amge coarsening strategies and application to the oseen equations we provide some extensions to the algebraic multigrid method based on element interpolation amge concerning the agglomeration process the application to nonconforming elements and the application to the mixed finite element discretization of the oseen linearized navier stokes equations this last point using amge for mixed finite elements becomes straightforward because of the availability of coarse level topologies we show this exemplarily for the crouzeix raviart element including a stability result the numerical results show that it really pays off to take a closer look at the agglomeration strategy a wrong choice can lead to insufficient convergence or even divergence of the overall multigrid method

instability of bound states of nonlinear schrodinger equations with morse index equal to two abstract we prove that every bound state of the nonlinear schrodinger equation nls with morse index equal to two with d 2 d 2 e q 0 is orbitally unstable we apply this result to two particular cases one is the nls equation with potential and the other is a system of three coupled nls equations in both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high

a new mixed finite element formulation and the mac method for the stokes equations a new mixed finite element method is formulated for the stokes equations in which the two components of the velocity and the pressure are defined on different meshes first order error estimates are obtained for both the velocity and the pressure also the well known mac method is derived from the resulting finite element method

a fully semi lagrangian discretization for the 2d incompressible navier stokes equations in the vorticity streamfunction formulation abstract a numerical method for the two dimensional incompressible navier stokes equations in vorticity streamfunction form is proposed which employs semi lagrangian discretizations for both the advection and diffusion terms thus achieving unconditional stability without the need to solve linear systems beyond that required by the poisson solver for the reconstruction of the streamfunction a description of the discretization of dirichlet boundary conditions for the semi lagrangian approach to diffusion terms is also presented numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

a block centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system abstract a block centered finite difference method is proposed for solving an unsteady asymptotic coupled model in which the flow is governed by darcy s law both in the one dimensional fracture and two dimensional porous media the second order error estimates in discrete norms are derived on nonuniform rectangular grids for both pressure and velocity the numerical scheme can be extended to nonmatching spatial and temporal grids without loss of accuracy numerical experiments are performed to verify the efficiency and accuracy of the proposed method it is shown that the pressure and velocity are discontinuous across the fracture interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

on periodic waves for sine and sinh gordon equations abstract results of existence as well as orbital stability instability of periodic waves associated with sine gordon and sinh gordon equations will be treated in this manuscript the sine gordon equation is known to be completely integrable exactly solvable and it has several special solutions as soliton and breather solutions we obtain in both cases a periodic sign changed wave which is smooth when posed on whole real line the main tool used is the grillakis shatah and strauss theory

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

a parallel oseen linearized algorithm for the stationary navier stokes equations abstract based on two grid discretizations and domain decomposition a parallel oseen linearized finite element algorithm for the stationary navier stokes equations with moderate or large viscosity parameter is proposed and analyzed the key idea of the algorithm is to first solve a nonlinear problem by picard iterative method on a coarse grid and then to solve an oseen problem in parallel on a fine grid to correct the coarse grid solution by using local a priori error estimate for the finite element solution and under the uniqueness condition error bounds of the corresponding finite element solution are analyzed numerical results are also given to demonstrate the high efficiency of the algorithm

on instability of excited states of the nonlinear schrodinger equation we introduce a new notion of linear stability for standing waves of the nonlinear schrodinger equation nls which requires not only that the spectrum of the linearization be real but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive we prove that excited states of the nls are not linearly stable in this more restrictive sense we then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability

stability of standing waves for nls with perturbed lame potential abstract we perturb a linear schrodinger equation with lame potential with a small positive or negative potential the new perturbed operator has one or more eigenvalues at most one in each spectral gap we then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

role of long waves in the stability of the plane wake this work is directed toward investigating the fate of three dimensional long perturbation waves in a plane incompressible wake the analysis is posed as an initial value problem in space more specifically input is made at an initial location in the downstream direction and then tracing the resulting behavior further downstream subject to the restriction of finite kinetic energy this presentation follows the outline given by criminale and drazin w o criminale and p g drazin stud appl math 83 123 1990 that describes the system in terms of perturbation vorticity and velocity the analysis is based on large scale waves and expansions using multiscales and multitimes for the partial differential equations the multiscaling is based on an approach where the small parameter is linked to the perturbation property independently from the flow control parameter solutions of the perturbative equations are determined numerically after the introduction of a regular perturbation scheme analytically deduced up to the second order numerically the complete linear system is also integrated since the results relevant to the complete problem are in very good agreement with the results of the first order analysis the numerical solution at the second order was deemed not necessary the use for an arbitrary initial value problem will be shown to contain a wealth of information for the different transient behaviors associated to the symmetry angle of obliquity and spatial decay of the long waves the amplification factor of transversal perturbations never presents the trend a growth followed by a long damping usually seen in waves with wave number of order one or less asymptotical instability is always observed

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

stability and superconvergence of mac scheme for stokes equations on nonuniform grids the marker and cell mac method a class of finite volume schemes based on staggered grids has been one of the simplest and most effective numerical schemes for solving the stokes and navier sto

field of values convergence analysis of augmented lagrangian preconditioners for the linearized navier stokes problem we study a block triangular preconditioner for finite element approximations of the linearized navier stokes equations the preconditioner is based on the augmented lagrangian formulation of the problem and was introduced by the authors in siam j sci comput 28 2006 pp 2095 2113 in this paper we prove field of values type estimates for the preconditioned system which lead to optimal convergence bounds for the gmres algorithm applied to solve the system two variants of the preconditioner are considered an ideal one based on exact solves for the velocity submatrix and a more practical variant based on block triangular approximations of the velocity submatrix

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

spectral stability analysis for special solutions of second order in time pdes the higher dimensional case abstract we develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary pde we apply these results to standing waves of the following problems the klein gordon equation for which we consider both ground states and certain excited states the klein gordon zakharov system and the beam equation we also discuss applications to excited states for the klein gordon model as well as multidimensional traveling waves not necessarily homoclinic to zero for general second order equations of this type in all cases our abstract results provide a complete characterization of the linear stability of such solutions

analysis and convergence of the mac scheme i the linear problem the mac marker and cell discretization of fluid flow is analysed for the stationary stokes equations it is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations estimates using mesh dependent norms analogous to the standard bf h 1 and l 2 norms are given for the velocity and pressure respectively

numerical investigation of effectivity indices of space time error indicators for navier stokes equations abstract in this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady navier stokes equations we define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error moreover we verify the reliability of the estimations through numerical experiments and we propose an effective space time adaptive strategy for the unsteady navier stokes equations such technique is based on two residual based error indicators that suitably drive the mesh and the timestep length modifications adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way

a block centered finite difference method for the darcy forchheimer model a block centered finite difference scheme is introduced to solve the nonlinear darcy forchheimer equation in which the velocity and pressure can be approximated simultaneously the second order error estimates for both pressure and velocity are established on a nonuniform rectangular grid numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

inclusion of microscopic rotation of powder particles during compaction in finite element method using cosserat continuum theory the effect of microscopic rotation of powder particles in compaction is included in the rigid plastic finite element method on the basis of the cosserat continuum theory in the cosserat continuum theory couple stress induced from the microscopic rotation is introduced and the equilibrium equations of moment for the couple stress are solved simultaneously with those of force a yield criterion for the cosserat porous continuum is proposed by taking the effect of the couple stress into consideration and constitutive equations for the rigid plastic porous material are derived from the yield criterion on the basis of the associated flow rule the equilibrium equations of force and moment for the cosserat continuum are formulated by the use of the galerkin method the effect of microscopic rotation of powder particles in plane strain closed die compaction is examined in addition the calculated result is compared with that for the conventional continuum without the microscopic rotation 1998 john wiley sons ltd

symmetry breaking bifurcation in the nonlinear schrodinger equation with symmetric potentials we consider the focusing attractive nonlinear schrodinger nls equation with an external symmetric potential which vanishes at infinity and supports a linear bound state we prove that the symmetric nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non degenerate local maxima at zero under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork in the particular case of double well potentials with large separation the power of nonlinearity determines the subcritical or supercritical character of the bifurcation the results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter

structural stability of non ground state traveling waves of coupled nonlinear schrodinger equations abstract we consider the linear stability and structural stability of non ground state traveling waves of a pair of coupled nonlinear schrodinger equations cnls which describe the evolution of co propagating polarized pulses in the presence of birefringence viewing the cnls equations as a hamiltonian perturbation of the manakov equations we find parameter regimes in which there are two stable families of traveling waves the usual variational methods for stability analysis of ground states do not apply instead we employ a liapunov schmidt type reduction to detect eigenvalues bifurcating from the imaginary axis we also demonstrate the instability of a family of vector solitons

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

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

stable and unstable manifolds for the nonlinear wave equation with dissipation on considere la stabilite pour des solutions solitons stationnaires du probleme de cauchy sur r n u tt u t u f u 0 u x 0 u t x 0 donnes

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

a tutorial in elementary finite element error analysis a systematic presentation of a priori and a posteriori error estimates abstract this paper provides a tutorial on the derivation of a priori and explicit residual based a posteriori error estimates for galerkin finite element discretizations of general linear elliptic operators for simplicity the presentation is in one dimension although extensions to multidimensions should be straightforward the structures of explicit residual based a posteriori error estimators and a priori error estimators are very similar they are each in the form of an upper bound on the error measured in some norm the structures of their derivations also follow similar lines it is therefore appropriate to present and discuss a priori and a posteriori error estimators in parallel in this paper error bounds are obtained for the error measured in the energy norm and the l2 norm in addition symmetric nonsymmetric positive definite and indefinite operators are discussed a priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm special attention is given to the advection diffusion equation a nonsymmetric operator and the helmholtz equation an indefinite operator we systematically describe the necessary steps in deriving a priori and a posteriori error estimates and provide a simplified understanding of their differences as well as their similarities

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

purely nonlinear instability of standing waves with minimal energy we consider hamiltonian systems with u 1 symmetry we prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable in spite of the absence of linear instability essentially the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system we apply our theory to the non

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

stabilized spectral methods for the navier stokes equations residual free bubbles and preconditioning in a series of papers appeared in the last few years the use of local bubble functions has been advocated as a stabilizing device for curing spurious oscillations in high order spectral legendre or chebyshev methods this work has been motivated by the success of the bubble approach in low order finite element methods as a general strategy to design stable approximations to saddle point problems such as the stokes problem or to convection dominated problems it has been shown that adding local bubbles to the usual piecewise polynomial trial and test functions and eliminating them by static condensation leads to a least square type approximation of the stokes problem and to an supg type approximation of the convection diffusion problem not only the bubble approach provides an elegant unifying framework in which different stabilization techniques may be cast but it also yields a path to the fully automatic choice of the stabilization parameters as well as to adaptivity via a posteriori error estimators a previous paper by the first two authors deals with the bubble stabilization of legendre galerkin methods for the incompressible navier stokes equations in velocity pressure formulation local bubbles allow the use of equal order polynomials for approximating the velocity and the pressure they dramatically reduce the oscillations related to poor resolution in the convection dominated regime in addition they enhance the robustness of a continuation method for treating the nonlinearity all this without giving up with the formal infinite order accuracy of a spectral legendre method while the paper clearly indicates the feasibility of the bubble approach it leaves room for improving the practical efficiency of the method both by fully exploiting the potential of the bubbles in the automatic user independent setting of the stabilization parameters and by finding appropriate techniques for the solution of the resulting algebraic system of discrete equations we address these two issues in the present paper we propose to adapt to spectral methods the recently developed idea of residual free bubbles and to resort to a suitably stabilized finite element preconditioner in solving the algebraic system iteratively the basic idea of the residual free bubble approach is as follows we enlarge the space where the solution is sought by auxiliary functions the bubbles whose support is localized in a cell or an element these functions are determined by the solution of local problems in such a way that in the enlarged space the governing equations are solved exactly within each element this motivates the name residual free bubbles if we consider a convection diffusion problem with piecewise constant coefficients and linear finite elements then the solution of the local problems in the convection dominated regime can be obtained essentially without approximation and at a small cost in more complicated situations some a

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

stability and convergence of characteristic mac scheme and post processing for the oseen equations on non uniform grids abstract in this paper we establish the lbb condition and stability for both velocity and pressure of characteristic mac scheme for the oseen equations on non uniform grids we obtain the second order convergence in discrete l2 norm for both velocity and pressure and the first order convergence in discrete h1 norm for velocity moreover we construct the post processing characteristic mac scheme to obtain second order accuracy in discrete h1 norm for the velocity finally some numerical experiments are presented to show the correctness and accuracy of the mac scheme

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

instability indices for matrix polynomials abstract there is a well established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are hermitian and skew hermitian this theory relates the number of negative directions for the matrix coefficients which are hermitian to the total number of unstable eigenvalues for the polynomial herein we extend the theory to even matrix polynomials of any finite degree in particular unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two we also consider hermitian matrix polynomials and derive an index which counts the number of eigenvalues with nonpositive imaginary part the results are refined if we consider the hermitian matrix polynomial to be a perturbation of a even polynomials however this refinement requires additional assumptions on the matrix coefficients

a remark on linearly unstable standing wave solutions to nls abstract we obtain an optimal growth estimate of a semigroup generated by a linearized operator around a standing wave solution nonlinear schrodinger equations in two dimension using the growth estimate of the semigroup we prove that a linearly unstable standing wave solution is orbitally unstable and that instability of the standing wave solution is mainly caused by a mode of an eigenfunction associated with the rightmost or the leftmost eigenvalues of the linearized operator our result is obtained by using the method of yajima and cuccagna that proved l p boundedness of the wave operator

orbital stability of standing waves for semilinear wave equations with indefinite energy the orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole r n with n 2 the stability is determined by the convexity on of the lowest energy d of standing waves with frequency the arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

finite difference methods for two phase incompressible flow in porous media two phase incompressible flow in porous media is governed by a system of nonlinear partial differential equations convection physically dominates diffusion and the object of this paper is to develop a finite difference procedure that reflects this dominance the pressure equation which is elliptic in appearance is discretized by a standard five point difference method the concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms a convergence analysis is given for the method

on classification of intrinsic localized modes for the discrete nonlinear schrodinger equation abstract we consider localized modes discrete breathers of the discrete nonlinear schrodinger equation i d n dt n 1 n 1 2 n n 2 n 1 n z we study the diversity of the steady state solutions of the form n t ei tvn and the intervals of the frequency of their existence the base for the analysis is provided by the anticontinuous limit negative and large enough where all the solutions can be coded by the sequences of three symbols 0 and using dynamical systems approach we show that this coding is valid for 3 4533 and the point is a point of accumulation of saddle node bifurcations also we study other bifurcations of intrinsic localized modes which take place for and give the complete table of them for the solutions with codes consisting of less than four symbols

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

a symplectic perspective on constrained eigenvalue problems abstract the maslov index is a powerful tool for computing spectra of selfadjoint elliptic boundary value problems this is done by counting intersections of a fixed lagrangian subspace which designates the boundary conditions with the set of cauchy data for the differential operator we apply this methodology to constrained eigenvalue problems in which the operator is restricted to a not necessarily invariant subspace the maslov index is defined and used to compute the morse index of the constrained operator we then prove a constrained morse index theorem which says that the morse index of the constrained problem equals the number of constrained conjugate points counted with multiplicity and give an application to the nonlinear schrodinger equation

a convergence analysis of yee s scheme on nonuniform grids the yee scheme is the principal finite difference method used in computing time domain solutions of maxwell s equations on a uniform grid the method is easily seen to be second order convergent in space this paper shows that the yee scheme is also second order convergent on a nonuniform mesh despite the fact that the local truncation error is nodally only of first order

bifurcation and stability for nonlinear schrodinger equations with double well potential in the semiclassical limit we consider the stationary solutions for a class of schrodinger equations with a symmetric double well potential and a nonlinear perturbation here in the semiclassical limit we prove that the reduction to a finite mode approximation give the stationary solutions up to an exponentially small term and that symmetry breaking bifurcation occurs at a given value for the strength of the nonlinear term the kind of bifurcation picture only depends on the nonlinearity power we then discuss the stability instability properties of each branch of the stationary solutions finally we consider an explicit one dimensional toy model where the double well potential is given by means of a couple of attractive dirac s delta pointwise interactions

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

adaptive discretization of stationary and incompressible navier stokes equations by stabilized finite element methods we have derived a residual based a posteriori error estimator for a stabilized finite element discretization of the stationary incompressible navier stokes equations with general boundary conditions an adaptive algorithm based on this error estimator is discussed and tested on some analytical and physical problems when possible we study precisely the behaviour of the effectivity index

asymptotic dynamics of nonlinear schrodinger equations with many bound states abstract we consider a nonlinear schrodinger equation with a bounded localized potential in r 3 the linear hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions suppose that the initial data is localized and small of order n in h1 and that its ground state component is larger than n3 e with e 0 small we prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory

augmented mixed finite element method for the oseen problem a priori and a posteriori error analyses abstract we propose a new augmented dual mixed method for the oseen problem based on the pseudostress velocity formulation the stabilized formulation is obtained by adding to the dual mixed approach suitable least squares terms that arise from the constitutive and equilibrium equations we prove that for appropriate values of the stabilization parameters the new variational formulation and the corresponding galerkin scheme are well posed and a cea estimate holds for any finite element subspaces we also provide the rate of convergence when each row of the pseudostress is approximated by raviart thomas or brezzi douglas marini elements and the velocity is approximated by continuous piecewise polynomials moreover we derive a simple a posteriori error estimator of residual type that consists of two residual terms and prove that it is reliable and locally efficient finally we include several numerical experiments that support the theoretical results

a second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

inertia law for spectral stability of solitary waves in coupled nonlinear schrodinger equations spectral stability analysis for solitary waves is developed in the context of the hamiltonian system of coupled nonlinear schrodinger equations the linear eigen value problem for a non self adjoint operator is studied with two self adjoint matrix schrodinger operators sharp bounds on the number and type of unstable eigen values in the spectral problem are found from the inertia law for quadratic forms associated with the two self adjoint operators symmetry breaking stability analysis is also developed with the same method

pressure schur complement preconditioners for the discrete oseen problem we consider several preconditioners for the pressure schur complement of the discrete steady oseen problem two of the preconditioners are well known from the literature and the other is new supplemented with an appropriate approximate solve for an auxiliary velocity subproblem these approaches give rise to a family of the block preconditioners for the matrix of the discrete oseen system in the paper we critically review possible advantages and difficulties of using various schur complement preconditioners we recall existing eigenvalue bounds for the preconditioned schur complement and prove such for the newly proposed preconditioner these bounds hold both for lbb stable and stabilized finite elements results of numerical experiments for several model two dimensional and three dimensional problems are presented in the experiments we use lbb stable finite element methods on uniform triangular and tetrahedral meshes one particular conclusion is that in spite of essential improvement in comparison with simple scaled mass matrix preconditioners in certain cases none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex e g circulating advection vector field

stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger equations in this paper we study the stability and evolution of solitary waves in perturbed generalized nonlinear schrodinger nls equations our method is based on the completeness of the bounded eigenstates of the associated linear operator in l2 space and a standard multiple scale perturbation technique unlike the adiabatic perturbation method our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur in particular our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator as an example we consider the perturbed cubic quintic nls equation in detail and determine the stability regions of its solitary waves in the instability region we also specify where the solitary waves decay collapse develop movingfronts or evolve into a stable spatially localized and temporally periodic state the generalization of this method to other perturbed nonlinear wave systems is also discussed

instability of bound states for abstract nonlinear schrodinger equations abstract we study the instability of bound states for abstract nonlinear schrodinger equations we prove a new instability result for a borderline case between stability and instability we also reprove some known results in a unified way

a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples