastic arc costs we propose a dynamic traveling salesman problem tsp with stochastic arc costs motivated by applications such as dynamic vehicle routing in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed we formulate this as a dynamic program dp and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach we then apply approximate linear programming alp to overcome the dp s curse of dimensionality obtain a semi infinite linear programming lower bound and discuss its tractability we also analyze a rollout version of the price directed policy implied by our alp and derive worst case guarantees for its performance our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound

arc reduction and path preference in stochastic acyclic networks the paper presents a heuristic for determining the path that maximizes the expected utility of a stochastic acyclic network the focus is on shortest route problems where a general nonlinear utility function is used to measure outcomes for such problems enumerating all feasible paths is the only way to guarantee that the global optimum has been found alternatively we develop a reduction algorithm based on stochastic dominance to speed up the computations monte carlo simulation is used to evaluate the approach in all 70 test problems comprising 20 to 60 nodes are randomly generated and analyzed the results indicate that the heuristic produces significant computational saving as the size of the network grows and that the quality of the reduced network solutions are better than those obtained from the original formulation

extendability of kirchhoff elastic rods in complete riemannian manifolds the kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods and is defined to be a solution of the euler lagrange equations associated to the energy with the effect of bending and twisting we consider the initial value problem for the euler lagrange equations in a riemannian manifold in a previous paper the author proved the existence and uniqueness of global solutions of the initial value problem in the case where the ambient space is a space form in the present paper we extend this result to the case where the ambient space is a general complete riemannian manifold this implies that an arbitrary kirchhoff elastic rod of finite length in a complete riemannian manifold extends to that of infinite length

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

error estimates for discontinuous galerkin method for nonlinear parabolic equations abstract we consider the nonlinear parabolic partial differential equations we construct a discontinuous galerkin approximation using a penalty term and obtain an optimal l l 2 error estimate

a discontinuous mixed covolume method for elliptic problems we develop a discontinuous mixed covolume method for elliptic problems on triangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

uniform convergence and a posteriori error estimators for the enhanced strain finite element method enhanced strain elements frequently employed in practice are known to improve the approximation of standard non enhanced displacement based elements in finite element computations the first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes it is shown that such schemes are locking free in the incompressible limit in the sense that the error bound in the a priori estimate is independent of the relevant lame constant the second contribution is a residual based a posteriori error estimate the l 2 norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals

least squares for the perturbed stokes equations and the reissner mindlin plate in this paper we develop two least squares approaches for the solution of the stokes equations perturbed by a laplacian term such perturbed stokes equations arise from finite element approximations of the reissner mindlin plate both are two stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain then for the rotation itself if desired one approach uses l2 norms and the other approach uses h 1 norms to define the least squares functionals it is shown that the h 1 norm approach under general assumptions and the l2 norm approach under certain h2 regularity assumptions admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners these methods do not degrade when the perturbed parameter the plate thickness approaches zero we also develop a three stage least squares method for the reissner mindlin plate which first solves for the curls of the rotation and the shear strain next for the rotation itself and then for the transverse displacement

new properties and examples of incomplete lorentzian tori a new procedure to construct incomplete metrics on a two torus is introduced relations between curvature and completeness are explored and examples showing a wide independence among them are given incompleteness appears as a usual property of the lorentzian tori and the density of the incomplete metrics in the set of all the lorentzian ones is discussed this article can be seen as a partial study of compact space times and the conformal lorentzian moduli space mod t2 as claimed by yurtsever j math phys 31 3064 1990 and as a continuation of the authors previous article j math phys 34 3768 1993

hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems in this paper we have analyzed a one parameter family of hp discontinuous galerkin methods for strongly nonlinear elliptic boundary value problems nabla cdot rm a u nabla u f u nabla u 0 with dirichlet boundary conditions these methods depend on the values of the parameter theta in 1 1 where 1 corresponds to the nonsymmetric and 1 corresponds to the symmetric interior penalty methods when rm a u nabla u nabla u and f u u f that is for the poisson problem the error estimate in the broken h 1 norm which is optimal in h mesh size and suboptimal in p degree of approximation is derived using piecewise polynomials of degree p 2 when the solution u in h 5 2 omega in the case of linear elliptic problems also this estimate is optimal in h and suboptimal in p further optimal error estimate in the l 2 norm when 1 is derived numerical experiments are presented to illustrate the theoretical results

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

a criterion for reduction of variables in the willmore chen variational problem and its applications we exhibit a criterion for a reduction of variables for willmorechen submanifolds in conformal classes associated with generalized kaluzaklein metrics on flat principal fibre bundles our method relates the variational problem of willmore chen with an elasticity functional defined for closed curves in the base space the main ideas involve the extrinsic conformal invariance of the willmore chen functional the large symmetry group of generalized kaluza klein metrics and the principle of symmetric criticality we also obtain interesting families of elasticae in both lens spaces and surfaces of revolution riemannian and lorentzian we give applications to the construction of explicit examples of isolated willmore chen submanifolds discrete families of willmore chen submanifolds and foliations whose leaves are willmore chen submanifolds

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

shortest paths in stochastic networks with correlated link costs the objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs each link is assumed to be in one of two possible conditions conditional probability density functions for link travel times are assumed known for each condition conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip this problem is treated as a multistage adaptive feedback control process each stage is described by the physical state the location of the current decision point and the information state the service level of the previously traversed links proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided

discontinuous finite volume element discretization for coupled flow transport problems arising in models of sedimentation the sedimentation consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the stokes problem written in terms of the mixture flow velocity and the pressure field here both the viscosity and the forcing term depend on the local solids concentration a semi discrete discontinuous finite volume element dfve scheme is proposed for this model the numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field whereas the pressure is approximated by piecewise constant elements the unique solvability of both the nonlinear continuous problem and the semi discrete dfve scheme is discussed and optimal convergence estimates in several spatial norms are derived properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples including flows under gravity with changing direction a secondary settling tank in an axisymmetric setting and batch sedimentation in a tilted cylindrical vessel

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

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

generalized dynamic programming for multicriteria optimization abstract applications of dynamic programming dp to multicriteria sequential decision problems involving the optimization of a multicriteria preference function have been rare this is due to the ease with which dp s monotonicity assumption can be violated in such situations generalized dp avoids the potential pitfalls created by this absence of monotonicity thereby guaranteeing optimality the methodology is applied to a prototypical multicriteria dp problem namely a multicriteria version of the shortest path problem

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

parallel replacement under capital rationing constraints contrary to serial replacement parallel replacement problems require a decision maker to evaluate a portfolio of replacement decisions in each time period because of economic interdependencies among assets in this paper we describe a parallel replacement problem in which the economic interdependence among assets is caused by capital rationing the research was motivated by the experience gained from a vehicle fleet replacement study where solutions to serial replacement problems could not be implemented since they violated management s budget plan when firms use budgets to control their expenditures competition for the limited funds creates interdependent problems in this paper we formulate the problem as a zero one integer program and develop a branch and bound algorithm based on lagrangian relaxation methodology a multiplier adjustment method is developed to solve one lagrangian dual

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 discontinuous finite volume method for the stokes problems we develop a new discontinuous finite volume method for solving the stokes equations on both triangular and rectangular meshes an optimal error estimate for the approximation of velocity is obtained in a mesh dependent norm first order l 2 error estimates are derived for the approximations of both velocity and pressure

mixed finite element discontinuous finite volume element discretization of a general class of multicontinuum models we present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua in the framework of mixture theory we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases the model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture barycentric velocities phase pressure and saturation we construct an accurate robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume element discretization of the mass or transport equations the scheme is devised for unstructured meshes and relies on mixed brezzi douglas marini approximations of phase and total velocities on piecewise constant elements for the approximation of phase or total pressures as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest such as volume fraction total density saturation etc as the discretization scheme is derived for a general formulation of multicontinuum physical systems it can be readily applied to a large class of simplified multiphase models on the other the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate an extensive set of numerical test cases involving two and three dimensional porous media are presented to demonstrate the accuracy of the method displaying an optimal convergence rate the physics preserving properties of the mixed primal scheme as well as the robustness of the method which is successfully used to simulate diverse physical phenomena such as density fingering terzaghi s consolidation deformation of a cantilever bracket and boycott effects the applicability of the method is not limited to flow in porous media but can also be employed to describe many other physical systems governed by a similar set of equations including e g multi component materials

a new discontinuous finite volume method for elliptic problems we develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients an optimal order error estimate is obtained in a mesh dependent norm an l2 error estimate is also obtained

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

equivariant willmore surfaces in conformal homogeneous three spaces abstract the complete classification of homogeneous three spaces is well known for some time of special interest are those with rigidity four which appear as riemannian submersions with geodesic fibres over surfaces with constant curvature consequently their geometries are completely encoded in two values the constant curvature c of the base space and the so called bundle curvature r in this paper we obtain the complete classification of equivariant willmore surfaces in homogeneous three spaces with rigidity four all these surfaces appear by lifting elastic curves of the base space once more the qualitative behaviour of these surfaces is encoded in the above mentioned parameters c r the case where the fibres are compact is obtained as a special case of a more general result that works via the principle of symmetric criticality for bundle like conformal structures in circle bundles however if the fibres are not compact a different approach is necessary we compute the differential equation satisfied by the equivariant willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the euler lagrange equation of 4 r 2 elasticae in surfaces with constant curvature c we also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature it allows us to give explicit parametrizations of willmore surfaces and willmore tori in those conformal homogeneous 3 spaces