the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

l infty error estimates on graded meshes with application to optimal control an l infty error estimate of the finite element approximation of an elliptic boundary value problem with dirichlet boundary conditions in domains with corners is given to achieve an approximation rate of h 2 ln h the mesh has to be appropriately graded near corners with an interior angle larger than omega 0 with omega 0 frac pi 2 for the poisson problem in contrast to previous publications the norm of the function that has to be approximated is separated from the constants in this estimate this result is applied to a linear quadratic optimal control problem with constraints on the control two approaches are considered one where the control is approximated by piecewise constant functions and improved by a postprocessing step and the other where the control is not discretized for both approaches a convergence rate of h 2 ln h in the maximum norm is shown

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

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

conjugacy of finite biprefix codes two languages x and y are called conjugates if they satisfy the conjugacy equation xz zy for some non empty language z we will compare solutions of this equation with those of the corresponding equation of words and study the case of finite biprefix codes x and y we show that the maximal z in this case is rational we will also characterize x and y in the case where they are both finite biprefix codes this yields the decidability of the conjugacy of two finite biprefix codes

on the stability of the l 2 projection in h 1 ω we prove the stability in h1 of the l2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions we give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions in particular stability of the l2 projection in h1 holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically

playing with conway s problem the centralizer of a language is the maximal language commuting with it the question raised by conway in j h conway regular algebra and finite machines chapman hall 1971 whether the centralizer of a rational language is always rational recently received a lot of attention in kunc m kunc the power of commuting with finite sets of words in proc of stacs 2005 in lncs vol 3404 springer 2005 pp 569 580 a strong negative answer to this problem was given by showing that even complete co recursively enumerable centralizers exist for finite languages using a combinatorial game approach we give here an incremental construction of rational languages embedding any recursive computation in their centralizers

extensions of the preiss differentiability theorem abstract the preiss differentiability theorem for lipschitz functions on banach spaces is generalized to locally lower upper semi lipschitz functions and several extensions are presented

uniform approximation of convex function in smooth banach spaces abstract this article is devoted to smooth approximation of convex functions on banach spaces with smooth norm we prove that if x is a smooth space and f is a w lower semicontinuous lipschitzian convex function on x then there exist two w lower semicontinuous g teaux differentiable convex function sequences f n n 1 and g n n 1 such that 1 f n f n 1 f and f g n 1 g n 2 f n f and g n f uniformly on x 3 cl x x d f n x x cl x x d g n x x x

convergence and quasi optimality of an adaptive finite element method for controlling l 2 errors in this paper a contraction property is proved for an adaptive finite element method for controlling the global l 2 error on convex polyhedral domains furthermore it is shown that the method converges in l 2 with the best possible rate the method that is analyzed is the standard adaptive method except that if necessary additional refinements are made to keep the meshes sufficiently mildly graded this modification does not compromise the quasi optimality of the resulting algorithm

a finite element method for elliptic optimal control problem on a non convex polygon with corner singularities abstract in this paper we study a finite element method overcoming corner singularities for elliptic optimal control problem posed on a polygon based on a corner singularity decomposition of singular solution for the associated elliptic system we derive an extraction formula for coefficients of singular parts called stress intensity factors and introduce a finite element discretization for regular parts of the solution pair furthermore we analyze the finite element solutions in l 2 and h 1 norms by numerical experiments we confirm the efficiency and reliability of the proposed method

quasi optimal approximation of surface based lagrange multipliers in finite element methods we show quasi optimal a priori convergence results in the l 2 and h 1 2 norm for the approximation of surface based lagrange multipliers such as those employed in the mortar finite element method we improve on the estimates obtained in the standard saddle point theory where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estimates for the dual variable are reached for the lowest order case i e k 1 an additional factor of sqrt h ln h and for higher order cases i e k 1 an additional factor of sqrt h in the a priori bound for the dual variable can be recovered the proof is based on the use of new estimates for the primal variable in strips of width o h near these surfaces

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

convergence of a finite element approximation to a state constrained elliptic control problem we consider an elliptic optimal control problem with pointwise state constraints the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation the corresponding minima are shown to converge in l 2 to the exact control as the discretization parameter tends to zero furthermore error bounds for the control and the state are obtained in both two and three space dimensions finally we present numerical examples which confirm our analytical findings

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

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

conway s problem for three word sets we prove two results on commutation of languages first we show that the maximal language commuting with a three element language i e its centralizer is rational thus giving an affirmative answer to a special case of a problem proposed by conway in 1971 second we characterize all languages commuting with a three element code the characterization is similar to the one proved by bergman for polynomials over noncommuting variables see trans am math soc 137 1969 327 and algebraic combinatorics on words cambridge university press cambridge 2000 a language commutes with a three element code x if and only if it is a union of powers of x

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

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

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

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

regularity estimates and optimal finite element methods in wp1 on polygonal domains abstract consider the poisson equation with the dirichlet boundary condition on a bounded convex polygonal domain r 2 we investigate the finite element approximation of singular solutions that are due to the non smoothness of the domain in the w p 1 norm 1 p in particular with analysis in weighted sobolev spaces and weighted holder spaces we provide regularity requirements on the given data and specific parameter selection criteria for graded meshes such that the resulting numerical approximation achieves the optimal convergence rate in w p 1 sample results from various numerical tests are provided to confirm the theory

two characterizations of super reflexive banach spaces by the behaviour of differences of convex functions abstract a function defined on a banach space x is called convex if it can be represented as a difference of two continuous convex functions in this work we study the relationship between some geometrical properties of a banach space x and the behaviour of the class of all convex functions defined on it more precisely we provide two new characterizations of super reflexivity in terms convex functions

generic frechet differentiability of convex functions dominated by a lower semicontinuous convex function abstract in this paper an extended real valued proper lower semicontinuous convex function f on a banach space is said to have the frechet differentiability property fdp if every proper lower semicontinuous convex function g with g f is frechet differentiable on a dense g subset of int dom g the interior of the effective domain of g we show that f has the fdp if and only if the w closed convex hull of the image of the subdifferential map of f has the radon nikod m property this is a generalization of the main theorem in a paper by lixin and shuzhong to appear according to this result it also gives several new criteria of asplund spaces

variational formulation of a model free boundary problem the purpose of this work is to present an error analysis of the numerical approximation by a finite element method of a free surface problem the analysis has been done in an abstract model which has many of the features of a free surface problem for a viscous liquid we study in this paper how the numerical approximation of the free boundary affects the approximation of the other variables of the problem and vice versa we present the numerical analysis of a free boundary problem that is intended to incorporate many of the difficulties found in a class of models of fluid flow phenomena with free surfaces one such phenomenon which motivates the current work is the flow of a liquid constrained only partly by a container that is in which a part of the boundary of the domain filled by the liquid is an interface with another liquid of much smaller density and for which surface tension plays a significant role in determining the shape of the free surface one model for the behavior of such liquids is based on the assumption that the surface tension between the two liquids is proportional to the curvature of the free surface the constant of proportionality is a physical property of the two fluids this model has been studied extensively in recently years both experimentally cf jean and pritchard 15 and pritchard 19 theoretically cf allain 2 beale 4 bemelmans 5 jean 14 pukhnachov 20 and solonnikov 27 asymptotically keller and miksis 16 and computationally cf cuvelier 10 ryskin and leal 23 and saito and scriven 24 our purpose here is to establish a framework for the analysis of convergence properties of the computational techniques being used the only previous work that we are aware of in this direction is by nitsche 18 in the first section of the paper we define our model problem in classical terms in the second section we construct a variational formulation for the problem that has two new features one is that it allows the existence of a solution to be proved with weaker assumptions on the data than has been possible received february 26 1990 revised november 20 1990 1980 mathematics subject classification 1985 revision primary 65n30

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

local and parallel finite element algorithms for the stokes problem based on two grid discretizations some local and parallel finite element algorithms for the stokes problem are proposed and analyzed in this paper these algorithms are motivated by the observation that for a solution to the stokes problem low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure one technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape regular grids

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

generic frechet differentiability of convex functions on non asplund spaces abstract let f be a continuous convex function on a banach space e this paper shows that every proper convex function g on e with g f is generically frechet differentiable if and only if the image of the subdifferential map f of f has the radon nikod m property and in this case it is equivalent to showing that the image of f is separable on each separable subspace of e

commutation with codes the centralizer of a set of words x is the largest set of words c x commuting with x xc x c x x it has been a long standing open question due to j h conway regular algebra and finite machines chapman hall london 1971 whether the centralizer of any rational set is rational while the answer turned out to be negative in general see m kunc proc of icalp 2004 lecture notes in computer science vol 3142 springer berlin 2004 pp 870 881 we prove here that the situation is different for codes the centralizer of any rational code is rational and if the code is finite then the centralizer is finitely generated this result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by b ratoandromanana rairo inform theor 23 4 1989 425 444 many of the techniques we use in this paper follow her ideas we also give in this paper an elementary proof for the prefix case

local energy estimates for the finite element method on sharply varying grids local energy error estimates for the nite element method for el liptic problems were originally proved in 1974 by nitsche and schatz these estimates show that the local energy error may be bounded by a local ap proximation term plus a global pollution term that measures the inuence of solution quality from outside the domain of interest and is heuristically of higher order however the original analysis of nitsche and schatz is restricted to quasi uniform grids we present local a priori energy estimates that are valid on shape regular grids an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation our chief technical innovation is an improved superapproximation result

conjugacy relations of prefix codes it is shown that if x and y are prefix codes and z is a non empty language satisfying the condition x z z y then z is the union of a non empty family p n i i of pairwise disjoint prefix sets such that x p i p i y for all i i consequently the conjugacy relations of prefix codes are explored and under the restriction that both of x and y are prefix codes the solutions of the conjugacy equation x z z y for languages are determined also the decidability of the conjugacy problem for finite prefix codes is confirmed

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

the commutation of finite sets a challenging problem we prove that given a set x of two nonempty words a set y of nonempty words commutes with x if and only if either y is a union of powers of x or x y subseteq t for some primitive word t we also show that the same holds for certain special types of codes but does not hold in general for sets of cardinality at least four

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

exponential convergence of hp finite element discretization of optimal boundary control problems with elliptic partial differential equations we investigate the numerical solution of boundary control problems with elliptic partial differential equations by the hp finite element method we prove exponential convergence with respect to the number of unknowns for an a priori chosen discretization here we have to prove that derivatives of arbitrary order of the solution belong to suitably chosen weighted sobolev spaces this result relies on the assumption that the number of switching points of the optimal control is finite numerical experiments confirm the theoretical findings

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

smooth approximation of convex functions in banach spaces abstract this paper shows that every extended real valued lower semi continuous proper respectively lipschitzian convex function defined on an asplund space can be represented as the point wise limit respectively uniform limit on every bounded set of a sequence of lipschitzian convex functions which are locally affine hence c at all points of a dense open subset and shows an analogous for w lower semi continuous proper respectively lipschitzian convex functions defined on dual spaces whose pre duals have the radon nikodym property

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

on effective construction of the greatest solution of language inequality xa bx in this paper we consider effective constructions of the greatest solution of the language inequality xa bx it has been proved by kunc in 2005 that the greatest solution of xa bx is regular provided that b is regular no matter what a is however this proof is based on kruskal s tree theorem and does not provide any effective way to construct the greatest solution we focus on this gap in this paper we give an effective construction of the greatest solution for the following two cases i a b are regular and there exists k 1 such that pref b a k b kpref b a where pref b is the set of prefixes of words in b ii a b are regular and b is a code with finite decoding delay our construction takes the point of view of games as shown by kunc in his regularity proof the construction of the greatest solution can be reduced to the construction of the winning region of a two player game our contribution is to show that the winning regions of the two player game for the two cases can be constructed effectively the main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy while the construction for the second case is based on the observation that the two player game can be reduced to a two player reachability game played on the transition graph of a one counter machine