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

the finite element method for the time dependent gross pitaevskii equation with angular momentum rotation we consider the time dependent gross pitaevskii equation describing the dynamics of rotating bose einstein condensates and its discretization with the finite element method we analyze a mass conserving crank nicolson type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the l2 and energy norm in space the estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the gross pitaevskii equation we demonstrate the performance of the method in numerical experiments by siam 2017

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

structure properties of min max systems and existence of global cycle time studies min max systems which are dynamic systems including three operations min max with unknown or stochastic parameters some sufficient conditions are given for the existence of global cycle times our results are based on structure properties of min max systems

optimal l error estimates for nonconforming and mixed finite element methods of lowest order for second order linear elliptic problems it is proved that thep1 nonconforming finite element method has the samel asymptotic accuracy as thep1 conforming one this result is applied to derive optimall error estimates for both the displacement and the stress fields of the lowest order raviart thomas mixed finite element method and a superconvergence result at the barycenter of each element

pointwise localized error estimates for parabolic finite element equations we derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations the results extend those obtained by a h schatz for stationary elliptic problems in particular they show that the error is more localized for higher order elements

an extension of the lax richtmyer theory we give a rigorous proof of the validity of the lax equivalence theorem when the true solution at timet and its approximation lie in different spaces thus modelling the practical situation where the former is a function of the space variables and the latter is only defined at grid points

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

a stability result for sectorial operators in branch spaces in this paper the author considers approximations of holomorphic semigroups on general banach spaces based on a acceptable rational functions a general stability theorem is proved it covers rational functions r z with r infty 1 and can be applied even to variable step cases in particular the question of the text l p stability p ne 2 of crank nicolson schemes for parabolic initial value problems is settled

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

a rapid method of evaluating the regulator and class number of a pure cubic field let 2 0 be the algebraic number field formed by adjoining 9 to the rationals 2 let r and h be respectively the regulator and class number of shanks has described a method of evaluating r for 2 vd where d is a positive integer his technique improved the speed of the usual continued fraction algorithm for finding r by allowing one to proceed almost directly from the nth to the mth step where m is approximately in in the continued fraction expansion of id this paper shows how shanks idea can be extended to the voronoi algorithm which is used to find r in cubic fields of negative discriminant it also discusses at length an algorithm for finding r and h for pure cubic fields 1 fd d an integer under a certain generalized riemann hypothesis the ideas developed here will provide a new method which will find r and h in 0 02 5 e operations when h is small this is an improvement over the 0 d h operations required by voronoi s algorithm to find r for example with d 200171999 it required only 5 minutes for an amdahl 470 vi computer to find that r 518594546 969083 and h 1 this same calculation would require about 8 days of computer time if it used only the standard voronoi algorithm a 0 and gcd w m2 m3 n2 n3 a 1 also if e m2n3 m3n2 then e a if u e we denote its conjugates by w and we also write the norm of as nia and define it as nico oco w from the simple fact that

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

an l error estimate for finite element solution of nonlinear elliptic problem with a source term the l error estimate for the finite element solution of nonlinear elliptic problem with a source term is discussed the lumped mass discretization method is used in the discretization and the error estimate o h 2 logh in l norm is obtained

finite element methods for nonlinear sobolev equations with nonlinear boundary conditions abstract we derive optimal l 2 error estimates for semi discrete finite element methods for nonlinear sobolev equations with nonlinear boundary conditions a new projection is introduced and used in the error analysis

on principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants let y be any totally complex quartic field two algorithms are described for determining whether or not any given ideal in y is principal one of these algorithms is very efficient in practice but its complexity is difficult to analyze the other algorithm is computationally more elaborate but in this case a complexity analysis can be provided these ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p 1 mod 5 given any quadratic or quintic nonresidue of p it is shown that these cyclotomic numbers can be efficiently computed in 0 log p 3 binary operations

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

an adaptive supg method for evolutionary convection diffusion equations abstract an adaptive algorithm for the numerical simulation of time dependent convection diffusion reaction equations will be proposed and studied the algorithm allows the use of the natural extension of any error estimator for the steady state problem for controlling local refinement and coarsening the main idea consists in considering the supg solution of the evolutionary problem as the supg solution of a particular steady state convection diffusion problem with data depending on the computed solution the application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order this argument is supported in the one dimensional case by numerical analysis in the numerical studies particularly the residual based error estimator from 18 will be applied which has proved to be robust in the supg norm the effectivity of this error estimator will be studied and the numerical results accuracy of the solution fineness of the meshes will be compared with results obtained by utilizing the adaptive algorithm proposed in 5

fully discrete convergence estimates for vortex methods in bounded domains in this paper the authors study vortex method for 2 dimensionsal euler equations of incompressible flow in bounded domains to approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy this field is extended smoothly to a small neighborhood of the boundary and the computation is carried out in the extended domain to construct the velocity field from vorticity a second order isoparametric finite element method is applied and to solve the characteristic equations the explicit euler s scheme is considered optimal error bounds for this fully discrete scheme are obtained

time separations of cyclic event rule systems with min max timing constraints the analysis of the time separations of events is a fundamental problem in the design and evaluation of discrete event systems important progresses have been made based on the event rule system model in the last decade the existing results for event rule systems with min and max constraints can be summarized briefly as the exact evaluation of time separations for acyclic systems is np complete for cyclic systems the structural condition of being tightly coupled is sufficient for long term time separations of events to be bounded in this paper we establish a necessary and sufficient structural boundedness condition uniformity for cyclic event rule systems with both min and max constraints tightly coupled systems are shown to be a special class of uniform systems the well known cas algorithm for finding bounds on long term time separations is adapted to find finite bounds for uniform systems our results are obtained by exploring the algebraic structures guiding the evolution of the systems

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

solving a backward heat conduction problem by variational method abstract the present paper is devoted to solve the backward heat conduction problem from the final temperature distribution the problem is transformed into an optimization problem and then a variational method is given a conjugate gradient method together with an appropriate stopping rule are used to solve this inverse problem several numerical examples are provided to show the high efficiency of the suggested method

analysis of fully discrete approximations to parabolic problems with rough or distribution valued initial data we consider fully discrete approximations to a parabolic initial boundary value problem with rough or distribution valued initial data in two space dimensions for discretization in time and space we apply single step methods and the standard galerkin method with piecewise linear test functions respectively for spatial discretization of the initial condition we are however forced to use more involved constructions our main result is stability and error estimates of the discrete solutions

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

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

conditions for the structural existence of an eigenvalue of a bipartite min max system in this paper we consider bipartite min max systems we present conditions for the structural existence of an eigenvalue and corresponding eigenvector for such systems where both the eigenvalue and eigenvector are supposed to be finite the conditions are stated in terms of the system matrices that describe a bipartite min max system structural in the previous means that not so much the numerical values of the finite entries in the system matrices are important rather than their locations within these matrices the conditions presented in this paper can be seen as an extension towards bipartite min max systems of known conditions for the structural existence of an eigenvalue of a max system involving the ir reducibility of the associated system matrix although developed for bipartite min max systems the conditions for the structural existence of an eigenvalue also can directly be applied to general min max systems when given in the so called conjunctive or disjunctive normal form

overcoming ho lder continuity in ill posed continuation problems many ill posed continuation problems in partial differential equations obey a logarithmic convexity inequality and can be stabilized in an appropriate banach space by imposing an a priori bound on

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

on the stability of variable stepsize rational approximations of holomorphic semigroups we consider variable stepsize time approximations of holomorphic semigroups on general banach spaces for strongly a o acceptable rational functions a general stability theorem is proved which does not impose any constraint on the ratios between stepsizes

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

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

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

optimal l error estimates for piecewise linear finite elements in 1 n 3 for 1 n 3 in this paper for two second order elliptic boundary value problems uniform convergence error estimates are derived using piecewise linear conforming and non conforming finite elements the estimates are valid for polyhedral convex regions rn with 1 n 3 the proof of the estimates is mainly based on sobolev s inequality by which the uniform estimates are reduced to estimates in the mean in this way it is possible to obtain the desired uniformo h2 estimate in the plane as a special case thus confirming theoretically natterer s numerical results

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

laplace transform method for parabolic problems with time dependent coefficients the laplace transform method has proven to be very efficient for dealing with parabolic problems whose coefficients are time independent and it is easily parallelizable however the method has not been proven to be applicable to linear problems whose coefficients are time dependent the reason is that the laplace transform of two time dependent functions leads to a convolution of the laplace transformed functions in the dual variable in this paper we propose a laplace transform method to linear parabolic problems with time dependent coefficients which is as efficient as the method for parabolic problems with time independent coefficients several numerical results are provided which support the efficiency of the proposed scheme

on the numerical recovery of a holomorphic mapping from a finite set of approximate values the least squares approach for the recovery of a holomorphic mapping from given perturbed nodal values is considered the mapping is assumed to be a priori bounded by a known quantity so that the recovery problem is well posed the present analysis shows that for nodes that are the zeroes of jacobi polynomials a quasioptimal algorithm results with a fairly moderate number of nodes the analysis also takes into account the effect of the numerical linear algebra involved some numerical experiments are presented

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

a kernel based method for the approximate solution of backward parabolic problems a method for approximating the solution of self adjoint parabolic equations backwards in time is presented the technique uses the green s function of the parabolic equation to formulate the problem as a first kind fredholm integral equation and then does a regularized least squares solution of the resulting discrete system error estimates and examples are presented direct knowledge of the kernel of the differential operator is not required

finite element approximation of a semilinear elliptic problem with a singular nonlinearity given nu in bbb r we consider the following problem find u 0 such that where omega subset bbb r d d 1 2 or 3 and c 0 in bar omega we prove h 1 and l infty error bounds for the standard continuous piecewise linear galerkin finite element approximation with a weakly acute triangulation our bounds are nearly optimal in addition for d 1 and 2 and c in bbb r we analyze a more practical scheme involving numerical integration on the nonlinear term we obtain nearly optimal h 1 and l infty error bounds for d 1 for this case we also present some numerical results

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

time discretization in the backward solution of parabolic equations ii the problem of solving a parabolic partial differential equation backwards in time by a method related to the tikhonov phillips regularization method is considered time discretizations based on pade approximations of the exponential function are studied and a priori estimates of the step length are given which guarantee an almost optimal error bound the computational efficiency of different discretizations is discussed some numerical examples are given in part ii of this paper we study the backward beam method and the same error estimates are obtained a new scheme for time descretization based on pade approximation is discussed

ultimate periodicity of orbits for min max systems the ultimate periodicity theorem is an important result in min max systems theory it was first proved by olsder and perennes in their unpublished work in this paper we present a new proof this proof is also based on two important theorems the existence of cycle time for any min max function and the nussbaum sine theorem however two different techniques pure min max function and conditional redundancy are used to obtain two important intermediate results the purpose of this paper is to provide a simple alternate proof to the ultimate periodicity theorem

effect of numerical integration for elliptic obstacle problems an elliptic obstacle problem is approximated by piecewise linear finite elements with numerical integration on the penalty and forcing terms this leads to diagonal nonlinearities and thereby to a practical scheme optimal error estimates in the maximum norm are derived the proof is based on constructing suitable super and subsolutions that exploit the special structure of the penalization and using quite precise pointwise error estimates for an associated linear elliptic problem with quadrature via the discrete maximum principle

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

finite element approximations for the stokes equations on curved domains and their errors we construct finite element spaces for the stokes equations on curved domains which satisfy the email protected ka condition moreover we estimate the errors of the finite element approximations for the stokes equations

computation of the class number and class group of a complex cubic field let h and g be respectively the class number and the class group of a complex cubic field of discriminant a a method is described which makes use of recent ideas of lenstra and schoof to develop fast algorithms for finding h and g under certain riemann hypotheses it is shown that these algorithms will compute h in o 1a 1 5 elementary operations and g in o 1a 1 4 elementary operations finally the results of running some computer programs to determine h and g for all pure cubic fields 9 vdi with 2 s d 30 000 are summarized

the trotter kato theorem and approximation of pdes we present formulations of the trotter kato theorem for approximation of linear c 0 semigroups which provide very useful framework when convergence of numerical approximations to solutions of pdes are studied applicability of our results is demonstrated using a first order hyperbolic equation a wave equation and stokes equation as illustrative examples

optimal filtering for the backward heat equation for the backward heat equation stabilized by an a priori initial bound an estimator is determined for intermediate values that is optimal with respect to the bound and the observation accuracy it is shown how this may be implemented computationally with error estimates for the computed approximation which can be made arbitrarily close to the uncertainty level induced by the ill posedness of the underlying problem thus the feasibility of this for practical computation inevitably severely limited by that inherent uncertainty is as good as possible

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis