a posteriori error estimates for mixed finite element approximations of elliptic problems we derive residual based a posteriori error estimates of the flux in l 2 norm for a general class of mixed methods for elliptic problems the estimate is applicable to standard mixed methods such as the raviart thomas nedelec and brezzi douglas marini elements as well as stabilized methods such as the galerkin least squares method the element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order

a recursive approach to local mesh refinement in two and three dimensions abstract the newest vertex strategy for local refinement in two dimensions is reviewed and discussed a recursive algorithm for local refinement of tetrahedral meshes in three dimensions based on similar bisection strategy is described the recursive approach requires certain restrictions on the initial mesh on the other hand under this condition the refinement process can be kept as local as needed and it can be fully inverted simple data structures and derefinement algorithms are also outlined

finite size scaling and the crossover to mean field critical behavior in the two dimensional ising model with medium ranged interactions critical amplitudes in finite size scaling relations show a singular dependence on the range of the interactions r the respective power laws are predicted from phenomenological crossover scaling considerations these predictions are tested by monte carlo simulations for medium ranged ising square lattices it is speculated that some deviations between the simulation results and corresponding predictions may be due to logarithmic corrections

a new a posteriori error estimate for the morley element in this paper we present a posteriori error analysis for the nonconforming morley element of the fourth order elliptic equation we propose a new residual based a posteriori error estimator and prove its reliability and efficiency these results refine those of beirao da veiga et al numer math 106 165 179 2007 by dropping two edge jump terms in both the energy norm of the error and the estimator and those of wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 by showing the efficiency in the sense of verfurth a review of a posteriori error estimation and adaptive mesh refinement techniques wiley teubner new york 1996 moreover the normal component in the estimators of beirao da veiga et al numer math 106 165 179 2007 and wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 is dropped and therefore only the tangential component of the stress on each edge comes into the estimator in addition we generalize these results to three dimensional case

eigenvalues of second order difference equations with periodic and antiperiodic boundary conditions this paper is concerned with periodic and antiperiodic boundary value problems for self adjoint second order difference equations existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained in addition a representation of solutions of a nonhomogeneous linear equation with initial conditions is given

continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem abstract this paper is concerned with continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem the n th eigenvalue is considered as a function on the space of the problems called the n th eigenvalue function a necessary and sufficient condition for all the n th eigenvalue functions to be continuous and several properties of the n th eigenvalue function on a subset of the space of the problems are given they play an important role in the study of continuous dependence of the n th eigenvalue function on the problem continuous dependence of the n th eigenvalue function on the equation and on the boundary condition is studied separately consequently the continuity and discontinuity of the n th eigenvalue function are completely characterized on the whole space of the problems especially asymptotic behavior of the n th eigenvalue function near each discontinuity point is given

energy norm a posteriori error estimates for mixed finite element methods this paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations it is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method the analysis is performed in two different ways under a saturation assumption and using a helmholtz decomposition for vector fields

an optimal adaptive finite element method for the stokes problem a new adaptive finite element method for solving the stokes equations is developed which is shown to converge with the best possible rate the method consists of 3 nested loops the outermost loop consists of an adaptive finite element method for solving the pressure from the elliptic schur complement system that arises by eliminating the velocity each of the arising finite element problems is a stokes type problem with the pressure being sought in the current trial space and the divergence free constraint being reduced to orthogonality of the divergence to this trial space such a problem is still continuous in the velocity field in the middle loop its solution is approximated using the uzawa scheme in the innermost loop the solution of the elliptic system for the velocity field that has to be solved in each uzawa iteration is approximated by an adaptive finite element method

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

a posteriori error estimates for nonconforming finite element methods computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model poisson problem in two and three space dimensions besides a refined residual based a posteriori error estimate an averaging estimator is established and an l 2 estimate is included the a posteriori error estimates are reliable and efficient the proof of reliability relies on a helmholtz decomposition

spectral theory of discrete linear hamiltonian systems this paper is concerned with spectral problems for a class of discrete linear hamiltonian systems with self adjoint boundary conditions where the existence and uniqueness of solutions of initial value problems may not hold a suitable admissible function space and a difference operator are constructed so that the operator is self adjoint in the space then a series of spectral results are obtained the reality of eigenvalues the completeness of the orthogonal normalized eigenfunction system rayleigh s principle the minimax theorem and the dual orthogonality especially the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

self adjoint extensions for second order symmetric linear difference equations abstract in this paper self adjoint extensions for second order symmetric linear difference equations with real coefficients are studied by applying the glazman krein naimark theory for hermitian subspaces both self adjoint subspace extensions and self adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions where the two endpoints may be regular or singular

an anti periodic lasalle oscillation theorem for a class of functional differential equations in this paper we establish a result on the existence of anti periodic solution for a class of functional differential equations which substantially extends and improves some important results in the literature

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

monte carlo investigations of phase transitions status and perspectives abstract using the concept of finite size scaling monte carlo calculations of various models have become a very useful tool for the study of critical phenomena with the system linear dimension as a variable as an example several recent studies of ising models are discussed as well as the extension to models of polymer mixtures and solutions it is shown that using appropriate cluster algorithms even the scaling functions describing the crossover from the ising universality class to the mean field behavior with increasing interaction range can be described additionally the issue of finite size scaling in ising models above the marginal dimension d 4 is discussed

an anti periodic solution for a class of recurrent neural networks in this paper recurrent neural networks with time varying delays and continuously distributed delays are considered sufficient conditions for the existence and exponential stability of the anti periodic solutions are established which are new and complement previously known results

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

an optimally convergent adaptive mixed finite element method we prove convergence and optimal complexity of an adaptive mixed finite element algorithm based on the lowest order raviart thomas finite element space in each step of the algorithm the local refinement is either performed using simple edge residuals or a data oscillation term depending on an adaptive marking strategy the inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator

quasi optimal convergence rate for an adaptive hybridizable c 0 discontinuous galerkin method for kirchhoff plates in this paper we present an adaptive hybridizable c 0 discontinuous galerkin hcdg method for kirchhoff plates a reliable and efficient a posteriori error estimator is produced for this hcdg method quasi orthogonality and discrete reliability are established with the help of a postprocessed bending moment and the discrete helmholtz decomposition based on these the contraction property between two consecutive loops and complexity of the adaptive hcdg method are studied thoroughly the key points in our analysis are a postprocessed normal normal continuous bending moment from the hcdg method solution and a lifting of jump residuals from inter element boundaries to element interiors

unbounded perturbations of nonlinear discrete periodic problem at resonance abstract in this paper we study the existence of solutions of nonlinear discrete boundary value problems 2 u t 1 k u t g t u t h t u 0 u t u 1 u t 1 where t 1 t h t r k is the k th eigenvalue of the linear problem 2 u t 1 u t 0 u 0 u t u 1 u t 1 g n r r satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u r

regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint abstract this paper is concerned with regular approximations of spectra of singular discrete linear hamiltonian systems with one singular endpoint for any given self adjoint subspace extension sse of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of induced regular sses generated by the same difference expression on smaller finite intervals it is shown that every sse of the minimal subspace has a pure discrete spectrum and the k th eigenvalue of any given sse is exactly the limit of the k th eigenvalues of the induced regular sses that is spectral exactness holds in the limit circle case furthermore error estimates for the approximations of eigenvalues are given in this case in addition in the limit point and intermediate cases spectral inclusive holds

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

discrete linear hamiltonian eigenvalue problems abstract this paper introduces general discrete linear hamiltonian eigenvalue problems and characterizes the eigenvalues assumptions are given among them the new notion of strict controllability of a discrete system that imply isolatedness and lower boundedness of the eigenvalues due to the quite general assumptions discrete sturm liouville eigenvalue problems of higher order are included in the presented theory

lyapunov type inequality for higher order difference equations abstract in this paper a new lyapunov type inequality for a class of higher order difference equations with anti periodic boundary conditions is established applying our inequality the lower bound of the eigenvalue for a related eigenvalue problem is given

eigenvalues of discrete linear second order periodic and antiperiodic eigenvalue problems with sign changing weight abstract by virtue of the eigenvalues of discrete linear second order neumann eigenvalue problems we study the eigenvalues of discrete linear second order periodic and antiperiodic eigenvalue problems with sign changing weight we find that these two problems have t real eigenvalues including the multiplicity respectively furthermore the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function furthermore these eigenvalues including the eigenvalues of neumann problem satisfy the order relation

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

a posteriori error analysis for the morley plate element with general boundary conditions this paper introduces and analyses a local residual based a posteriori error indicator for the morley finite element method of the biharmonic kirchhoff plate bending problem in the theoretical part of the paper a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions the error indicator is proven to be both reliable and efficient the numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions these tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements copyright 2009 john wiley sons ltd

a posteriori error estimates for nonconforming finite element methods for fourth order problems on rectangles the a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established but nonconforming discretisations are more easy to implement in practice the a posteriori error analysis for the morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously this crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular morley finite element the incomplete biquadratic finite element and the adini finite element this paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual based a posteriori error estimator an application to the morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor product meshes

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

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

eigenvalues of discrete sturm liouville problems with eigenparameter dependent boundary conditions abstract we consider the discrete right definite sturm liouville problems p t 1 y t 1 q t y t m t y t t 1 t z a 0 b 0 y 0 c 0 d 0 y 0 a 1 b 1 y t 1 c 1 d 1 y t 1 where 1 t z 1 2 t m t 0 for t 1 t z 1 i i 0 where i a i d i b i c i for i 0 1 we obtain the existence of the eigenvalues the sign changing times of the eigenfunctions and the interlacing results of the eigenvalues of the above problem the dirichlet problem and the neumann problem

convergence and quasi optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems recently an adaptive nonconforming finite element method anfem has been developed by carstensen and hoppe in numer math 103 251 266 2006 in this paper we extend the result to some nonsymmetric and indefinite problems the main tools in our analysis are a posteriori error estimators and a quasi orthogonality property in this case we need to overcome two main difficulties one stems from the nonconformity of the finite element space the other is how to handle the effect of a nonsymmetric and indefinite bilinear form an appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order crouzeix raviart element it is shown that the anfem is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops moreover quasi optimality in the sense of quasi optimal algorithmic complexity can be shown for the anfem the results of numerical experiments confirm the theoretical findings

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

residual based a posteriori error analysis for symmetric mixed arnold winther fem this paper introduces an explicit residual based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after arnold winther with pointwise symmetric and h text div conforming stress approximation the residual based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof hence it is straightforward to implement an adaptive mesh refining algorithm numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh refining a higher convergence rate for piecewise affine data is observed in the l 2 stress error and reproduced in non smooth situations by the adaptive mesh refining strategy

eigenvalues of second order difference equations with coupled boundary conditions abstract this paper is concerned with coupled boundary value problems for self adjoint second order difference equations existence of eigenvalues is proved numbers of eigenvalues are calculated and relationships between the eigenvalues of a self adjoint second order difference equation with three different coupled boundary conditions are established these results extend the relevant existing results of periodic and antiperiodic boundary value problems

regular approximations of spectra of singular second order symmetric linear difference equations abstract this paper is concerned with regular approximations of spectra of singular second order symmetric linear difference equations for any given self adjoint subspace extension of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of regular self adjoint subspace extensions generated by the same difference expression on smaller finite intervals

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

a posteriori error analysis of a fully discrete finite element method for the wave equation abstract a fully discrete finite element method is proposed for linear wave equations the spatial discretization is carried out using p l conforming element while the temporal discretization is obtained using p 1 continuous discontinuous galerkin method borrowing a novel time reconstruction operator combined with the technique of elliptic reconstruction a reliable posteriori error bound is developed based on this error bound two adaptive algorithms in time and space separately are proposed a series of numerical experiments are reported to illustrate the performance of the error bound and the adaptive algorithms proposed

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

crossover scaling from classical to non classical critical behaviour interacting physical systems in the neighborhood of criticality and massive continuum field theories can often be characterized by just two physical scales a macroscopic correlation length and a microscopic interaction range related to the coupling and measured by the ginzburg number g a critical crossover limit can be defined when both scales become large while their ratio stays finite the corresponding scaling functions are universal and they are related to the standard field theory renormalization group functions the critical crossover describes the unique flow from the gaussian to the nonclassical fixed point

convergence of an adaptive mixed finite element method for kirchhoff plate bending problems some reliable and efficient a posteriori error estimators are produced for a mixed finite element method the hellan herrmann johnson h h j method for kirchhoff plate bending problems cf k hellan acta polytech scand civil engrg ser 46 1967 pp 1 28 l herrmann j eng mech div asce 93 1967 pp 13 26 c johnson numer math 21 1973 pp 43 62 based on these results with k 0 1 where k denotes the polynomial order of the discrete moment field space an adaptive mixed finite element method amfem is set up and its convergence and complexity are studied thoroughly the key points of the theoretical analysis include achieving a discrete helmholtz decomposition and a discrete inf sup condition which serve as the main tools in deducing the quasi orthogonality of the moment field and the discrete reliability of the estimator it is shown that the amfem is a contraction for the sum of the moment field error in an energy norm and the scaled error estimator between two consecutive adaptive loops moreover an estimate for the amfem s complexity via the number of elements is developed

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

a convergent nonconforming adaptive finite element method with quasi optimal complexity in this paper we prove convergence and quasi optimal complexity of a simple adaptive nonconforming finite element method in each step of the algorithm the iterative solution of the discrete system is controlled by an adaptive stopping criterion and the local refinement is based on either a simple edge residual or a volume term depending on an adaptive marking strategy we prove that this marking strategy guarantees a strict reduction of the error augmented by the volume term and an additional oscillation term and quasi optimal complexity of the generated sequence of meshes

a comparison of a posteriori error estimators for mixed finite element discretizations by raviart thomas elements we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations in particular we present and analyze four different kinds of error estimators a residual based estimator a hierarchical one error estimators relying on the solution of local subproblems and on a superconvergence result respectively finally we examine the relationship between the presented error estimators and compare their local components

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

eigenvalues of second order symmetric equations on time scales with periodic and antiperiodic boundary conditions this paper studies periodic and antiperiodic boundary value problems for second order symmetric linear equations on time scales by properties of eigenvalues of the dirichlet boundary value problem and some oscillation results existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained these results not only unify the existing ones of periodic and antiperiodic boundary value problems for second order symmetric differential and difference equations but also contain more complicated time scales

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

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored