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

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

attractors and error estimates for discretizations of incompressible navier stokes equations by embedding sets of nodal values to function spaces variational arguments are applied to the finite difference approximations to the two dimensional incompressible navier stokes equations the error estimates for trajectories and the it time free error estimates with tolerance for attractors are proved the upper semicontinuity of the attractors with respect to the finite difference discretizations are also proved arguments of applying the techniques to the finite element approximations are also given

bifurcation from infinity for reaction diffusion equations under nonlinear boundary conditions we consider reaction diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter we prove that as the parameter crosses some critical values a resonance type phenomenon provides solutions that bifurcate from infinity we characterize the bifurcated branches when they are sub or supercritical we obtain both landesman lazer type conditions that guarantee the existence of solutions in the resonant case and an anti maximum principle

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight abstract the bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described in a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case for each value of the parameter we prove existence of an unique nonconstant equilibrium solution it belongs to a globally parametrized bifurcation curve constructed using a lyapunov schmidt type reduction combined with the morse lemma that solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi orbits the behavior of its trace when the parameter is large is established and complete diagrams containing all bifurcation and stability information are also provided

on the stability of integral equations of the first kind with logarithmic kernels this paper discusses the stability of the galerkin method for a class of boundary integral equations of the first kind these integral equations arise in acoustics elasticity and hydrodynamics and the kernels of the principal parts of the corresponding integral operators all have logarithmic singularities it is shown that an optimal choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence of the approximate solutions the results here are consistent with those obtained by the tikhonov regularization procedure

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

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

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

error estimates for the semidiscrete galerkin approximations of the fitzhugh nagumo equations l 2 error estimates are obtained for the semidiscrete approximation of the fitzhugh nagumo equations with nonsmooth initial data the equations consist of a nonlinear parabolic equation coupled with an ordinary differential equation smoothing due to the parabolic equation is used to show increased accuracy for the approximation for positive time

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

unbounded nonlinear perturbations of linear elliptic problems at resonance 435 where lj is an eigenvalue of l a is the derivative in the direction outward normal to the boundary 2 of 0 and f h q recently shapiro lo established an existence result for the problem 1 1 in the case that g has unbounded nonlinearities and satisfies a landesman lazer type condition shapiro s result however does not cover the case that g has linear growths our first result is an extension of shapiro s result to the case that g has a linear growth we next consider the case g does not satisfies any landesman lazer type condition that is we study the case lim inf oc g x t lim supi up g x t a e on q several authors considered this case with g having bounded nonlinearities cf 9 fucik and krebec 5 and hess 6 gave existence results for 1 1 in case that j l q and g is a continuous function satisfying g x t g t g t g r and g lim fm g t 0 we extend these results for 1 1 to the case g has unbounded nonlinearities and f l2 our method employed here is different from those in s 6 or lo 2

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity we consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter as the parameter crosses some critical values there appear certain resonances in the equation producing solutions that bifurcate from infinity we study the bifurcation branches characterize when they are sub or supercritical and analyse the stability type of the solutions furthermore we apply these results and techniques to obtain landesman lazer type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti maximum principle

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

an elliptic system with bifurcation parameters on the boundary conditions in this paper we consider the elliptic system u a x upvq v b x urvs in a smooth bounded domain with boundary conditions u u v v on here and are regarded as parameters and p s 1 q r 0 verify p 1 s 1 qr we consider the case where a x 0 in and a x is allowed to vanish in an interior subdomain 0 while b x 0 in our main results include existence of nonnegative nontrivial solutions in the range 0 0 where 1 is characterized by means of an eigenvalue problem and the uniqueness of such solutions we also study their asymptotic behavior in all possible cases as both 0 as 1 for fixed respectively for fixed and when both in case 1

on a fem bem formulation for an exterior quasilinear problem in the plane we use a version of the fem bem method introduced by costabel boundary elements ix vol 1 c a brebbia et al eds springer verlag 1987 and han j comput math 8 1990 pp 223 232 to discretize an exterior quasilinear problem we provide error estimates for the galerkin method and propose a fully discrete scheme based on simple quadrature formulas furthermore we show that these numerical integration schemes preserve the optimal rates of convergence finally we present results of numerical experiments involving our discretization method

coupling of the finite volume element method and the boundary element method an a priori convergence result the coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain this discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems we show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary provide a convergence and an a priori analysis in an energy semi norm and provide an existence and an uniqueness result for the discrete system all results are also valid for the upwind version numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems which can also be convection dominated

a galerkin characteristic unified finite element method for moving thermal fronts in porous media abstract we investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non isothermal flow conditions the governing equations consist of coupling the darcy equation for the pressure to two convection diffusion reaction equations for the temperature and depth of conversion the aim is to develop a non oscillatory unified galerkin characteristic method for efficient simulation of moving fronts in porous media the method is based on combining the modified method of characteristics with a galerkin finite element discretization of the governing equations the main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure velocity temperature and concentration analysis of convergence and stability is also presented in this study and error estimates in the l 2 norm are established for the numerical solutions in addition due to the lagrangian treatment of convection terms the standard courant friedrichs lewy condition is relaxed and the time truncation errors are reduced in the diffusion reaction part we verify the method for the benchmark problem of moving fronts around an array of cylinders the numerical results obtained demonstrate the ability of the proposed method to capture the main flow features

a posteriori boundary element error estimation an a posteriori error estimator is presented for the boundary element method in a general framework it is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators the estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for symm s and hypersingular integral equations numerical results are also given to demonstrate the eectiveness of the estimator for these equations it can be used for adaptive h p and hp methods c 1999 elsevier science b v all rights reserved msc 65n35 65n15 65r20 65d07 45l10

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

numerical analysis of a modified finite element nonlinear galerkin method a fully discrete modified finite element nonlinear galerkin method is presented for the two dimensional equation of navier stokes type the spatial discretization is based on two finite element spaces xh and xh defined on a coarse grid with grid size h and a fine grid with grid size h h respectively the time discretization is based on the euler explicit scheme with respect to the nonlinear term we analyze the stability and convergence rate of the method comparing with the standard finite element galerkin method and the nonlinear galerkin method this method can admit a larger time step under the same convergence rate of same order hence this method can save a large amount of computational time finally we provide some numerical tests on this method the standard finite element galerkin method and the nonlinear galerkin method which are in a good agreement with the theoretical analysis

a convergent finite element formulation for transonic flow a finite element formulation for the full potential equation in the case of two dimensional transonic flow is presented the formulation is based on an optimal control approach developed by glowinski and pironneau the solution of the full potential equation is obtained by a minimization problem using a new compactness result it is possible to prove convergence for the solutions of the minimization problem the a priori assumption of existence and uniqueness of a weak solution of the full potential equation satisfying an entropy condition implies that the limit function must be the solution it is possible to extend the convergence result to the case of three dimensional transonic potential flow

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

convergence analysis of three finite element iterative methods for the 2d 3d stationary incompressible magnetohydrodynamics abstract three finite element iterative methods are designed and analyzed for solving 2d 3d stationary incompressible magnetohydrodynamics mhd by a new technique strong uniqueness conditions for both stokes type iterative method iterative method i and newton iterative method iterative method ii are obtained which are weaker than the ones reported in open literature stability and optimal convergence rates for the above two methods are derived where the iterative method ii has an exponential convergent part with respect to iterative step m moreover oseen type iterative method iterative method iii is unconditionally stable and convergent under a uniqueness condition finally performance of the three proposed methods is investigated by numerical experiments

an energy and helicity conserving finite element scheme for the navier stokes equations we present a new finite element scheme for solving the navier stokes equations that exactly conserves both energy int omega u 2 and helicity int omega u cdot nabla times u in the absence of viscosity and external force we prove stability exact conservation and convergence for the scheme energy and helicity are exactly conserved by using a combination of the usual convective form with the rotational form of the nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization numerical results are presented that compare the scheme to the usual trapezoidal schemes

on the convergence of a time discretization scheme for the navier stokes equations version linearisee du schema d euler implicite utilisee pour l approximation des solutions des equations de navier stokes dans un domaine bidimensionnel

unconditional convergence and optimal l2 error estimates of the crank nicolson extrapolation fem for the nonstationary navier stokes equations abstract in this paper we study stability and convergence of fully discrete finite element method on large timestep which used crank nicolson extrapolation scheme for the nonstationary navier stokes equations this approach bases on a finite element approximation for the space discretization and the crank nicolson extrapolation scheme for the time discretization it reduces nonlinear equations to linear equations thus can greatly increase the computational efficiency we prove that this method is unconditionally stable and unconditionally convergent moreover taking the negative norm technique we derive the l 2 h 1 unconditionally optimal error estimates for the velocity and the l 2 unconditionally optimal error estimate for the pressure also numerical simulations on unconditional l 2 stability and convergent rates of this method are shown

asymptotic behavior of semidiscrete finite element approximations of biot s consolidation problem error estimates for spatially discrete galerkin finite element approximations of biot s model for consolidation of saturated porous media are presented the short and long time behaviors of such approximations based on both stable and unstable combinations of finite element spaces of displacement and pore pressure fields are discussed

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

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

a fully eulerian formulation for fluid structure interaction problems in this work we present a fully eulerian framework for fluid structure interaction fsi problems coupling the incompressible navier stokes equations with a hyperelastic solid the fully eulerian framework is a monolithic implicit variational formulation for the coupled problem in contrast to the well established arbitrary lagrangian eulerian ale coordinates the fully eulerian framework formulates both subproblems fluid and solid in eulerian coordinates this concept circumvents various difficulties connected to ale coordinates since no artificial domain mapping is used the formulation is an interface capturing method and uses an extension of the solid s deformation the initial point set to detect the interface location by construction very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible

a nonconforming a posteriori estimator for the coupling of cell centered finite volume and boundary element methods the coupling of the cell centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain where local flux conservation is important based on the piecewise constant interior finite volume solution we define a morley type interpolant built on a nonconforming finite element together with the cauchy data of the exterior boundary element solution this allows us to define a residual based a posteriori error estimator with respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh refining algorithm in two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming morley interpolant

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

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

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

equilibria and global dynamics of a problem with bifurcation from infinity we consider a parabolic equation ut u u 0 with nonlinear boundary conditions where as s in j m arrieta r pardo a rodriguez bernal bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity proc roy soc edinburgh sect a 137 2 2007 225 252 the authors proved the existence of unbounded branches of equilibria for close to a steklov eigenvalue of odd multiplicity in this work we characterize the stability of such equilibria and analyze several features of the bifurcating branches we also investigate several question related to the global dynamical properties of the system for different values of the parameter including the behavior of the attractor of the system when the parameter crosses the first steklov eigenvalue and the existence of extremal equilibria we include appendix a where we prove a uniform antimaximum principle and several results related to the spectral behavior when the potential at the boundary is perturbed

coupling of finite and boundary element methods for an elastoplastic interface problem a class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain a typical example is a problem from three dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material the nonlinear problem is given in variational form with a strictly convex functional the linear elliptic problem is described by boundary integral equations on the coupling boundary the typical saddle point structure of such problems is analyzed galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary the convergence of the galerkin approximation is based on the saddle point structure which is shown to hold for the exact as well as the discretized problems

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

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

discrete dirichlet to neumann maps for unbounded domains abstract it is shown how to construct a discrete counterpart of the dirichlet to neumann dtn map for use on an artificial boundary b introduced in exterior boundary value problems following the idea of deakin and dryden this discrete map provides an approximate non reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain bounded by b different discrete maps are constructed for use with finite difference and finite element methods in the solution of some simple problems shows that use of the discrete dirichlet to neumann ddtn map yields nearly the same accuracy as that of using the continuous dtn map

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

nonlinear balance for reaction diffusion equations under nonlinear boundary conditions dissipativity and blow up the authors consider a reaction diffusion equation in a bounded smooth domain r n with nonlinear flux terms on the boundary of they derive suitable conditions on the nonlinear terms of the problem which imply its dissipativity the authors explore different situations in which there exist solutions of the problem that blow up in finite time the main tools in the proof of the results are either energy estimates or sub and super solution techniques

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

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

the tangential variation of a localized flux type eigenvalue problem in this work the differentiability of the principal eigenvalue 1 to the localized steklov problem u qu 0 in u x u on where is a smooth subdomain of and is its characteristic function relative to is shown as a key point the flux subdomain is regarded here as the variable with respect to which such differentiation is performed an explicit formula for the derivative of 1 with respect to is obtained the lack of regularity up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem therefore the whole analysis must be done in the weak sense of h1 the study is of interest in mathematical models in morphogenesis

a dual dual formulation for the coupling of mixed fem and bem in hyperelasticity we combine a dual mixed finite element method with the boundary integral equation method to study the solvability and galerkin approximations of a linear nonlinear transmission problem arising in plane elastostatics our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain and it leads to what we call a dual dual mixed formulation since the corresponding operator equation becomes of a twofold saddle point structure we derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis in particular we extend the finite element space for plane elasticity given by the peers element and define an explicit finite element boundary element subspace satisfying all the required compatibility conditions most of our analysis makes use of an extension of the classical babuska brezzi theory to a class of nonlinear saddle point problems

on the adaptive coupling of fem and bem in 2 d elasticity this paper concerns the combination of the finite element method fem and the boundary element method bem using the symmetric coupling as a model problem in two dimensions we consider the hencky material a certain nonlinear elastic material in a bounded domain with navier lame differential equation in the unbounded complementary domain using some boundary integral operators the problem is rewritten such that the galerkin procedure leads to a fem bem coupling and quasi optimally convergent discrete solutions beside this a priori information we derive an a posteriori error estimate which allows up to a constant factor the error control in the energy norm since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh refinement we state adaptive algorithms for the h version of the fem bem coupling illustrating numerical results are included

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

a banach spaces based analysis of a new mixed primal finite element method for a coupled flow transport problem abstract in this paper we introduce and analyze a new finite element method for a strongly coupled flow and transport problem in r n n 2 3 whose governing equations are given by a scalar nonlinear convection diffusion equation coupled with the stokes equations the variational formulation for this model is obtained by applying a suitable dual mixed method for the stokes system and the usual primal procedure for the transport equation in this way and differently from the techniques previously developed for this and related coupled problems no augmentation procedure needs to be incorporated now into the solvability analysis which constitutes the main advantage of the present approach the resulting continuous and discrete schemes which involve the cauchy fluid stress the velocity of the fluid and the concentration as the only unknowns are then equivalently reformulated as fixed point operator equations consequently the well known schauder banach and brouwer theorems combined with babuska brezzi s theory in banach spaces monotone operator theory regularity assumptions and sobolev imbedding theorems allow to establish the corresponding well posedness of them in particular raviart thomas approximations of order k 0 for the stress discontinuous piecewise polynomials of degree k for the velocity and continuous piecewise polynomials of degree k 1 for the concentration becomes a feasible choice for the galerkin scheme next suitable strang type lemmas are employed to derive optimal a priori error estimates finally several numerical results illustrating the performance of the mixed primal scheme and confirming the theoretical rates of convergence are provided

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

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 primal mixed formulation for exterior transmission problems in bf r 2 we combine a primal mixed finite element approach with a dirichlet to neumann mapping arising from the boundary integral equation method to study the weak solvability and galerkin approximations of a class of linear exterior transmission problems in potential theory our results are mainly based on the babuska brezzi theory for variational problems with constraints we establish the uniqueness of solution for the continuous and discrete formulations and show that finite element subspac es of lagrange type satisfy the discrete compatibility conditions in addition we provide the error analysis including polygonal approximations of the domain and prove strong convergence of the galerkin solutions moreover under additional regularity assumptions on the solution of the continuous formulation we obtain the asymptotic rate of convergence o h

an efficient two step algorithm for the stationary incompressible magnetohydrodynamic equations a new highly efficient two step algorithm for the stationary incompressible magnetohydrodynamic equations is studied in this paper the algorithm uses a lower order finite element pair i e p 1 b p 1 p 1 to compute an initial approximation that is using the mini element i e p 1 b p 1 to approximate the velocity and pressure and p1 element to approximate the magnetic field then applies a higher order finite element pair i e p 2 p 1 p 2 to solve a linear system on the same mesh furthermore the convergence analyses of standard galerkin finite element method and the two step algorithm are addressed lastly numerical experiments are presented to verify both the theory and the efficiency of the algorithm

on the finite element approximation of a cascade flow problem the finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented the nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements we deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains including the effect of numerical integration

a new conservative numerical scheme for flow problems on unstructured grids and unbounded domains abstract we consider the prototype of a flow and transport problem in two dimensions in particular in many fluid dynamics problems the boundary conditions may be unknown or the domain may be unbounded furthermore only numerical schemes which conserve mass and are stable with respect to dominating convection are appropriate to solve these problems therefore we develop a new discrete scheme to address these issues more precisely we couple the cell centered finite volume method and the boundary element method and introduce suitable transmission conditions also semi robust a posteriori estimates of residual type are developed which allow us to use an adaptive mesh refinement algorithm to treat efficiently problems which have singularities or boundary internal layers several examples illustrate the effectiveness of our new adaptive coupling method

existence results for a nonlinear transmission problem abstract let o and i be open bounded regular subsets of r n such that the closure of i is contained in o let f o be a regular function on o and let f and g be continuous functions from i r to r by exploiting an argument based on potential theory and on the leray schauder principle we show that under suitable and completely explicit conditions on f and g there exists at least one pair of continuous functions u o u i such that u o 0 in o cl i u i 0 in i u o x f o x for all x o u o x f x u i x for all x i i u o x i u i x g x u i x for all x i where the last equality is attained in certain weak sense a simple example shows that such a pair of functions u o u i is in general neither unique nor locally unique if instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition then we prove the existence of at least one classical solution which is in addition locally unique

the coupling of boundary integral and finite element methods for the exterior steady oseen problem abstract in this article we present a new numerical method for solving the steady oseen equations in an unbounded plane domain the technique consists in coupling the boundary integral and the finite element methods an artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one the solution in exterior region is represented by an integral equation over the artificial boundary this integral equation is incorporated into a velocity pressure formulation for the interior region and a finite element method is used to approximate the resulting variational problem finally the optimal error estimates of the numerical solution are derived computer results will be discussed in a forthcoming paper

analysis of the pressure stabilized petrov galerkin method for the evolutionary stokes equations avoiding time step restrictions optimal error estimates for the pressure stabilized petrov galerkin method for the evolutionary stokes equations are proved in the case of regular solutions without restriction on the length of the time step these results clarify the instability of the discrete pressure for small time steps reported in the literature first the limit situation of the continuous in time discretization is considered and second the numerical analysis for the backward euler scheme is carried out the main results are applicable to higher order finite elements the analytical results are strongly based on an appropriate approach for computing the initial velocity which is suggested in this paper numerical studies confirm the theoretical results showing in particular that this instability does not occur for the proposed initial condition

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior

projection method iii spatial discretization on the staggered grid in e liu siam j numer anal 1995 we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous we call this the semi discrete case in this paper we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid we prove that the numerical solution in velocity has full accuracy up to the boundary despite the fact that there are numerical boundary layers present in the semi discrete solutions

an efficient spectral projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases abstract an efficient and accurate numerical scheme is presented for the axisymmetric navier stokes equations in primitive variables in a cylinder the scheme is based on a new spectral galerkin approximation for the space variables and a second order projection scheme for the time variable the new spectral projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition a sensible comparison is made with a standard second order in time and space finite difference scheme based on a stream function vorticity formulation and with available experimental data the numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition the spectral projection scheme is still more accurate in terms of a fixed number of unknowns and more efficient in terms of cpu time required for resolving the flow at a fixed reynolds number to within a prescribed accuracy than the finite difference scheme more importantly the spectral projection scheme can be readily extended to three dimensional nonaxisymmetric cases