least squares mixed finite element methods for non selfadjoint elliptic problems i error estimates a least squares mixed finite element method for general second order non selfadjoint elliptic problems in two and three dimensional domains is formulated and analyzed the finite element spaces for the primary solution approximation u h and the flux approximation sigma h consist of piecewise polynomials of degree k and r respectively the method is mildly nonconforming on the boundary the cases k r and k 1 r are studied it is proved that the method is not subject to the lbb condition optimal l 2 and h 1 error estimates are derived for regular finite element partitions numerical experiments confirming the theoretical rates of convergence are presented

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

least squares methods for stokes equations based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for stokes and elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied we provide two methods the first is posed in terms of the velocity pressure pair without the introduction of additional variables the second adds a vorticity variable in both cases we employ least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak 1994

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

convergence analysis for least squares finite element approximations of second order two point boundary value problems in this paper a c 0 least squares finite element method for second order two point boundary value problems is considered the problem is recast as a first order system standard and improved optimal error estimates in maximum norms are established superconvergence estimates at interelement lobatto and gauss points are developed numerical experiments are given to illustrate theoretical results

div first order system ll fosll for second order elliptic partial differential equations the first order system ll fosll approach for general second order elliptic partial differential equations was proposed and analyzed in z cai et al siam j numer anal 39 2001 pp 1418 1445 in order to retain the full efficiency of the l 2 norm first order system least squares fosls approach while exhibiting the generality of the inverse norm fosls approach the fosll approach of cai et al was applied to the div curl system with added slack variables and hence it is quite complicated in this paper we apply the fosll approach to the div system and establish its well posedness for the corresponding finite element approximation we obtain a quasi optimal a priori error bound under the same regularity assumption as the standard galerkin method but without the restriction to sufficiently small mesh size unlike the fosls approach the fosll approach does not have a free a posteriori error estimator we then propose an explicit residual error estimator and establish its reliability a

domain decomposition for a mixed finite element method in three dimensions we consider the solution of the discrete linear system resulting from a mixed finite element discretization applied to a second order elliptic boundary value problem in three dimensions based on a decomposition of the velocity space these equations can be reduced to a discrete elliptic problem by eliminating the pressure through the use of substructures of the domain the practicality of the reduction relies on a local basis presented here for the divergence free subspace of the velocity space we consider additive and multiplicative domain decomposition methods for solving the reduced elliptic problem and their uniform convergence is established

a mixed finite element method for the stokes problem an acceleration pressure formulation the existing finite element methods 6 9 11 for the stokes equation lead to a saddle point problem for removing certain restrictions on the elements and improving the accuracy this paper develops a new method the stokes equation is treated as a first order linear system using a least squares method to minimize the defect in the differential operator we find that the result converges asymptotically to optimal order in sobolev spaces in the 2 d case if the domain is subdivided into regular triangles and the piecewise linear functions in the space h 1 are chosen to be trial functions for both velocity and pressure then we have error estimates p p h 0 u u h 0 ch 2

the l 2 norm error estimates for the div least squares method this paper studies l 2 norm error estimates for the div least squares method for which the associated homogeneous least squares functional is equivalent to the h mbox div times h 1 norm for the respective dual and primal variables least squares of this type for the second order elliptic equations elasticity and the stokes equations are an active area of research and error estimates in the h mbox div times h 1 norm were previously established in this paper we establish optimal l 2 norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

pointwise error estimates for first order div least squares finite element methods and applications to superconvergence and a posteriori error estimators first order div least squares finite element methods for second order elliptic partial differential equations are considered while there has been significant progress in error estimates for the methods to the best of our knowledge these estimates are based on the global norm of the error in this paper we provide highly localized pointwise error estimates for the primary variable u by establishing that the least squares solutions are locally higher order perturbations of the standard galerkin solutions as elementary consequences we identify the superconvergent points the set of superconvergent points for the primary function of the least squares solution is the same as that of the standard galerkin solution also we present a class of average type a posteriori error estimators for the method and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum norm on each single element of the underlying mesh

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

analysis of least squares finite element methods for the navier stokes equations in this paper we study finite element methods of least squares type for the stationary incompressible navier stokes equations in two and three dimensions we consider methods based on velocity vorticity pressure form of the navier stokes equations augmented with several nonstandard boundary conditions least squares minimization principles for these boundary value problems are developed with the aid of the agmon douglis nirenberg adn elliptic theory among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions results of several computational experiments with least squares methods which illustrate among other things the optimal convergence rates are also reported

weak regularity and finitely forcible graph limits graphons are analytic objects representing limits of convergent sequences of graphs lovasz and szegedy conjectured that every finitely forcible graphon i e any graphon determined by finitely many graph densities has a simple structure in particular one of their conjectures would imply that every finitely forcible graphon has a weak varepsilon regular partition with the number of parts bounded by a polynomial in varepsilon 1 we construct a finitely forcible graphon w such that the number of parts in any weak varepsilon regular partition of w is at least exponential in varepsilon 2 2 5 log varepsilon 2 this bound almost matches the known upper bound for graphs and in a certain sense is the best possible for graphons

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

least squares methods for elliptic systems a weighted least squares method is given for the numerical solution of elliptic partial differential equations of agmon douglis nirenberg type and an error analysis is provided some examples are given 1 introduction the use of least squares methods for the approximate solution of equations dates back at least to gauss the modern theory of least squares methods in the numerical solution of elliptic boundary value problems starts in 1970 with the papers of bramble and schatz 5 6 this work uses a finite dimensional space s of approximating functions similar to the spaces used in finite element methods the approximate solution is defined to be the minimizer of a least squares functional that is a weighted sum of the least squares residual in the differential equation and the least squares residual in the boundary condition the paper 5 has an historical importance for the following reason it appeared during the time when numerical analysts were shifting attention from finite difference methods to finite element methods and it provided for the first time a family of approximation methods for the solution of the dirichlet problem whose order of accuracy could be made arbitrarily large the paper 6 provided an extension to an elliptic equation of order 2m and 3 gave important simplifications in the analysis the principal advantages of the method are that one need not satisfy exactly the dirichlet boundary condi tions and that the mathematical analysis dictates in a natural way the relative weights that are given to the boundary and interior terms in the least squares functional also the method provides in a quasioptimal sense as good a solution as can be expected from the space s on the other hand the method requires that s consist of functions which are smooth enough to lie in the domain of the elliptic operator also the method seems to produce matrices with large condition number for various reasons it is of interest to extend the theory of least squares methods to include elliptic systems first if a second order elliptic equation is written as a first order system it would seem and this is borne out by our analysis that the smoothness requirements for the spaces of approximating functions would be reduced thus eliminating one of the disadvantages of the method a second motivation for extending the least squares method to elliptic systems is that elliptic systems occur frequently in applications an example of an elliptic system is the system of equations for stokes flow for this system the least squares method does not require the space of approximating vector fields to be incompressible instead

weak coupling of solutions of first order least squares method a theoretical analysis of a first order least squares finite element method for second order self adjoint elliptic problems is presented we investigate the coupling effect of the approximate solutions u h for the primary function u and h for the flux a u we prove that the accuracy of the approximate solution u h for the primary function u is weakly affected by the flux a u that is the bound for u u h 1 is dependent on but only through the best approximation for multiplied by a factor of meshsize h similarly we provide that the bound for h h div is dependent on u but only through the best approximation for u multiplied by a factor of the meshsize h this weak coupling is not true for the non selfadjoint case we provide the numerical experiment supporting the theorems in this paper

perturbations of forms and error estimates for the finite element method at a point with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point we first derive a variety of local error estimates for u uh at a point x0 where uh belongs to a finite element space shr and is an approximation to u satisfying the local equations a u u h varphi f varphi for all varphi in shr with compact support in a neighborhood of x0 here the a cdot cdot are bilinear forms associated with second order elliptic equations and the f are linear functionals in the case that f equiv 0 our results coincide with those of schatz siam j numer anal 38 2000 pp 1269 1293 but are improvements when f neq 0 we apply these results to improve the superconvergence error estimates obtained by schatz sloan and wahlbin siam j numer anal 33 1996 pp 505 521 at points x0 where the subspaces are symmetric with respect to x0

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

an adi pe approach for modeling radio transmission loss in tunnels alternate direction implicit adi method is used to study radio wave propagation in tunnels using the parabolic equation pe we formulate the adi technique for use in tunnels with rectangular circular and arched cross sections and with lossy walls the electrical parameters of the lossy walls are characterized by an equivalent surface impedance a vector pe is also formulated for use in tunnels with lossy walls it is shown that the adi is more computationally efficient than the crank nicolson method however boundary conditions become more difficult to model the boundary conditions of the adi intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced also when implementing the adi in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important to validate the adi pe we show simulation results for tunnel test cases with known analytical solutions furthermore the adi pe is used to simulate real tunnels in order to compare with experimental data it is shown that the pe models the electric fields most accurately in real tunnels at large distances where the lower order modes dominate

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes part 1 a smooth problem and globally quasi uniform meshes a class of a posteriori estimators is studied for the error in the maximum norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems the meshes are unstructured and in particular it is not assumed that there are any known superconvergent points the estimators are based on averaging operators which are approximate gradients recovered gradients which are then compared to the actual gradient of the approximation on each element conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large in a controlled fashion to be detailed below compared to the size of the elements

high accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients abstract in this article two level compact implicit difference methods of o k2 kh2 h4 using 9 spatial grid points are proposed for the numerical solution of the system of two dimensional nonlinear parabolic equations with variable coefficients subject to the dirichlet boundary conditions where k 0 and h 0 are step lengths in time and space directions respectively the proposed difference method for scalar equation is applied for the solution of the heat conduction equation in polar coordinates to obtain the two level unconditionally stable adi method of o k2 h4 the method having two variables also has been successfully applied on two dimensional unsteady navier stokes model equations in polar coordinates some numerical examples are presented to demonstrate the accuracy of the implementation

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

infinite dimensional finitely forcible graphon graphons are analytic objects associated with convergent sequences of dense graphs finitely forcible graphons that is those determined by finitely many subgraph densities are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science lovasz and szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension which would have implications on the minimum number of parts in its weak e regular partition we disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension

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

first order system least squares for the stress displacement formulation linear elasticity this paper develops a least squares finite element method for linear elasticity in both two and three dimensions the least squares functional is based on the stress displacement formulation with the symmetry condition of the stress tensor imposed in the first order system for the respective displacement and stress using the crouzeix raviart and raviart thomas finite element spaces our least squares finite element method is shown to be optimal in the broken h1 and h div norms uniform in the incompressible limit

a new two level implicit discretization of o k 2 kh 2 h 4 for the solution of singularly perturbed two space dimensional non linear parabolic equations we propose a new two level implicit difference method of o k2 kh2 h4 for the solution of singularly perturbed non linear parabolic differential equation e uxx uyy f x y t u ux u y ut 0 0 subject to appropriate initial and dirichlet boundary conditions where k 0 and h 0 are grid sizes in time and space directions respectively and e 0 is a small parameter we also develop new methods of o kh2 h4 for the estimates of u x and u y in all cases we use 9 spatial grid points and a single computational cell the proposed methods are directly applicable to singular problems we do not require any special scheme to solve singular problems we also discuss alternating direction implicit adi method for solving diffusion equation in polar cylindrical coordinates this method permits multiple use of the one dimensional tri diagonal algorithm with a considerable saving in computing time and produces a very efficient solver it is shown that the adi method is unconditionally stable numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions

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

testing permutation properties through subpermutations there has been great interest in deciding whether a combinatorial structure satisfies some property or in estimating the value of some numerical function associated with this combinatorial structure by considering only a randomly chosen substructure of sufficiently large but constant size these problems are called property testing and parameter testing where a property or parameter is said to be testable if it can be estimated accurately in this way the algorithmic appeal is evident as conditional on sampling this leads to reliable constant time randomized estimators our paper addresses property testing and parameter testing for permutations in a subpermutation perspective more precisely we investigate permutation properties and parameters that can be well approximated based on a randomly chosen subpermutation of much smaller size in this context we use a theory of convergence of permutation sequences developed by the present authors c hoppen y kohayakawa c g moreira r m sampaio limits of permutation sequences through permutation regularity manuscript 2010 34pp to characterize testable permutation parameters along the lines of the work of borgs et al c borgs j chayes l lovasz v t sos b szegedy k vesztergombi graph limits and parameter testing in stoc 06 proceedings of the 38th annual acm symposium on theory of computing acm new york 2006 pp 261 270 in the case of graphs moreover we obtain a permutation result in the direction of a famous result of alon and shapira n alon a shapira a characterization of the natural graph properties testable with one sided error siam j comput 37 6 2008 1703 1727 stating that every hereditary graph property is testable

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

analysis of least squares mixed finite element methods for nonlinear nonstationary convection diffusion problems some least squares mixed finite element methods for convection diffusion problems steady or nonstationary are formulated and convergence of these schemes is analyzed the main results are that a new optimal a priori l 2 error estimate of a least squares mixed finite element method for a steady convection diffusion problem is developed and that four fully discrete least squares mixed finite element schemes for an initial boundary value problem of a nonlinear nonstationary convection diffusion equation are formulated also some systematic theories on convergence of these schemes are established

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

extrapolation algorithm of compact adi approximation for two dimensional parabolic equation abstract in this paper we propose a compact alternating direction implicit adi scheme for solving the two dimensional parabolic equation the maximum norm convergence is proven and the convergence rate is second order in time and fourth order in space we develop a richardson extrapolation algorithm to increase the accuracy to sixth order both in time and space numerical experiments show the effectiveness of the method

finitely forcible graph limits are universal abstract the theory of graph limits represents large graphs by analytic objects called graphons graph limits determined by finitely many graph densities which are represented by finitely forcible graphons arise in various scenarios particularly within extremal combinatorics lovasz and szegedy conjectured that all such graphons possess a simple structure e g the space of their typical vertices is always finite dimensional this was disproved by several ad hoc constructions of complex finitely forcible graphons we prove that any graphon is a subgraphon of a finitely forcible graphon this dismisses any hope for a result showing that finitely forcible graphons possess a simple structure and is surprising when contrasted with the fact that finitely forcible graphons form a meager set in the space of all graphons in addition since any finitely forcible graphon represents the unique minimizer of some linear combination of densities of subgraphs our result also shows that such minimization problems which conceptually are among the simplest kind within extremal graph theory may in fact have unique optimal solutions with arbitrarily complex structure

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

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

on the accuracy of least squares methods in the presence of corner singularities abstract this paper treats elliptic problems with corner singularities finite element approximations based on variational principles of the least squares type tend to display poor convergence properties in such contexts moreover mesh refinement or the use of special singular elements do not appreciably improve matters here we show that if the least squares formulation is done in appropriately weighted spaces then optimal convergence results in unweighted spaces like l2

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

optimal error estimate for the div least squares method with data f inl 2 and application to nonlinear problems the div least squares methods have been studied by many researchers for the second order elliptic equations elasticity and the stokes equations and optimal error estimates have been obtained in the h mathrm div times h 1 norm however there is no known convergence rate when the given data f belongs only to l 2 space in this paper we will establish an optimal error estimate in the l 2 times h 1 norm with the given data f in l 2 and hence fill a theoretical gap of least squares methods as a consequence of this estimate we will provide a convergence analysis for the linearization process on solving navier stokes equations which uses the div least squares method for solving the corresponding stokes equations

on least squares finite element methods for the poisson equation and their connection to the dirichlet and kelvin principles least squares finite element methods for first order formulations of the poisson equation are not subject to the inf sup condition and lead to stable solutions even when all variables are approximated by equal order continuous finite element spaces for such elements one can also prove optimal convergence in the energy norm equivalent to a norm on h 1 omega times h omega mbox rm div for all variables and optimal l2 convergence for the scalar variable however showing optimal l2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first order system of partial differential equations in fact numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal l2 accuracy in this paper we show that optimal l2 error rates for the flux can be achieved without the curl constraint provided that one uses the div conforming family of brezzi douglas marini or brezzi douglas duran fortin elements then we proceed to reveal an interesting connection between a least squares finite element method involving h omega mbox rm div conforming flux approximations and mixed finite element methods based on the classical dirichlet and kelvin principles we show that such least squares finite element methods can be obtained by approximating through an l2 projection the hodge operator that connects the kelvin and dirichlet principles our principal conclusion is that when implemented in this way a least squares finite element method combines the best computational properties of finite element methods based on each of the classical principles

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

a new high accuracy two level implicit off step discretization for the system of two space dimensional quasi linear parabolic partial differential equations abstract in this article we propose a two level compact implicit off step discretization for the solution of the system of 2d quasi linear parabolic partial differential equations subject to suitable initial and boundary conditions new methods for the estimates of first order space derivatives of the solution are also derived these methods are fourth order accurate in space and second order accurate in time and involve only nine spatial grid points of a single compact cell we further develop the alternating direction implicit adi scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates the proposed methods are directly applicable to singular problems without the need of any special technique which is the main advantage of this work the method is effectively applied to the time dependent navier stokes model equations in polar coordinates numerical examples are provided to illustrate the accuracy of the methods

multilevel boundary functionals for least squares mixed finite element methods for least squares mixed finite element methods for the first order system formulation of second order elliptic problems a technique for the weak enforcement of boundary conditions is presented this approach is based on least squares boundary functionals which are equivalent to the h 1 2 and h1 2 norms on the trace spaces of lowest order raviart thomas elements for the flux and standard continuous piecewise linear elements for the pressure respectively continuity and coercivity of the resulting bilinear form is proved implying optimal order convergence of the resulting galerkin approximation the boundary least squares functional is implemented using multilevel principles and the technique is tested numerically for a model problem

sharp l_2 norm error estimates for first order div least squares methods a theoretical analysis of a first order div least squares finite element method is presented our main interest is providing l 2 norm error estimates for the primary function u and the flux mbox boldmath mathcal a nabla u while there has been significant progress in the error estimates for the flux in l 2 norm these estimates have drawbacks such as requiring smooth solutions or that the error estimate for mbox boldmath mbox boldmath h 0 is coupled with the error estimate for u u h 1 where u h mbox boldmath h is the least squares approximate solution for u mbox boldmath mathcal a nabla u in this paper with the minimum regularity assumption u in h 1 alpha alpha 1 2 we separate the error estimate for mbox boldmath mbox boldmath h 0 from u u h 1 and establish that mbox boldmath mbox boldmath h 0 is bounded by the best approximation for mbox boldmath in the finite element space and u u h 0 then we proceed to obtain two new estimates for u u h 0 the first estimate is useful to understand how the accuracy of u u h 0 affects the accuracy of mbox boldmath mbox boldmath h 0 and the other sheds new light on the convergence behavior of u u h 0 our analysis does not require smooth solutions and the domain is allowed to be nonconvex the resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the raviart thomas spaces for the flux mbox boldmath

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives

a new low order least squares nonconforming characteristics mixed finite element method for burgers equation in this paper a new low order least squares nonconforming characteristics mixed finite element method mfem is considered for two dimensional burgers equation by use of two typical characters of the elements for approximating the velocity and flux variables a the consistency errors of the elements can be estimated as order o h 2 with respect to the mesh size h in the broken h 1 norm one order higher than their interpolation errors b the elements interpolation operators satisfy certain orthogonalities which lead to o h 2 order error estimates in l 2 norm and some superclose results the accuracy of corresponding variables in l 2 norm is improved by one order compared with other mfems in the existing literature it seems that the results provided herein have never been derived before