a new mixed finite element for the stokes and elasticity problems the stokes problem is approximated by a mixed finite element method using a new finite el ement which has properties analogous to the finite volume methods namely the local conservation of the momentum and the mass estimates of optimal order are derived for the errors in the velocity the pressure and the gradient of the velocity this new finite element also works for the elasticity problem and all estimates are valid uniformly with respect to the compressibility finally some numerical results for the incompressible navier stokes equations are presented

a new dual mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions abstract in this paper we consider the stress displacement rotation formulation of the plane linear elasticity problem with pure traction boundary conditions and develop a new dual mixed finite element method for approximating its solution the main novelty of our approach lies on the weak enforcement of the non homogeneous neumann boundary condition through the introduction of the boundary trace of the displacement as a lagrange multiplier a suitable combination of peers and continuous piecewise linear functions on the boundary are employed to define the dual mixed finite element scheme we apply the classical babuska brezzi theory to show the well posedness of the continuous and discrete formulations then we derive a priori rates of convergence of the method including an estimate for the global error when the stresses are measured with the l2 norm finally several numerical results illustrating the good performance of the mixed finite element scheme are reported

a dual mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions abstract in this paper we consider the hu washizu principle and propose a new dual mixed finite element method for nonlinear incompressible plane elasticity with mixed boundary conditions the approach extends a related previous work on the dirichlet problem and imposes the neumann essential boundary condition in a weak sense by means of an additional lagrange multiplier the resulting variational formulation becomes a twofold saddle point operator equation which for convenience of the subsequent analysis is shown to be equivalent to a nonlinear threefold saddle point problem in this way a slight generalization of the classical babuska brezzi theory is applied to show the well posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimates in particular the classical peers space is suitably enriched to define the associated galerkin scheme next we develop a local problems based a posteriori error analysis and derive an implicit reliable and quasi efficient estimate and a fully explicit reliable one finally several numerical results illustrating the good performance of the explicit a posteriori estimate for the adaptive computation of the discrete solutions are provided

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

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

finite element error analysis of a quasi newtonian flow obeying the carreau or power law we consider the finite element approximation of a quasi newtonian flow where the viscosity obeys the carreau or power law for sufficiently regular solutions we prove energy type error bounds for the velocity and pressure these bounds improve on existing results in the literature

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

inf sup conditions for twofold saddle point problems necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics this work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented

discrete boundary finite element schemes for an exterior problem for the time harmonic maxwell s equation abstract a general boundary integral formulation using galerkin procedure is applied to compute the scattering electric field produced by the diffraction of an incident electromagnetic wave by a perfectly conducting obstacle this electric field satisfies the three dimensional time harmonic maxwell s equations for which the skin currents and charges are to be approximated using boundary finite element method with the help of linear and quadratic finite elements of lagrange type we introduce an approximate surface on which the discrete formulation is defined and construct approximate surface currents and charges in addition we study the existence and the uniqueness of the solution of the discrete problem and develop some error estimates for the currents and charges numerical results are also presented in order to validate our numerical approach

enriched finite element subspaces for dual dual mixed formulations in fluid mechanics and elasticity abstract in this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the galerkin schemes for a new class of dual mixed variational formulations the approach which has been applied to several linear and nonlinear boundary value problems is based on the introduction of additional unknowns given by the flux and the gradient of velocity and by the stress and strain tensors and rotations for fluid mechanics and elasticity problems respectively in this way the procedure yields twofold saddle point operator equations as the resulting weak formulations also named dual dual ones which are analyzed by means of a slight generalization of the well known babuska brezzi theory then in order to introduce well posed galerkin schemes we extend the arguments used in the continuous case to the discrete one and show that some usual finite elements need to be suitably enriched depending on the nature of the problem this leads to piecewise constant functions raviart thomas of lowest order peers elements and the deviators of them as the appropriate subspaces

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

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

hypergeometric identities arising from the elephant random walk abstract a probabilistic approach is provided to establish new hypergeometric identities it is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime

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

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

mixed finite element methods for linear elasticity with weakly imposed symmetry in this paper we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima tions to both stresses and displacements the methods are based on a modified form of the hellinger reissner variational principle that only weakly imposes the symmetry condition on the stresses although this approach has been previously used by a number of authors a key new ingredient here is a constructive derivation of the elasticity complex starting from the de rham complex by mimicking this construction in the discrete case we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de rham complex these elements appear to be simpler than the ones previously derived for example we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field

characterizing the inf sup condition on product spaces in this paper we establish characterization results for the continuous and discrete inf sup conditions on product spaces the inf sup condition for each component of the bilinear form involved and suitable decompositions of the pivot space in terms of the associated null spaces are the key ingredients of our theorems we illustrate the theory through its application to bilinear forms arising from the variational formulations of several boundary value problems

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

quasi norm error bounds for the finite element approximation of a non newtonian flow we consider the finite element approximation of a non newtonian flow where the viscosity obeys a general law including the carreau or power law for sufficiently regular solutions we prove energy type error bounds for the velocity and pressure these bounds improve on existing results in the literature a key step in the analysis is to prove abstract error bounds initially in a quasi norm which naturally arises in degenerate problems of this type

dual hybrid methods for the elasticity and the stokes problems a unified approach a unified approach to construct finite elements based on a dual hybrid formulation of the linear elasticity problem is given in this formulation the stress tensor is considered but its symmetry is relaxed by a lagrange multiplier which is nothing else than the rotation this construction is linked to the approximations of the stokes problem in the primitive variables and it leads to a new interpretation of known elements and to new finite elements moreover all estimates are valid uniformly with respect to compressibility and apply in the incompressible case which is close to the stokes problem