an inf sup stable and residual free bubble element for the oseen equations we investigate the residual free bubble method for the linearized incompressible navier stokes equations starting with a nonconforming inf sup stable element pair for approximating the velocity and pressure we enrich the velocity space by discretely divergence free bubble functions to handle the influence of strong convection an important feature of the method is that the stabilization does not generate an additional coupling between the mass equation and the momentum equation as is the case for the streamline upwind petrov galerkin method applied to equal order interpolation furthermore the discrete solution is piecewise divergence free a property which is useful for the mass balance in transport equations coupled with the incompressible navier stokes equations

pattern recognition by graph matching using the potts mft neural networks this paper is concerned with programming of the potts mean field theory neural networks for pattern recognition by homomorphic mapping of the attributed relational graphs arg in order to generate the homomorphic mapping from the scene relational graph to the model graph we make use of the recently introduced suganthan technical report nanyang technical university 1994 compatibility functions in relation to the hopfield network an efficient pose clustering algorithm is used to separate and localize different occurrences of any particular object model in the scene the pose clustering algorithm also eliminates spurious hypotheses generated by the network and resolves ambiguities in the final interpretation the performance of the proposed approach to pattern recognition by homomorphism is demonstrated using a number of line patterns silhouette images and circle patterns

a posteriori l 2 error estimates for the nonconforming p 1 p 0 finite element discretization of the stokes equations abstract this paper focusses on a residual based a posteriori error estimator for the l 2 error of the velocity for the nonconforming p 1 p 0 finite element discretization of the stokes equations we derive an a posteriori error estimator which yields a local lower as well as a global upper bound on the error numerical tests demonstrate the efficiency of the global error estimator and give a comparison with respect to the adaptive grid refinement to an a posteriori error estimator in the discrete energy norm proposed by dari et al 1995

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

boundary dynamics driven entanglement we will show how to generate entangled states out of unentangled states on a bipartite system by means of dynamical boundary conditions the auxiliary system is defined by a symmetric but not self adjoint hamiltonian and we will also study the space of self adjoint extensions of the bipartite system we will show that only a small set of these extensions leads to separable dynamics and we will characterize these extensions various simple examples illustrating this phenomenon are discussed in particular we will analyze the hybrid system consisting of a planar quantum rotor and a spin system under a wide class of boundary conditions

robustness of an elementwise parallel finite element method for convection diffusion problems we consider an elementwise data parallel finite element procedure recently proposed by layton and rabier appl math lett 5 1992 pp 67 70 j numer linear algebra appl 2 1995 pp 363 394 applied to singularly perturbed convection diffusion equations with possibly highly anisotropic diffusion it is shown that the number of iterations required for the solution of the linear algebraic system is proportional to the inverse of the smallest grid element diameter uniformly in the diffusion parameter and the degree of anisotropy this is optimal since the method can in some cases use only element matrices and load vectors and the algorithm requires only local communication on the physical mesh between adjacent elemental subdomains our analysis includes both the usual galerkin formulation and the streamline upwind finite element formulations the convergence result holds with conforming elements of any order or elements of arbitrary order from a new family of nonconforming elements

numerical approximation of a one dimensional space fractional advection dispersion equation with boundary layer finite element computations for singularly perturbed convection diffusion equations have long been an attractive theme for numerical analysis in this article we consider the singularly perturbed fractional advection dispersion equation fade with boundary layer behavior we derive a theoretical estimate which shows that the under resolved case corresponds to e

image sequence analysis using relational structures abstract time varying images are usually analyzed to obtain a description of the observed objects and of their actions an abstraction process is necessary which converts the input images into symbolic descriptions the attributes of image symbols and the relationship between symbols are computed from the sampled intensity or colour measurements relational structures are a formal tool to describe not only the symbolic image representation but furthermore stored models as prototypes of objects and objects as instantiations of such prototypes following this paradigm an essential task of image sequence analysis is the generation of mappings between images to establish the correspondence relationship and between images and prototypes to identify objects different types of morphisms between relational structures rs morphisms are proposed together with an approach for computing them by searching for cliques in a compatibility graph of hypothetical assignments of elements from both structures the common formalism presented here aims at a unification of methods for those steps necessary to instantiate objects and follow them through the sequence one practical reason for this attempt is to make relational database systems available to store and retrieve the large amount of data whidch might be generated during the exploration of a real world image sequence

convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction diffusion problems the numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems the quasi optimal order error estimates are proved in the e weighted h 1 norm valid uniformly up to a logarithmic factor in the singular perturbation parameter by using the interpolation postprocessing technique the global superconvergent error estimates in e weighted h 1 norm are obtained numerical experiments are given to demonstrate validity of our theoretical analysis

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

pictorial representations of fuzzy connectives part ii cases of compensatory operators and self dual operators abstract as the continuation of part i in which pictorial representations of t norms t conorms and averaging operators were made with the aid of a computer this paper discusses compensatory operators and proposes generalized compensatory operators which can be obtained from averaging operators self dual operators are discussed which can be defined by using t norms t conorms and averaging operators finally symmetric sums are reviewed which are also self dual operators the pictorial representations of these operators are also made

on the stable solution of transient convection diffusion equations a transient convection diffusion equation is considered which particularly arises in optimization problems for the asymmetric flow field flow fractionation af4 process a time dependent generalization of the monotone edge averaged finite element eafe scheme is used to obtain a stable discretization in the convection dominated regime new error estimates are proved for this scheme and a comparison with the popular supg method is presented numerical experiments demonstrate that the eafe method is more suitable for problems where boundary layers are formed we consider transient convection diffusion equation with dominating convection the model has an application in optimal control of the asymmetric flow field flow fractionation process for a stable discretization we propose a monotone edge averaged finite element eafe scheme eafe is generalized to a time dependent case and a new error estimate is proved we present numerical results for comparison with a popular supg method

casimir effect and global theory of boundary conditions the consistency of quantum field theories defined on domains with external borders imposes very restrictive constraints on the type of boundary conditions that the fields can satisfy we analyse the global geometrical and topological properties of the space of all possible boundary conditions for scalar quantum field theories the variation of the casimir energy under the change of boundary conditions reveals the existence of singularities generically associated with boundary conditions which either involve topology changes of the underlying physical space or edge states with unbounded below classical energy the effect can be understood in terms of a new type of maslov index associated with the non trivial topology of the space of boundary conditions we also analyse the global aspects of the renormalization group flow t duality and the conformal invariance of the corresponding fixed points

a numerical study of a posteriori error estimators for convection diffusion equations this paper presents a numerical study of a posteriori error estimators for convection diausion equations the study involves the gradient indicator an a posteriori error estimator which is based on gradient recovery three residual based error estimators for diaerent norms and two error estimators which are defined by solutions of local neumann problems they are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids the numerical study shows for both criteria of comparison that none of the considered error estimators works satisfactorily in all tests 2000 elsevier science s a all rights reserved msc 65n50 65n30

nonconforming streamline diffusion finite element methods for convection diffusion problems we analyze nonconforming finite element approximations of streamline diffusion type for solving convection diffusion problems both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same o h k 1 2 order of convergence in l 2 as in the conforming case for convection dominated problems a rigorous error analysis supported by numerical experiments is given

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

kronecker product graph matching abstract in this paper the interpolator based kronecker product graph matching ibkpgm algorithm for performing attributed graph matching is presented the ibkpgm algorithm is based on the kronecker product graph matching kpgm formulation this new formulation incorporates a general approach to a wide class of graph matching problems based on attributed graphs allowing the structure of the graphs to be based on multiple sets of attributes salient features of the ibkpgm algorithm are that no assumption is made about the adjacency structure of the graphs to be matched and that the explicit calculation of compatibility values between all vertices of the reference and input graphs as well as between all edges of the reference and input graphs are avoided

a posteriori error analysis of nonconforming finite element methods for convection diffusion problems abstract a unified framework is established for the a posteriori error analysis of nonconforming finite element approximations to convection diffusion problems under some certain conditions the theory assures the semi robustness of residual error estimates in the usual energy norm and the robustness in a modified norm and applies to several nonconforming finite elements such as the crouzeix raviart triangular element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element etc based on the general error decomposition in different norms we show that the key ingredients of error estimation are the existence of a bounded linear operator v h c v h nc with some elementary properties and the estimation on the consistency error related to the particular numerical scheme the numerical results are presented to illustrate the practical behavior of the error estimator and check the theoretical predictions

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

a posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection diffusion equations this paper is devoted to the study of a posteriori and a priori error estimates for the scalar nonlinear convection diffusion equation c t nabla cdot u f c varepsilon delta c 0 the estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the l 1 norm in the situation where the diffusion parameter varepsilon is smaller or comparable to the mesh size numerical experiments underline the theoretical results

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

matching invariant to translations rotations and scale changes abstract an optimization approach to invariant matching is proposed in this approach an object or a pattern is invariantly represented by an object centred description called an attributed relational structure ars embedding invariant properties and relations between the primitives of the pattern such as line segments and points noise effect is taken into account such that a scene can consist of noisy sub parts of a model the matching is then to find the optimal mapping between the arss of the scene and the model a gain functional is formulated to measure the goodness of fit and is to be maximized by using the relaxation labelling method experiments are shown to illustrate the matching algorithm and to demonstrate that the approach is truly invariant to arbitrary translations rotations and scale changes under noise

vacuum structure and boundary renormalization group the vacuum structure is probed by boundary conditions the behaviour of thermodynamical quantities such as free energy boundary entropy and entanglement entropy under the boundary renormalization group flow are analysed in 2d conformal field theories the results show that wherever vacuum energy and boundary entropy turn out to be very sensitive to boundary conditions the vacuum entanglement entropy is independent of boundary properties when the boundary of the entanglement domain does not overlap the boundary of the physical space in all cases the second law of thermodynamics holds along the boundary renormalization group flow

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

optimal encoding of graph homomorphism energy using fuzzy information aggregation operators abstract the attributed relational graph matching arg strategy is a well known approach to object pattern recognition in the past for the parallel solution of arg matching problem an overall objective function was constructed using linearly weighted information aggregation function and one set of parameter values was chosen for all models by trial and error for the parameters in the function in this paper the compatibility between every pair of model and scene attributes is interpreted as a fuzzy value and subsequently the nonlinear fuzzy information aggregation operators are used to fuse the information captured in the chosen attributes to learn the parameters in the fuzzy information aggregation operators the learning from samples strategy is used the learning of weight parameters is formulated as an optimisation problem and solved using the gradient projection algorithm based learning procedure the learning procedure implicitly evaluates ambiguity robustness and discriminatory power of the relational attributes chosen for graph matching and assigns weights appropriately to the chosen attributes the learning procedure also enables us to compute a distinct set of optimal parameters for every model to reflect the characteristics of the model so that the homomorphic arg matching problem can be optimally encoded in the energy function for the model experimental results are presented to illustrate effectiveness and necessity of the parameter estimation and learning procedures

a pocs based graph matching algorithm a novel projections onto convex sets pocs graph matching algorithm is presented two way assignment constraints are enforced without using elaborate penalty terms graduated nonconvexity or sophisticated annealing mechanisms to escape from poor local minima results indicate that the presented algorithm is robust and compares favorably to other well known algorithms

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

finite element method to solve the spectral problem for arbitrary self adjoint extensions of the laplace beltrami operator on manifolds with a boundary abstract a numerical scheme to compute the spectrum of a large class of self adjoint extensions of the laplace beltrami operator on manifolds with boundary in any dimension is presented the algorithm is based on the characterisation of a large class of self adjoint extensions of laplace beltrami operators in terms of their associated quadratic forms the convergence of the scheme is proved a two dimensional version of the algorithm is implemented effectively and several numerical examples are computed showing that the algorithm treats in a unified way a wide variety of boundary conditions

a posteriori optimization of parameters in stabilized methods for convection diffusion problems part i abstract stabilized finite element methods for convection dominated problems require the choice of appropriate stabilization parameters from numerical analysis often only their asymptotic values are known this paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional exemplarily this framework is applied to the supg finite element method and the minimization of a residual based error estimator an error indicator and a functional including the crosswind derivative of the computed solution benefits of the basic approach are demonstrated by means of numerical results

a stabilized non conforming finite element method for incompressible flow in this paper we extend the recently introduced edge stabilization method to the case of non conforming finite element approximations of the linearized navier stokes equation to get stability also

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

two point one dimensional delta delta prime interactions non abelian addition law and decoupling limit in this contribution to the study of one dimensional point potentials we prove that if we take the limit on a potential of the type we obtain a new point potential of the type when u0 and u1 are related to v0 v1 w0 and w1 by a law with the structure of a group this is the borel subgroup of we also obtain the non abelian addition law from the scattering data the spectra of the hamiltonian in the decoupling cases emerging in the study are also described in full it is shown that for the values of the couplings the singular kurasov matrices become equivalent to dirichlet at one side of the point interaction and robin boundary conditions at the other side

an assessment of discretizations for convection dominated convection diffusion equations the performance of several numerical schemes for discretizing convection dominated convection diffusion equations will be investigated with respect to accuracy and efficiency accuracy is considered in measures which are of interest in applications the study includes an exponentially fitted finite volume scheme the streamline upwind petrov galerkin supg finite element method a spurious oscillations at layers diminishing sold finite element method a finite element method with continuous interior penalty cip stabilization a discontinuous galerkin dg finite element method and a total variation diminishing finite element method femtvd a detailed assessment of the schemes based on the hemker example will be presented

robust a posteriori error estimates for conforming and nonconforming finite element methods for convection diffusion problems a posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection diffusion problems our main contribution is finding an appropriate norm to measure the error which incorporates a discrete energy norm a discrete dual semi norm of the convective derivative and jumps of the approximate solution over element faces edges in two dimensions the error estimator is shown to be robust with respect to the peclet number in the sense of the modified norm based on a general error decomposition we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme and the remaining terms can be bounded in a unified way the numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy

a robust supg norm a posteriori error estimator for stationary convection diffusion equations a robust residual based a posteriori estimator is proposed for the supg finite element method applied to stationary convection diffusion reaction equations the error in the natural supg norm is estimated the main concern of this paper is the consideration of the convection dominated regime a global upper bound and a local lower bound for the error are derived where the global upper estimate relies on some hypotheses numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses a comparison to other residual based estimators with respect to the adaptive grid refinement is also provided

on the color image segmentation algorithm based on the thresholding and the fuzzy c means techniques abstract in this paper a segmentation algorithm for color images based on the thresholding and the fuzzy c means fcm techniques is presented the scale space filter is used as a tool for analyzing the histograms of three color components the methodology uses a coarse fine concept to reduce the computational burden required for the fcm the coarse segmentation attempts to segment coarsely using the thresholding technique while the fine segmentation assigns the pixels which remain unclassified after the coarse segmentation to the closest class using the fcm attempts also have been made to compare the performance of the proposed algorithm with other existing algorithms ohlander s rosenfeld s and bezdek s intensive computer simulation has been performed and the results are discussed in this paper the simulation results indicate that the proposed algorithm yields the most accurate segmented image on the color coordinate proposed by ohta et al while requiring a reasonable amount of computational effort

a streamline diffusion method for nonconforming finite element approximations applied to convection diffusion problems abstract we consider a nonconforming streamline diffusion finite element method for solving convection diffusion problems the theoretical and numerical investigation for triangular and tetrahedral meshes recently given by john maubach and tobiska has shown that the usual application of the sdfem gives not a sufficient stabilization additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case the error analysis has been essentially based on the existence of a conforming finite element subspace of the nonconforming space thus the analysis can be applied for example to the crouzeix raviart element but not to the nonconforming quadrilateral elements proposed by rannacher and turek in this paper parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case numerical experiments support the theoretical predictions

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

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

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985