the discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of poisson equation by using the direct method of lines the numerical simulation for the exterior problem of poisson equation is considered we introduced a polygonal artificial boundary ce and designed a discrete artificial boundary condition on it by using the direct method of lines then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary the finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed furthermore numerical examples show that the discrete artificial boundary condition is very eaective and more accurate than the neumann boundary condition which is often used in engineering literatures 1999 elsevier science s a all rights reserved

a posteriori finite element error control for the p laplace problem three a posteriori error estimates are discussed for the p laplace problem as a model equation for nonquadratic growth of order p 2 the standard residual based error estimate is addressed with emphasis on involved constants determined as local eigenvalues up to errors in their numerical computation they provide a guaranteed upper bound for the w1 p seminorm and a weighted w1 2 seminorm of u uh under the major assumption that nabla u h is positive a second sharper a posteriori estimate is based on the numerical solution of local interface problems averaging techniques for a posteriori error control on the flux error motivate a third estimator the results are presented for any shape regular but unstructured triangulation into triangles they partly rely on a positive weight function rho nabla u p 2 nabla u h p 2 2 similar to the notion of quasi norms due to barrett and liu numerical experiments indicate the surprising accuracy of our averaging error estimators confirm our estimates and allow for a comparison

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

a posteriori error estimators gradient recovery by averaging and superconvergence for the linear finite element solution to a linear elliptic model problem we derive an error estimator based upon appropriate gradient recovery by local averaging in contrast to popular variants like the zz estimator our estimator contains some additional terms that ensure reliability also on coarse meshes moreover the enhanced estimator is proved to be locally efficient and asymptotically exact whenever the recovered gradient is superconvergent we formulate an adaptive algorithm that is directed by this estimator and illustrate its aforementioned properties as well as their importance in numerical tests

topsis for modm abstract in this study we extend topsis to solve a multiple objective decision making problem the principle of compromise of topsis for multiple criteria decision making is that the chosen solution should have the shortest distance from the positive ideal solution as well as the longest distance from the negative ideal solution thus we reduce a k dimensional objective space to a two dimensional objective space by a first order compromise procedure we then use membership functions of fuzzy set theory to represent the satisfaction level for both criteria we obtain a single objective programming problem by using the max min operator for the second order compromise operation to illustrate the procedure the bow river valley water quality management problem is solved by use of topsis

refined fully explicit a posteriori residual based error control the explicit residual based a posteriori error estimator for elliptic partial differential equations is reliable up to the multiplication of some generic constant which needs to be involved for full error control the present mathematical literature takes this constant from the stability and approximation properties of clement type quasi interpolation operators and so results in an overestimation of the error which is bigger than for implicit and more expensive a posterori error estimators this paper propagates a paradigm shift to start with an equilibration error estimator technique followed by its efficiency analysis the outcome is a refined residual based a posteriori error estimate with explicit constants which leads to slightly sharper error control than the work of veeser and verfurth in 2009 a first application to guaranteed explicit error estimation for two dimensional nonconforming and a generalization to higher order finite element methods conclude the paper

a modified topsis with a different ranking index abstract as a tool for decision analysis the topsis technique for order preference by similarity to ideal solution attempts to choose alternative that should simultaneously have the closest distance from the positive ideal solution pis and the farthest distance from the negative ideal solution nis although the ranking index of topsis is reasonable it contains a flaw that is this ranking index is irrespective of the weights of separations of an alternative from the pis and the nis in other words no matter what weights the decision maker assigns to these two separations the ranking results would not differ as if he has no preference for these two separations this flaw will certainly limit the applicability of topsis by treating the separations of an alternative from the pis and the nis as a cost criterion and a benefit criterion respectively we reduced the original mcdm multi criteria decision making problem to a new mcdm problem with these two criteria only by proposing w and w as the weights of the cost criterion and the benefit criterion respectively we defined a new ranking index experimental results showed that if the number of alternatives exceeds two or if the relative importance of the two separations should be considered the proposed ranking index would be a better choice finally two numerical examples of a real life case are given for illustration in summary the proposed ranking index is intelligible and intrinsically superior to the original ranking index in seeking compromised solutions

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part ii higher order fem averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing in the second paper of our analysis of their reliability we consider conforming h fem of higher i e not of lowest order in two or three space dimensions in this paper reliablility is shown for conforming higher order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of local averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

a local a posteriori error estimator based on equilibrated fluxes we present and analyze a new a posteriori error estimator for lowest order conforming finite elements it is based on raviart thomas finite elements and can be obtained locally by a postprocessing technique involving for each vertex a local subproblem associated with a dual mesh under certain regularity assumptions on the right hand side we obtain an error estimator where the constant in the upper bound for the true error tends to one replacing the conforming finite element solution by a postprocessed one the error estimator is asymptotically exact the local equivalence between our estimator and the standard residual based error estimator is established numerical results illustrate the performance of the error estimator

solving multi period project selection problems with fuzzy goal programming based on topsis and a fuzzy preference relation project portfolio managers are multi objective decision makers dms who are expected to select the best mix of projects by maximizing profits and minimizing risks over a multi period planning horizon however project portfolio decisions are complex multi objective problems with a high number of projects from which a subset has to be chosen subject to various constraints and a multitude of priorities and preferences we propose a goal programming gp approach for project portfolio selection that embraces conflicting fuzzy goals with imprecise priorities a fuzzy goal with an aspiration level and a predefined membership function is defined for each objective the impreciseness in the priorities of the membership values of the fuzzy goals is modeled with fuzzy relations this leads to type ii fuzzy sets since fuzzy relations are organized between the membership values of the fuzzy goals which are themselves fuzzy sets the proposed model is based on the technique for order preference by similarity to ideal solution topsis and fuzzy preference relations topsis is used to reduce the multi objective problem into a bi objective problem the resulting bi objective problem is solved with fuzzy gp fgp the fuzzy preference relations are used to help dms express their preferences with respect to the membership values of the fuzzy goals the proposed approach is used to solve a real life problem characterized as a fuzzy multi objective project selection with multi period planning horizon mops mpph the performance of the proposed approach is compared with a competing method in the literature we show that our approach generates high quality solutions with minimal computational efforts

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

averaging techniques for reliable a posteriori fe error control in elastoplasticity with hardening averaging techniques are popular tools in adaptive finite element methods for numerical simulation in continuum mechanics since they provide efficient a posteriori error control in this paper the reliability of any averaging estimator is shown for low order finite element methods in one time step of elastoplasticity with hardening the constants and higher order terms are effected by the hardening and the smoothness of given right hand sides but are independent of the structure of a shape regular mesh since it involves a different functional analytical framework the case of perfect plasticity is excluded from this paper

singular sturmian comparison theorems for linear hamiltonian systems abstract in this paper we prove singular comparison theorems on unbounded intervals for two nonoscillatory linear hamiltonian systems satisfying the sturmian majorant condition and the legendre condition at the same time we do not impose any controllability condition the results are phrased in terms of the comparative index and the numbers of proper focal points of the minimal principal solutions of these systems at both endpoints of the considered interval the main idea is based on an application of new transformation theorems for principal and antiprincipal solutions at infinity and on new limit properties of the comparative index involving these solutions this work generalizes the recently obtained sturmian separation theorems on unbounded intervals for one system by the authors 2019 as well as the sturmian comparison theorems and transformation theorems on compact intervals by j elyseeva 2016 and 2018 we note that all the results are new even in the completely controllable case

residual based a posteriori error estimate for hypersingular equation on surfaces the hypersingular integral equation of the first kind equivalently describes screen and neumann problems on an open surface piece the paper establishes a computable upper error bound for its galerkin approximation and so motivates adaptive mesh refining algorithms numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh refining the numerical realisation requires the evaluation of the hypersingular integral operator at a source point this and other details on the algorithm are included

a posteriori error estimate and h adaptive algorithm on surfaces for symm s integral equation a residual based a posteriori error estimate for boundary integral equations on surfaces is derived in this paper a localisation argument involves a lipschitz partition of unity such as nodal basis functions known from finite element methods the abstract estimate does not use any property of the discrete solution but simplifies for the galerkin discretisation of symm s integral equation if piecewise constants belong to the test space the estimate suggests an isotropic adaptive algorithm for automatic mesh refinement an alternative motivation from a two level error estimate is possible but then requires a saturation assumption the efficiency of an anisotropic version is discussed and supported by numerical experiments

a posteriori error estimator and error control for contact problems in this paper we consider two error estimators for one body contact problems the first error estimator is defined in terms of h div conforming stress approximations and equilibrated fluxes while the second is a standard edge based residual error estimator without any modification with respect to the contact we show reliability and efficiency for both estimators more over the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order the second estimator is used in a control based afem refinement strategy and the decay of the error in the energy is shown several numerical tests demonstrate the performance of both estimators

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

adaptive computational methods for variational inequalities based on mixed formulations this work describes concepts for a posteriori error estimation and adaptive mesh design for finite element models where the solution is subjected to inequality constraints these methods are developed here for several model problems based on these examples unified frameworks are proposed which provide a systematic way of adaptive error control for problems stated in form of variational inequalities copyright 2006 john wiley sons ltd

a modified weighted topsis to identify influential nodes in complex networks abstract identifying influential nodes in complex networks is still an open issue although various centrality measures have been proposed to address this problem such as degree betweenness and closeness centralities they all have some limitations recently technique for order performance by similarity to ideal solution topsis as a tradeoff between the existing metrics has been proposed to rank nodes effectively and efficiently it regards the centrality measures as the multi attribute of the complex network and connects the multi attribute to synthesize the evaluation of node importance of each node however each attribute plays an equally important part in this method which is not reasonable in this paper we improve the method to ranking the node s spreading ability a new method named as weighted technique for order performance by similarity to ideal solution weighted topsis is proposed in our method we not only consider different centrality measures as the multi attribute to the network but also propose a new algorithm to calculate the weight of each attribute to evaluate the performance of our method we use the susceptible infected recovered sir model to do the simulation on four real networks the experiments on four real networks show that the proposed method can rank the spreading ability of nodes more accurately than the original method

sturmian comparison theorems for completely controllable linear hamiltonian systems in singular case abstract in this paper we consider two linear hamiltonian differential systems on an open unbounded interval we assume that the systems are completely controllable and satisfy the sturmian majorant condition and the legendre condition we derive a singular sturmian comparison theorem for the case when the minorant system has a solution which is principal at both endpoints of the considered interval the main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by aharonov and elias 2010 we illustrate our new theory by several examples

estimation of higher sobolev norm from lower order approximation given an accurate piecewise constant flux approximation q of a smooth but unknown exact flux p how can one compute a sobolev norm of p over a small domain omega this question arises in a duality argument of goal oriented finite element a posteriori error analysis two equivalent practical estimators eta m and eta mathcal e are presented for an approximation q of p h 1 omega and model situations are addressed where q is asymptotically an upper and lower bound of p h 1 omega

superconvergence for the gradient of finite element approximations by l 2 projections a gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy the recovery technique is based on the method of least squares surface fitting in a finite dimensional space corresponding to a coarse mesh it is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces the recovery technique is robust efficient and applicable to a wide class of problems such as the stokes and elasticity equations

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

singular sturm comparison theorems sturm s comparison theorem is discussed for differential equations whose coefficients are continuous in an open finite or infinite interval but the coefficients cannot necessarily be extended continuously to the boundary points of the interval

superconvergence property of an over penalized discontinuous galerkin finite element gradient recovery method a polynomial preserving recovery method is introduced for over penalized symmetric interior penalty discontinuous galerkin solutions to a quasi linear elliptic problem as a post processing method the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns as well as general meshes satisfying condition by means of the averaging technique we prove the polynomial preserving recovery method for averaged solutions is superconvergent satisfying similar estimates as those for conforming finite element methods we deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator consequently the a posteriori error estimator based on the recovered gradient is asymptotically exact extensive numerical results consistent with our analysis are presented

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

compromise solution by mcdm methods a comparative analysis of vikor and topsis the multiple criteria decision making mcdm methods vikor and topsis are based on an aggregating function representing closeness to the ideal which originated in the compromise programming method in vikor linear normalization and in topsis vector normalization is used to eliminate the units of criterion functions the vikor method of compromise ranking determines a compromise solution providing a maximum group utility for the majority and a minimum of an individual regret for the opponent the topsis method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution but it does not consider the relative importance of these distances a comparative analysis of these two methods is illustrated with a numerical example showing their similarity and some differences 2003 elsevier b v all rights reserved

a new method of identifying influential nodes in complex networks based on topsis abstract in complex networks identifying influential nodes is the very important part of reliability analysis which has been a key issue in analyzing the structural organization of a network in this paper a new evaluation method of node importance in complex networks based on technique for order performance by similarity to ideal solution topsis approach is proposed topsis as a multiple attribute decision making madm technique has been an important branch of decision making since then in addition topsis is first applied to identify influential nodes in a complex network in this open issue in different types of networks in which the information goes by different ways we consider several different centrality measures as the multi attribute of complex network in topsis application topsis is utilized to aggregate the multi attribute to obtain the evaluation of node importance of each node it is not limited to only one centrality measure but considers different centrality measures because every centrality measure has its own disadvantage and limitation then we use the susceptible infected si model to evaluate the performance numerical examples are given to show the efficiency and practicability of the proposed method

extension of the topsis method for decision making problems with fuzzy data decision making problem is the process of finding the best option from all of the feasible alternatives in this paper from among multicriteria models in making complex decisions and multiple attribute models for the most preferable choice technique for order preference by similarity to ideal solution topsis approach has been dealt with in real word situation because of incomplete or non obtainable information the data attributes are often not so deterministic there for they usually are fuzzy imprecise therefore the aim of this paper is to extend the topsis method to decision making problems with fuzzy data in this paper the rating of each alternative and the weight of each criterion are expressed in triangular fuzzy numbers the normalized fuzzy numbers is calculated by using the concept of a cuts finally a numerical experiment is used to illustrate the procedure of the proposed approach at the end of this paper

adaptive approximation of young measure solutions in scalar nonconvex variational problems this paper addresses the numerical approximation of young measures appearing as generalized solutions to scalar nonconvex variational problems we prove a priori and a posteriori error estimates for a macroscopic quantity the stress for a scalar three well problem we show convergence of other quantities such as young measure support and microstructure region numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to carstensen and roubicek numer math 84 2000 pp 395 414

an approach to generalization of fuzzy topsis method the topsis method is a technique for establishing order preference by similarity to the ideal solution and was primarily developed for dealing with real valued data this technique is currently one of most popular methods for multiple criteria decision making mcdm in many cases it is hard to present precisely exact ratings of alternatives with respect to local criteria and as a result these ratings are seen as fuzzy values a number of papers have been devoted to fuzzy extensions of the topsis method in the literature but these extensions are not complete since the ideal solutions are usually presented as real values not by fuzzy values or as fuzzy values which are not attainable in the decision matrix in most of these papers a defuzzification of elements of the fuzzy decision matrix is used which leads inevitably to a loss of important information and may even produce the wrong results in this paper we propose a new direct approach to the fuzzy extension of the topsis method which is free of the limitations of other known approaches we show that the distances of the alternatives from the ideal solutions may be treated in some sense as modified weighted sums of local criteria it is known that using weighted sums is not the best approach to the aggregation of local criteria in many real world situations therefore here we propose the use in addition to weighted sums some other types of local criteria aggregation in the topsis method and we develop a method for the generalization of different aggregation modes providing compromised final results

a posteriori error analysis for elliptic pdes on domains with complicated structures the discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores the model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh width at most and mixed boundary conditions for the poisson equation reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

superconvergence and gradient recovery of linear finite elements for the laplace beltrami operator on general surfaces superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the laplace beltrami equation on general surfaces with mildly structured triangular meshes for a large class of practically useful grids the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the laplace beltrami equation and as a result various postprocessing gradient recovery including simple and weighted averaging local and global l 2 projections and zienkiewicz and zhu z z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution numerical experiments are presented to confirm the theoretical results

singular sturmian separation theorems on unbounded intervals for linear hamiltonian systems abstract in this paper we develop new fundamental results in the sturmian theory for nonoscillatory linear hamiltonian systems on an unbounded interval we introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and based on this notion we prove singular sturmian separation theorems on an unbounded interval the main results are formulated in terms of the minimal principal solutions at both endpoints of the considered interval and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases as a natural tool we use the comparative index which was recently implemented into the theory of linear hamiltonian systems by the authors and independently by j elyseeva throughout the paper we do not assume any controllability condition on the system our results turn out to be new even in the completely controllable case

averaging technique for fe a posteriori error control in elasticity part ii λ independent estimates in the second part of our investigation on a posteriori error estimates and a posteriori error control in finite element analysis in elasticity we focus on robust a posteriori error bounds first we establish a residual based a posteriori error estimate which is reliable and efficient up to higher order terms and independent multiplicative constants the lame constant steers the incompressibility second we show the robust efficiency and reliability of averaging techniques in certain norms numerical evidence supports that the reliability of depends on the smoothness of given right hand sides and is independent of the structure of a shape regular mesh

adaptive multilevel inexact sqp methods for pde constrained optimization we present a class of inexact adaptive multilevel trust region sqp methods for the efficient solution of optimization problems governed by nonlinear pdes the algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process 1 implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2 implementable accuracy requirements for iterative solvers of the linearized pde and adjoint pde on the current grid we prove global convergence to a stationary point of the infinite dimensional problem moreover we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations numerical results are presented

all first order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable all first order averaging or gradient recovery operators for lowest order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic elliptic model problem in a bounded lipschitz domain in r d given a piecewise constant discrete flux p h e p h that is the gradient of a discrete displacement as an approximation to the unknown exact flux p that is the gradient of the exact displacement recent results verify efficiency and reliability of m min p h q h l 2 q h qh in the sense that m is a lower and upper bound of the flux error p p h l 2 up to multiplicative constants and higher order terms the averaging space q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions the minimal value m is frequently replaced by some averaging operator a p h q h applied within a simple post processing to p h the result q h ap h e q h provides a reliable error bound with m a p h ap h l 2 this paper establishes a c eff m and so equivalence of m and a this implies efficiency of a for a large class of patchwise averaging techniques which includes the zz gradient recovery technique the bound c eff 3 88 established for tetrahedral p 1 finite elements appears striking in that the shape of the elements does not enter the equivalence a m is robust with respect to anisotropic meshes the main arguments in the proof are ascoli s lemma a strengthened cauchy inequality and elementary calculations with mass matrices

inhomogeneous dirichlet conditions in a priori and a posteriori finite element error analysis the numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given dirichlet data ud by functions ud h in the trace space of a finite element space on d in this paper quantitative a priori and a posteriori estimates are presented for two choices of ud h namely the nodal interpolation and the orthogonal projection in l2 d onto the trace space two corresponding extension operators allow for an estimate of the boundary data approximation in global h1 and l2 a priori and a posteriori error estimates the results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation