postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials this paper addresses the use of isogeometric analysis to solve solid mechanics problems involving nearly incompressible materials the present work is focused on extension of two field mixed variational formulations in both small and large strains to isogeometric analysis inf sup stable displacement pressure combinations for mixed formulations are developed based on the subdivision property of nurbs stability and convergence properties of the proposed displacement pressure combinations are illustrated by computing numerical inf sup constants and error norms the performance of the proposed formulations is assessed by studying several benchmark examples involving nearly incompressible and incompressible elastic and elasto plastic materials in both small and large strain regime

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

uniform preconditioners for a parameter dependent saddle point problem with application to generalized stokes interface equations we consider an abstract parameter dependent saddle point problem and present a general framework for analyzing robust schur complement preconditioners the abstract analysis is applied to a generalized stokes problem which yields robustness of the cahouet chabard preconditioner motivated by models for two phase incompressible flows we consider a generalized stokes interface problem application of the general theory results in a new schur complement preconditioner for this class of problems the robustness of this preconditioner with respect to several parameters is treated results of numerical experiments are given that illustrate robustness properties of the preconditioner

wave propagation in a fitzhugh nagumo type model with modified excitability we examine a generalized fitzhugh nagumo fhn type model with modified excitability derived from the diffusive morris lecar equations for neuronal activity we obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms we study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms fronts and pulses while the activator wave maintains the usual kink front shape the nonequilibrium ising bloch bifurcation for the wave speed that occurs in the fhn model where the control parameter is the ratio of inhibitor to activator time scales persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter

a posteriori error estimators for the stokes and oseen equations the problem of obtaining posteriori estimates of the discretization error when one uses finite element methods to approximate problems with an incompressibility constraint is discussed a general approach to the treatment of the constraint condition and to the possible non self adjointness of the associated momentum equations is presented a posteriori error estimates are derived for adaptive h p and h p type finite element schemes key features are that the local error residual problems are not subject to an incompressibility constraint thereby avoiding the need for special finite element schemes and that the analysis is valid for essentially any discretization scheme including continuous and discontinuous pressure spaces the estimator bounds the actual error measured in an energy like norm

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

a unified mixed finite element approximations of the stokes darcy coupled problem abstract in this paper we develop and analyze a unified approximation of the velocity pressure pair for the stokes darcy coupled problem in a plane domain it is well known that stable finite element approximations for the stokes problem may not be appropriate for darcy problem and for the coupling of fluid flow modeled by the stokes equations with porous media flow modeled by the darcy equation and therefore different spaces are commonly used for the discretizations of the darcy and the stokes problems in this work we proposed a modification of the darcy problem which allows us to apply the classical mini element to the whole coupled stokes darcy problem the proposed method is probably one of the cheapest method for continuous approximation of the coupled system has optimal accuracy with respect to solution regularity and has simple and straightforward implementations numerical experiments are also presented which confirm the excellent stability and accuracy of our method

a new nonconforming galerkin scheme for the stokes problem partially circumventing the discrete babǔska brezzi condition abstract in this paper we introduce a new stable nonconforming galerkin scheme for the interior stokes problem our approach uses a dirichllety mapping d from the pressure space q l 0 2 into the velocity space h h 0 1 2 in order to make a suitable change of variables in the original mixed formulation this transformation yields an equivalent variational formulation whose associated bilinear form is strongly coercive on the product space h q we then replace the mapping d by its finite element approximation on a subspace h h of h and define the corresponding discrete scheme on a finite element subspace h h q h of h q the leads to a nonconforming galerkin method whose bilinear form is uniformly strongly coercive on any family h h q h h h 0 provided that h o h consequently although h o h implies the satisfaction of the discrete bab ska brezzi condition between h h and q h it turns out that this stability condition is not needed between the approximating spaces h h and q h therefore all the combinations of velocity and pressure subspaces with continuous lagrange elements including certainly equal order interpolations become stable in addition the cea estimate is valid for all h h 0 and hence the usual rates of convergence for the error are also derived finally we discuss the matrix formulation of the algorithm and show that the resulting galerkin equations reduce to two uncoupled linear systems whose stiffness matrices are positive definite

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

local and parallel finite element algorithms based on the partition of unity for the stokes problem by combining the techniques of the two grid method and the partition of unity we derive two local and parallel finite element algorithms for the stokes problem the most interesting features of these algorithms are 1 the partition of unity technique introduces a framework for domain decomposition 2 only a series of local residual problems need to be solved on these subdomains in parallel meanwhile requiring very little communication 3 a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions the optimal error estimates in l 2 and energy norms are proved under some assumptions also several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms

multi level spectral galerkin method for the navier stokes problem i spatial discretization a multi level spectral galerkin method for the two dimensional non stationary navier stokes equations is presented the method proposed here is a multiscale method in which the fully nonlinear navier stokes equations are solved only on a low dimensional space subsequent approximations are generated on a succession of higher dimensional spaces j 2 j by solving a linearized navier stokes problem around the solution on the previous level error estimates depending on the kinematic viscosity 0 1 are also presented for the j level spectral galerkin method the optimal accuracy is achieved when we demonstrate theoretically that the j level spectral galerkin method is much more efficient than the standard one level spectral galerkin method on the highest dimensional space

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

a mixed finite element method for nearly incompressible multiple network poroelasticity in this paper we present and analyze a new mixed finite element formulation of a general family of quasi static multiple network poroelasticity mpet equations the mpet equations describe flow a

wavy fronts and speed bifurcation in excitable systems with cross diffusion a bifurcation of excitation fronts induced by cross diffusion in two component bistable reaction diffusion systems of activator inhibitor type is discovered this bifurcation is similar to the nonequilibrium ising bloch bifurcation a different type of fronts whose spatial profiles are characterized by oscillating tails are associated with this bifurcation these fronts are described using exact analytical solutions of piecewise linear reaction diffusion equations

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

a two level method in time and space for solving the navier stokes equations based on newton iteration a two level method in time and space for solving the time dependent navier stokes equations based on newton iteration is considered in this paper considering that the small eddy components carry little part of the whole energy we use a larger time step size to obtain the small eddy component as compared with the large eddy components numerical analysis shows that it is possible to choose a proper time step size without compromising the overall accuracy moreover some numerical examples are provided to complement our theoretical analysis

an expandable local and parallel two grid finite element scheme an expandable local and parallel two grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of poisson equation compared with the usual local and parallel finite element schemes the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of cpus convergence results based on h 1 and l 2 a priori error estimation of the scheme are obtained which show that the scheme can reach the optimal convergence orders within ln h 2 or ln h two grid iterations if the coarse mesh size h and the fine mesh size h are properly configured in 2 d or 3 d case respectively some numerical results are presented at the end of the paper to support our analysis

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

adaptive inexact iterative algorithms based on polynomial degree robust a posteriori estimates for the stokes problem in this paper we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear stokes problem we base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components namely the discretization error component the inner algebraic solver error component and possibly the outer algebraic solver error component for algorithms of the uzawa type we prove that our estimate gives a guaranteed upper bound on the total error as well as a polynomial degree robust local efficiency and this on each step of the employed iterative algorithm our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error the developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver implementation into the freefem programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies as example we choose here the unpreconditioned and preconditioned uzawa algorithm and the preconditioned minimum residual minres algorithm in combination with the taylor hood discretization

postprocessing fourier galerkin method for the navier stokes equations a full discrete two level postprocessing fourier galerkin scheme for the unsteady navier stokes equations with a periodic boundary condition is proposed in this paper by defining a new projection the interaction between the large and small eddies is reflected by the associated space splitting to some extent therefore a weakly coupled system of the large and small eddies is obtained stability and error estimates for the weakly coupled two level scheme are established in the paper the proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse level subspace by solving the coarse level standard galerkin equation and the matrix of the linear algebraic equations arising in each fine level time stepping is a sparse matrix especially for large scale computations as a result the new scheme saves a lot of cpu time and memory in comparison with the standard fourier galerkin method for deriving an approximation of prescribed accuracy

local and parallel finite element post processing scheme for the stokes problem abstract a local and parallel finite element post processing scheme based on partition of unity method is proposed and analyzed in this paper for the stokes problem firstly a standard galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components secondly the higher frequency components are computed on fine grids by some local and parallel procedure to post process the standard galerkin approximation the motivation of the proposed local and parallel finite element post processing scheme is based on the superposition principle finally to eliminate the effect of the dirichlet boundary conditions which are imposed on the internal artificial boundaries a global coarse grid correction is done to improve the l 2 accuracy of the approximation

analytic solutions for monotonic and oscillating fronts in a reaction diffusion system under external fields abstract a piecewise linear reaction diffusion system including an external field is considered analytic solutions are obtained for the propagating front the front velocity the perturbed front and the growth rate of perturbations it is found that when the ratio of the time scales null cline parameter and the external field are small enough the moving front has an oscillating behaviour similar to that of a front in an oscillatory medium a comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases in particular we find that for fronts oscillating about a stable state both concentrations must oscillate in two variable models whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

the postprocessing galerkin and nonlinear galerkin methods a truncation analysis point of view we revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view we assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation keeping terms to certain orders we then generate approximate systems which correspond to numerical schemes we show that the first two leading order methods are in fact the postprocessed galerkin and postprocessed nonlinear galerkin methods respectively hence postprocessed galerkin is a natural leading order method more natural than the standard galerkin method for approximating solutions of parabolic dissipative pdes the analysis is presented in the framework of the two dimensional navier stokes equation nse however similar analysis may be done for any parabolic dissipative nonlinear pde the truncation analysis is based on asymptotic estimates in time for the low and high mode components we also introduce and investigate an alternative postprocessing scheme whic

biased random walks and propagation failure the critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non markovian biased random walks with reactions from the hamilton jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation if the reaction rate is larger than the jump frequency invasion always succeeds even in the case of maximal bias numerical simulations support our analytical predictions

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

multilevel methods in space and time for the navier stokes equations we investigate the discretization in time of numerical schemes based on multilevel spatial splittings for the two dimensional periodic navier stokes equations the approximate solution is computed as the sum of a low frequency component and a high frequency one these two terms are advanced in time using different stepsizes we show improved stability conditions with respect to the classical galerkin method we derive error estimates that indicate that the high frequency term can be integrated less often we address implementation issues and show that the method should yield a significant gain in computing time

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

tangent space correction method for the galerkin approximation based on two grid finite element abstract a novel fully discrete two grid finite element method for certain semi linear parabolic equations is presented in this paper the new scheme is based on two different finite element subspaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h h nonlinearity is treated in the coarse grid subspace by solving the standard galerkin equation while in the fine grid subspace only a linear equation has to be solved differing from the usual two grid method the splitting of the coarse grid subspace and its related fine grid incremental subspace is based on a new projection in which sense the incremental subspace is closely identified with the tangent space of certain operator on each time step with linear finite element discretization the stability and error estimate results for such new scheme are derived the results show that the difference between this new approximation and the fine grid standard galerkin approximation in h 1 norm is of the order h 4 d 2 where d 1 2 3 is the space dimension

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

a posteriori error estimations for mixed finite element approximations to the navier stokes equations based on newton type linearization abstract in this paper we derive a posteriori error estimates for inf sup stable mixed finite element approximations to the evolutionary navier stokes equations we reduce the problem of getting a posteriori error estimations of a non linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem the main idea is based on the fact that the solutions of some newton type linearized navier stokes equations around the plain galerkin approximations approach the solution of the original navier stokes equations with a bigger rate of convergence than the plain galerkin method as a consequence the difference between the galerkin approximations and the solution of the linearized problem is an estimator of the galerkin error moreover since the galerkin approximation of the evolutionary navier stokes equations is also the galerkin approximation of the linearized equations any a posteriori estimator of the error in the newton type linearized navier stokes equations is also an a posteriori error estimator of the full navier stokes equations

a posteriori error control of h p finite elements for variational inequalities of the first and second kind in this paper residual based a posteriori error estimates for variational inequalities including those of the second kind are proposed the variational formulation and its discretizations are considered in an abstract framework of general banach spaces the main idea is to express the residual in terms of some lagrange multipliers which can be obtained for instance by some post processing the residual itself is defined for an arbitrary element of the trial space the error of this element and of the solution of the variational inequality is then estimated by the dual norm of the residual plus some computable remainder terms this concept is applied to a variety of frictional contact problems such as signorini and obstacle problems as well as to the bingham fluid problem the applicability of the estimates is confirmed by several numerical experiments in particular the general framework allows for the discretization with h p adaptivity which leads to nearly exponential convergence rates in most cases

stationary and oscillatory fronts in a two component genetic regulatory network model abstract we investigate a two component gene network model originally used to describe the spatiotemporal patterning of the gene products in early drosophila development by considering a particular mode of interaction between the two gene products denoted proteins a and b we find both stable stationary and time oscillatory fronts can occur in the reaction diffusion system we reduce the system by replacing b with its spatial average shadow system and assume an abrupt on and off switch for the genes in doing so explicit formula are obtained for all steady state solutions and their linear eigenvalues using the diffusion of a d a and the basal production rate r as bifurcation parameters we explore ranges in which a monotone stationary front is stable and show it can lose stability through a hopf bifurcation giving rise to oscillatory fronts we also discuss the existence and stability of steady state and time oscillatory solutions with multiple extrema an intuitive explanation for the occurrence of stable stationary and oscillatory front solutions is provided based on the behavior of a in the absence of b and the opposite regulation between a and b such behavior is also interpreted in terms of the biological parameters in the model including those governing the connection of the gene network

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

stability analysis of the nonlinear galerkin method our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations and to study the stability of the schemes spatial discretization can be performed by either spectral or pseudospectral methods finite elements or finite differences time discretization is done by two level schemes partly or fully explicit the algorithms that we present stem from the study of the evolution equations from the dynamical systems point of view they are based on a differentiated treatment of the small and large wave lengths and they are particularly adapted to the integration of such equations on large intervals of time introduction the long time integration of dissipative evolution equations has emerged as a new type of problem in numerical analysis and its appearance is due to the considerable increase in computing power that we have seen during the last years much effort has been devoted in the past to the approximation of evolution equations on finite intervals of time or in the dynamically simple case where the solution converges to a steady stationary state as time goes to infinity however in many physically relevant situations the solutions to a dissipative evolution equation do not converge to a stationary solution they rather remain time dependent and they converge to a compact attractor that encompasses the nonlinear dynamics hence and whether this is addressed implicitly or explicitly the numerical integration of the evolution equation is then closely related to the approximation of the attractor that may be a complicated set even a fractal inertial algorithms are new integration algorithms that are based on the approximation of the attractor by smooth simple manifolds namely the approximate inertial manifolds these algorithms will be recalled below but at this point let us point out that one of their main features is a differentiated treatment of small and large wave lengths an inertial manifold is an exact law of dependence of small wave lengths in terms of the large ones the small wave received april 3 1990 1980 mathematics subject classification 1985 revision primary 65f10 65f35 65n20 35a40 1991 american mathematical society 0025 5718 91 1 00 25 perpage

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

generalized postprocessed approximations to the navier stokes equations based on two grids abstract in this paper we generalize previous postprocessed approximations to the navier stokes equations the idea of postprocessing in the context of mixed finite element approximations is the following once a standard galerkin mixed finite element approximation is computed using a mesh we will call coarse of size h at a fixed time t we postprocess the approximation using a finer mesh of size h h by solving a steady problem with data based on the galerkin approximation in the literature one can find different ways to postprocess the galerkin approximation depending on the problem that is solved over the fine mesh at the final time t basically one can solve a stokes problem an oseen type problem or a newton type problem in the present paper we present a method based on several parameters that generalize the above postprocessing techniques depending on the values chosen for the different parameters one can recover one of the old known postprocessing procedures but also produce a new different method we get error bounds for the generalized method valid for any of the values of the different parameters in all the cases the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain galerkin method in terms of the coarse mesh h and optimal in terms of the fine mesh h the computational added cost of postprocessing is however negligible for the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the navier stokes equations some numerical experiments show the performance of the method for different values of the parameters

a posteriori error estimates for fully discrete nonlinear parabolic problems abstract we present estimates for the spatial error in fully discrete approximations to nonlinear parabolic problems that extend the a posteriori estimates for the continuous time semi discretization introduced in de frutos and novo j de frutos j novo a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations comput methods appl mech engrg 191 2002 4893 4904 the error indicator is defined as the difference between the fully discrete galerkin and the fully discrete postprocessed approximations we prove that the main term of the temporal error in both the fully discrete galerkin and the postprocessed approximation is the same as a consequence the spatial approximation error can be independently estimated with regard to the time approximation error some numerical experiments are provided

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

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

an approximate inertial manifolds approach to postprocessing the galerkin method for the navier stokes equation in a recent paper we have introduced a postprocessing procedure for the galerkin method for dissipative evolution partial differential equations with periodic boundary conditions the postprocessing technique uses approximate inertial manifolds to approximate the high modes the small scale components in the exact solutions in terms of the galerkin approximations which in this case play the role of the lower modes large scale components this procedure can be seen as a defect correction technique but contrary to standard procedures the correction is computed only when the time evolution is completed here we extend these results to more realistic boundary conditions specifically we study in detail the two dimensional navier stokes equations subject to homogeneous nonslip dirichlet boundary conditions we also discuss other equations such as reaction diffusion systems and the cahn hilliard equations

a posteriori error estimates of stabilization of low order mixed finite elements for incompressible flow in this paper we derive a posteriori error estimates for the stabilization of low order mixed finite element methods for the stokes equations by defining different projection estimators we prove that up to higher order perturbation terms the estimators yield global upper and lower bounds on the error of stabilized finite element methods in numerical tests each error estimator is shown to be equivalent to the discretization error it is also shown that the adaptive strategy based on both projection estimators is efficient to detect local singularities in the flow problems

a simple construction of a fortin operator for the two dimensional taylor hood element abstract a fortin operator is constructed to verify the discrete inf sup condition for the lowest order taylor hood element and its variant in two dimensions the approach is closely related to the recent work by mardal et al 2013 that is based on the isomorphism of the tangential edge bubble function space to a subspace of the lowest order edge element space a more precise characterization of this subspace and a numerical quadrature are introduced to simplify the analysis and remove the mesh restriction the constructed fortin operator is stable in both h 1 and l 2 norm for general shape regular triangulations

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection abstract in this paper we present a stabilized finite element formulation for the transient incompressible navier stokes equations the main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the laplacian of the pressure and the divergence of this new vector field this leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures in the case of the transient equations we consider the possibility of treating the pressure gradient projection either implicitly or explicity in the first case the number of unknowns of the problem is substantially increased with respect to the standard galerkin formulation nevertheless iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns velocity and pressure when this vector field is treated explicitly the increment of computational cost of the stabilized formulation with respect to the galerkin method is very low we provide a stability estimate for the case of the simple backward euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection

pointwise error estimates for finite element solutions of the stokes problem in this paper pointwise error estimates for general finite element approximations of the stokes problem are established on quasi uniform grids in r n the results obtained in this paper improve and extend the existing error estimates in the maximum norm for the stokes problem the new pointwise error estimates exhibit a more local dependence of the errors on the true solution and as a by product provide logarithm free bounds for all errors except the error of the velocity approximation of the lowest order

multilevel algorithms for mixed problems ii treatment of the mini element we present a multilevel algorithm for the mixed finite element approximation of the stokes problem by the mini element a local l 2 projection of the bubble functions and a strengthened cauchy inequality are essential tools for the convergence analysis of the multigrid algorithm

a posteriori error estimation of steady state finite element solutions of the navier stokes equations by a subdomain residual method abstract we present a subdomain residual method as well as its mathematical basis for estimating errors in steady state finite element solutions of the incompressible navier stokes equations the estimated errors are obtained by solving a series of local problems in which velocity boundary condition is used wherever the exact traction boundary condition is not available an iterative procedure similar to the newton method is employed to improve the error estimates the performance of the method is demonstrated in two numerical examples i e the channel flow over a backward facing step and past a circular cylinder at low reynolds numbers

a posteriori error estimates of finite element method for the time dependent navier stokes equations abstract in this paper we consider the posteriori error estimates of galerkin finite element method for the unsteady navier stokes equations by constructing the approximate navier stokes reconstructions the errors of velocity and pressure are split into two parts for the estimates of time part the energy method and other skills are used for the estimates of spatial part the well developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted more important the error estimates of time part can be controlled by the estimates of spatial part as a consequence the posteriori error estimates in l 0 t l 2 l 0 t h 1 and l 2 0 t l 2 norms for velocity and pressure are derived in both spatial discrete and time space fully discrete schemes

a local projection stabilization continuous galerkin petrov method for incompressible flow problems abstract a local projection stabilization lps method in space is considered to approximate the evolutionary oseen equations optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous in time case for both the velocity and pressure approximation in addition the fully discrete case in combination with higher order continuous galerkin petrov cgp methods is studied error estimates of order k 1 are proved where k denotes the polynomial degree in time assuming that the convective term is time independent numerical results show that the predicted order is also achieved in the general case of time dependent convective terms

a posteriori error estimation with the p version of the finite element method for nonlinear parabolic differential equations we analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions the spatial discretization is carried out using the p version of the finite element method the error estimates are obtained by solving an elliptic problem at the desired times when the estimation is wanted some numerical experiments prove the efficiency of the error estimation

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded