a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

coercive boundary value problems for regular degenerate differential operator equations abstract this study focuses on nonlocal boundary value problems bvp for degenerate elliptic differential operator equations doe that are defined in banach valued function spaces where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients several conditions obtained that guarantee the maximal l p regularity and fredholmness these results are also applied to nonlocal bvp for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

on the virtual element method for topology optimization on polygonal meshes it is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh which can steer the minimization process towards local and non physical minima and by the accuracy of the method employed to discretize the underlying differential problem which may not be able to correctly capture the physics of the problem in light of the above remarks in this paper we consider polygonal meshes and employ the virtual element method vem to solve two classes of paradigmatic topology optimization problems one governed by nearly incompressible and compressible linear elasticity and the other by stokes equations several numerical results show the virtues of our polygonal vem based approach with respect to more standard methods

error estimates of the weakly over penalized symmetric interior penalty method for two variational inequalities in this paper we apply the weakly over penalized symmetric interior penalty method to solve some variational inequalities which include the signorini problem and the obstacle problem optimal a priori error estimates in energy norm are derived some numerical tests are provided to confirm our theoretical analysis

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

on fourier multiplier transformations of banach valued functions we obtain analogues of the mihlin multiplier theorem and littlewood paley inequalities for functions with values in a suitable banach space b the requirement on b is that it have the unconditionality property for martingale difference sequences

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

order preserving supg stabilization for the virtual element formulation of advection diffusion problems abstract in the framework of the discretization of advection diffusion problems by means of the virtual element method we consider stabilization issues herein stabilization is pursued by adding a consistent supg like term for this approach we prove optimal rates of convergence numerical results clearly show the stabilizing effect of the method up to very large peclet numbers and are in very good agreement with the expected rate of convergence

a virtual element method for the acoustic vibration problem we analyze in this paper a virtual element approximation for the acoustic vibration problem we consider a variational formulation relying only on the fluid displacement and propose a discretization by means of mathrm h mathrm div h div virtual elements with vanishing rotor under standard assumptions on the meshes we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates with this end we prove approximation properties of the proposed virtual elements we also report some numerical tests supporting our theoretical results

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

relationship between the vertex centered linearity preserving scheme and the lowest order virtual element method for diffusion problems on star shaped polygons abstract as a generalization of the finite element method the virtual element method vem has made remarkable achievements in the algorithm and analysis of simulating various problems recently vertex centered linearity preserving scheme vlps wu et al 2016 is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems in this paper we study the relationship between vlps and the lowest order vem for diffusion problems on star shaped polygonal meshes from an algebraic point of view the global stiffness matrix of vlps with a special stabilization term coincides with that of the lowest order vem while the load terms are generally different specifically we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts and the consistency parts are the same while the stability parts coincide under some assumptions as a by product a new stability term is obtained for vlps besides a post processing procedure is suggested for the lowest order vem to preserve the positivity of the cell centered unknowns and maintain at the same time the local conservation on the primary meshes the positivity existence and uniqueness of the cell centered unknowns are studied numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post processing procedure on various polygonal meshes

flux reconstruction and solution post processing in mimetic finite difference methods abstract we present a post processing technique for the mimetic finite difference solution of diffusion problems in mixed form our post processing method yields a piecewise linear approximation of the scalar variable that is second order accurate in the l2 norm on quite general polyhedral meshes including non convex and non matching elements the post processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables this technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied the post processing method is computationally inexpensive optimal performance is confirmed by numerical experiments

automatically inf sup compliant diamond mixed finite elements for kirchhoff plates we develop a mixed finite element approximation scheme for kirchhoff plate theory based on the reformulation of kirchhoff plate theory of ortiz and morris 1 in this reformulation the moment equilibrium problem for the rotations is in direct analogy to the problem of incompressible two dimensional elasticity this analogy in turn opens the way for the application of diamond approximation schemes hauret et al 2 to kirchhoff plate theory we show that a special class of meshes derived from an arbitrary triangulation of the domain the diamond meshes results in the automatic satisfaction of the corresponding inf sup condition for kirchhoff plate theory the attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests we also provide a reinterpretation of the diamond approximation scheme for kirchhoff plate theory within the framework of discrete mechanics in this interpretation the discrete moment equilibrium problem is formally identical to the classical continuous problem and the two differ only in the choice of differential structures it also follows that the satisfaction of the inf sup condition is a property of the cohomology of a certain discrete transverse differential complex this close connection between the classical inf sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems

convergence analysis of the mimetic finite difference method for elliptic problems with staggered discretizations of diffusion coefficients we study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in k lipnikov g manzini j d moulton and m shashkov j comput phys 305 2016 pp 111 126 in contrast to the conventional approach the diffusion coefficient enters both the primary mimetic operator i e the discrete divergence and the inner product in the space of gradients the diffusion coefficient is therefore evaluated on different mesh locations i e inside mesh cells and on mesh faces such a staggered discretization may provide the flexibility necessary for future development of efficient numerical schemes for nonlinear problems especially for problems with degenerate coefficients these new mimetic schemes preserve symmetry and positive definiteness of the continuum problem which allow us to use efficient algebraic solvers such as the preconditioned conjugate gradient method we show that these schemes are inf sup stable and establish a priori error

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

mixed interpolated elements for reissner mindlin plates dcpartement de mathtmatiyue l nit ersitt 1 aral quthec canado summary we present in this paper a procedure to establish reissner mindlin plate bending elements the procedure is based on the idea to combine known resuits on the approximation of stokes problems with known results on the approximation of elliptic problems the proposed elements satisfy the mathematical conditions of stability and convergence and some of them promise to provide efficient elements for practical solutions

on mixed finite element methods for the reissner mindlin plate model in this paper we analyze the convergence of mixed finite element approximations to the solution of the reissner mindlin plate problem we show that several known elements fall into our analysis thus providing a unified approach we also introduce a low order triangular element which is optimalorder convergent uniformly in the plate thickness

a residual based error estimator for the mimetic finite difference method we present a local error indicator for the mimetic finite difference method for diffusion type problems on polyhedral meshes under essentially the same general hypotheses used in siam j numer anal 43 1872 1896 2005 to show the convergence of the method we prove the global reliability and local efficiency of the proposed estimator

a posteriori error analysis of a mixed virtual element method for a nonlinear brinkman model of porous media flow abstract in this paper we present an a posteriori error analysis of a mixed vem discretization for a nonlinear brinkman model of porous media flow which has been proposed by the authors in a previous work therein the system is formulated in terms of a pseudostress tensor and the velocity gradient whereas the velocity and the pressure of the fluid are computed via postprocessing formulas furthermore the well posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established we now propose reliable and efficient residual based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem the resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes for the analysis we make use of a global inf sup condition helmholtz decomposition local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions finally we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

b separable boundary value problems in banach valued function spaces in this paper the nonlocal boundary value problems for anisotropic partial differential operator equations with dependent coefficients in banach valued besov b spaces are studied the principal parts of the appropriate differential operators are non self adjoint several conditions for separability and fredholmness are given these results permit us to establish that the inverse of the corresponding differential operators belong to the schatten q class the spectral properties of the appropriate differential operators are also investigated in addition we study the maximal regularity of nonlocal initial boundary value problems for abstract parabolic equations finite or infinite systems of parabolic equations and the separability of nonlocal boundary value problems for finite or infinite systems of quasi elliptic equations in b spaces

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

asymptotically sharp inequalities for polynomials involving mixed laguerre norms abstract the paper concerns best constants in markov type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself the norm of the polynomial and its derivative is taken in l 2 on the half line with the weight t e t and t e t respectively under the assumption that is larger than the order of the derivative we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity thus proving a conjecture raised by bottcher and dorfler in 2010 4

a mimetic discretization of the stokes problem with selected edge bubbles a new mimetic finite difference method for the stokes problem is proposed and analyzed the mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons polyhedrons in this paper the mimetic generalization of the unstable p 1 p 0 and the conditionally stable q1 p0 finite element is shown to be fully stable when applied to a large range of polygonal meshes moreover we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges a simple strategy for selecting such edges is proposed and verified with numerical experiments

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

convergence analysis of symmetric dual wind discontinuous galerkin approximation methods for the obstacle problem abstract this paper formulates and analyzes symmetric dual wind discontinuous galerkin dg methods for second order elliptic obstacle problem these new methods follow from the dg differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation pde we establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular these results are also shown to hold for some non positive penalty parameters with the emphasis on zero penalization across all interior and boundary edges numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods

conforming polygonal finite elements summary in this paper conforming finite elements on polygon meshes are developed polygonal finite elements provide greater flexibility in mesh generation and are better suited for applications in solid mechanics which involve a significant change in the topology of the material domain in this study recent advances in meshfree approximations computational geometry and computer graphics are used to construct different trial and test approximations on polygonal elements a particular and notable contribution is the use of meshfree natural neighbour nn basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons this numerical formulation enables the construction of conforming approximation on n gons n 3 and hence extends the potential applications of finite elements to convex polygons of arbitrary order numerical experiments on second order elliptic boundary value problems are presented to demonstrate the accuracy and convergence of the proposed method copyright 2004 john wiley sons ltd

hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems we present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems under a saturation assumption the global reliability and efficiency of the proposed a posteriori estimator are proved several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies finally we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures

maximal b regular boundary value problems with parameters this study focuses on non local boundary value problems bvp for elliptic differential operator equations doe defined in banach valued besov b spaces here equations and boundary conditions contain certain parameters this study found some conditions that guarantee the maximal regularity and fredholmness in banach valued b spaces uniformly with respect to these parameters these results are applied to non local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

a posteriori error estimates for a virtual element method for the steklov eigenvalue problem abstract the paper deals with the a posteriori error analysis of a virtual element method for the steklov eigenvalue problem the virtual element method has the advantage of using general polygonal meshes which allows implementing efficiently mesh refinement strategies we introduce a residual type a posteriori error estimator and prove its reliability and global efficiency local efficiency estimates also hold although in some elements they involve boundary terms that are not known to be locally negligible we use the corresponding error estimator to drive an adaptive scheme finally we report the results of a couple of numerical tests that allow us to assess the performance of this approach

a four node plate bending element based on mindlin reissner plate theory and a mixed interpolation this communication discusses a 4 node plate bending element for linear elastic analysis which is obtained as a special case from a general nonlinear continuum mechanics based 4 node shell element formulation the formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate and shell element

a low order locking free virtual element for linear elasticity problems abstract we propose and analyze a low order virtual element for linear elasticity problems it can be seen as an evolution of the bernardi raugel element to general polygonal meshes like the bernardi raugel element the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge the proposed method is well posed and optimal error estimates are obtained in the l 2 and h 1 norms moreover the generic constants in these estimates are shown to be uniform with respect to the lame coefficient numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions

virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation abstract we propose and analyze a virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation that is to say stresses are not require to be symmetric but only to satisfy a weaker condition based on lagrange multipliers the proposed method is well posed and the error bounds are shown to be uniform with the incompressibility parameter numerical tests confirm the convergence rate that is expected from the theory

a divergence free reconstruction of the nonconforming virtual element method for the stokes problem abstract in this paper we propose and investigate a divergence free reconstruction of the nonconforming virtual element for the stokes problem by constructing the computable raviart thomas like interpolation operator we guarantee the independence between the velocity error estimation u u h 1 h and the continuous pressure p as it happens for the divergence free flow solver moreover this modified scheme can also inherit the advantages of the classical nonconforming virtual element method such as very general meshes including non convex and degenerate elements a unified scheme for an arbitrary order approximation accuracy k etc then we provide the optimal l 2 error estimates for the velocity gradient and the pressure by taking advantage of the raviart thomas like interpolation operator and avoiding the use of a trace inequality finally three numerical experiments are presented to conform the theoretical analysis

on the best constants in inequalities of the markov and wirtinger types for polynomials on the half line abstract the main topic of the paper is best constants in markov type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves the norm is the l 2 norm with laguerre weight the leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term which is the operator norm of a volterra operator are given for best constants in inequalities of the wirtinger type the limit is computed and an asymptotic formula for the error term is presented

a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

convergence analysis of the mimetic finite difference method for elliptic problems we propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes the a priori error analysis relies on the connection between the mimetic formulation and the lowest order raviart thomas mixed finite element method the theoretical results are confirmed by numerical experiments

a nystrom based finite element method on polygonal elements abstract we consider families of finite elements on polygonal meshes that are defined implicitly on each mesh cell as solutions of local poisson problems with polynomial data functions in the local space on each mesh cell are evaluated via nystrom discretizations of associated integral equations allowing for curvilinear polygons and non polynomial boundary data several experiments demonstrate the approximation quality of interpolated functions in these spaces

an analysis of some mixed enhanced finite element for plane linear elasticity the paper investigates mixed enhanced strain finite elements developed within the context of the u p formulation for nearly incompressible linear elasticity problems a rigorous convergence and stability analysis is detailed providing also l2 error estimates for the displacement field extensive numerical tests are developed showing in particular the accordance of the computational results with the theoretical predictions

the nonconforming virtual element method for elasticity problems abstract we present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation this method is uniformly convergent for the nearly incompressible case the optimal convergence in the h 1 norm is proved under some regularity assumptions we mention that for the lowest order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by brenner and sung 1992 19 up to an approximation of the right hand side besides we also present the nonconforming virtual element method for the pure traction problem which is proved to be uniformly convergent with respect to the lame constant finally we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

extended finite element method on polygonal and quadtree meshes in this paper we present mesh independent modeling of discontinuous flelds on polygonal and quadtree flnite element meshes this approach falls within the class of extended and generalized flnite element methods where the partition of unity framework is used to introduce additional enrichment functions within the classical displacement based flnite element approximation for crack modeling a discontinuous function and the two dimensional asymptotic crack tip flelds are used as enrichment functions linearly complete partition of unity approximations are adopted on polygonal convex and nonconvex elements and quadtree meshes excellent agreement with reference solution results is obtained for mixed mode stress intensity factors on benchmark crack problems and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a higher order formulation of the mimetic finite difference method a new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes this approach provides a higher order accurate approximation to the flux of the exact solution in analogy with the original formulation a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability the scalar solution field is approximated by a piecewise constant function a computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable finally optimal convergence rates and accuracy improvement with respect to the lower order formulation are shown by numerical examples

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

a mimetic discretization of the reissner mindlin plate bending problem we present a mimetic approximation of the reissner mindlin plate bending problem which uses deflections and rotations as discrete variables the method applies to very general polygonal meshes even with non matching or non convex elements we prove linear convergence for the method uniformly in the plate thickness

polygonal finite elements for topology optimization a unifying paradigm in topology optimization literature the parameterization of design is commonly carried out on uniform grids consisting of lagrangian type finite elements e g linear quads these formulations however suffer from numerical anomalies such as checkerboard patterns and one node connections which has prompted extensive research on these topics a problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members leading to mesh dependent sub optimal designs thus to address the geometric features of the spatial discretization we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions more specifically we consider polygonal meshes constructed from voronoi tessellations which in addition to possessing higher degree of geometric isotropy allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities copyright 2009 john wiley sons ltd

error estimates for low order isoparametric quadrilateral finite elements for plates this paper deals with the numerical approximation of the bending of a plate modeled by reissner mindlin equations it is well known that in order to avoid locking some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods in particular one of the most widely used procedures is based on the family of elements called mitc mixed interpolation of tensorial components we consider two lowest order methods of this family on quadrilateral meshes under mild assumptions we obtain optimal h1 and l2 error estimates for both methods these estimates are valid with constants independent of the plate thickness we also obtain error estimates for the approximation of the plate vibration problem finally we report some numerical experiments showing the very good behavior of the methods even in some cases not covered by our theory

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

finite element methods for a modified reissner mindlin free plate model the solution of the reissner mindlin plate problem with free boundary conditions presents a strong layer effect near the free edges as a consequence the solution is not even uniformly bounded even in h3 2 which implies that at most an o h1 2 uniform convergence rate can be reached by finite element methods in the h1 norm following instead the modified free boundary model presented by beirao da veiga and brezzi which gives more regular solutions better error estimates can be obtained in principle in this paper we present and analyze the extension of different families of well known optimal plate methods to this new model all the modified methods presented are proved to be optimal and free of locking

a two level method for mimetic finite difference discretizations of elliptic problems we propose and analyze a two level method for mimetic finite difference approximations of second order elliptic boundary value problems we prove that the two level algorithm is uniformly convergent i e the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underlying partition we also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom numerical results that validate the theory are also presented

design principles and error analysis for reduced shear plate bending finite elements in this paper we consider the problem of designing plate bending elements which are free of shear locking this phenomenon is known to afflict several elements for the reissner mindlin plate model when the thickness d of the plate is small due to the inability of the approximating subspaces to satisfy the kirchhoff constraint to avoid locking a reduction operator is often applied to the stress to modify the variational formulation and reduce the effect of this constraint we investigate the conditions required on such reduction operators to ensure that the approximability and consistency errors are of the right order a set of sufficient conditions is presented under which optimal errors can be obtained these are derived directly without transforming the problem via a hemholtz decomposition or considering it as a mixed method our analysis explicitly takes into account boundary layers and their resolution and we prove via an asymptotic analysis that convergence of the finite element approximations will occur uniformly as d rightarrow 0 even on quasiuniform meshes the analysis is carried out in the case of a free boundary where the boundary layer is known to be strong we also propose and analyze a simple post processing scheme for the shear stress our general theory is used to analyze the well known mitc elements for the reissner mindlin plate as we show the theory makes it possible to analyze both straight and curved elements we also analyze some other elements

gradient schemes for the signorini and the obstacle problems and application to hybrid mimetic mixed methods gradient schemes is a framework which enables the unified convergence analysis of many different methods such as finite elements conforming non conforming and mixed and finite volumes methods for 2nd order diffusion equations we show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed dirichlet neumann and signorini boundary conditions this extension allows us to provide error estimates for numerical approximations of such models recovering known convergence rates for some methods and establishing new convergence rates for schemes not previously studied for variational inequalities the general framework we develop also enables us to design a new numerical method for the obstacle and signorini problems based on hybrid mimetic mixed schemes we provide numerical results that demonstrate the accuracy of these schemes and confirm our theoretical rates of convergence

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

weighted l2 analogs of bernstein s inequality and classical orthogonal polynomials abstract let p n be the class of algebraic polynomials of degree at most n some weighted l 2 analogues of the bernstein s inequality for polynomials p p n are investigated and a connection with the classical orthogonal polynomials is given

necessary and sufficient condition for a vmo function abstract the purpose of this paper is to prove a necessary and sufficient condition for a vanishing mean oscillation function to obtain this result we use an extension to morrey spaces of an inequality by fefferman and stein concerning the maximal and the sharp maximal function the fractional maximal function used by muckenhoupt and wheeden and the spaces generated by blocks well studied by shirai komori and mizuhara

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

virtual element methods for hyperbolic problems on polygonal meshes in the present paper we develop the virtual element method for hyperbolic problems on polygonal meshes considering the linear wave equations as our model problem after presenting the semi discrete scheme we derive the convergence estimates in h 1 semi norm and l 2 norm moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the newmark method and the bathe method finally we show the practical behaviour of the proposed method through a large set of numerical tests

virtual element for the buckling problem of kirchhoff love plates abstract in this paper we develop a virtual element method vem of high order to solve the fourth order plate buckling eigenvalue problem on polygonal meshes we write a variational formulation based on the kirchhoff love model depending on the transverse displacement of the plate we propose a c 1 conforming virtual element discretization of arbitrary order k 2 and we use the so called babuska osborn abstract spectral approximation theory to show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the buckling modes eigenfunctions and a double order for the buckling coefficients eigenvalues finally we report some numerical experiments illustrating the behavior of the proposed scheme and confirming our theoretical results on different families of meshes

virtual element method for a nonlocal elliptic problem of kirchhoff type on polygonal meshes abstract we consider the discretization of the nonlocal elliptic problem of kirchhoff type using the virtual element method vem over polygonal meshes the nonlocal diffusion coefficient is approximated by using the l 2 projection operator which is directly computable from the degrees of freedom however the presence of the nonlocal term reduces the sparsity of the jacobian matrix which would increase the computational burden to avoid this and to retain the sparsity of the jacobian the nonlinear system is replaced by an equivalent system inspired by the work on the fem by gudi 2012 the numerical results show that the proposed methodology yields optimal convergence rate in the l 2 norm theoretical estimates are derived for h 1 norms and are verified by solving a numerical example

on the coupling of vem and bem in two and three dimensions this paper introduces and analyzes the combined use of the virtual element method vem and the boundary element method bem to numerically solve linear transmission problems in two and three dime

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

a multilevel multiscale mimetic m3 method for two phase flows in porous media we describe a multilevel multiscale mimetic m 3 method for solving two phase flow water and oil in a heterogeneous reservoir the governing equations are the elliptic equation for the reservoir pressure and the hyperbolic equation for the water saturation on each time step we first solve the pressure equation and then use the computed flux in an explicit upwind finite volume method to update the saturation to reduce the computational cost the pressure equation is solved on a much coarser grid than the saturation equation the coarse grid pressure discretization captures the influence of multiple scales via the subgrid modeling technique for single phase flow recently proposed in yu a kuznetsov mixed finite element method for diffusion equations on polygonal meshes with mixed cells j numer math 14 4 2006 305 315 v gvozdev discretization of the diffusion and maxwell equations on polyhedral meshes technical report ph d thesis university of houston 2007 yu kuznetsov mixed finite element methods on polyhedral meshes for diffusion equations in computational modeling with pdes in science and engineering springer verlag berlin in press we extend significantly the applicability of this technique by developing a new robust and efficient method for estimating the flux coarsening parameters specifically with this advance the m 3 method can handle full permeability tensors and general coarsening strategies which may generate polygonal meshes on the coarse grid these problem dependent coarsening parameters also play a critical role in the interpolation of the flux and hence in the advection of saturation for two phase flow numerical experiments for two phase flow in highly heterogeneous permeability fields including layer 68 of the spe tenth comparative solution project demonstrate that the m 3 method retains good accuracy for high coarsening factors in both directions up to 64 for the considered models moreover we demonstrate that with a simple and efficient temporal updating strategy for the coarsening parameters we achieve accuracy comparable to the fine scale solution but at a fraction of the cost

high order discontinuous galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes abstract we propose and analyze a high order discontinuous galerkin finite element method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements we analyze the well posedness of the resulting formulation prove h p version error a priori estimates and present a dispersion analysis showing that polygonal meshes behave as classical simplicial quadrilateral grids in terms of dispersion properties the theoretical estimates are confirmed through various two dimensional numerical verifications

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

extremal problems inequalities and classical orthogonal polynomials in this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities besides the basic properties of the classical orthogonal polynomials we consider polynomial inequalities of landau and kolmogoroff type some weighted polynomial inequalities in l 2 norm of markov bernstein type as well as the corresponding connections with the classical orthogonal polynomials

exponential convergence of the hp virtual element method in presence of corner singularities in the present work we analyze the hp version of virtual element methods for the 2d poisson problem we prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined thus extending the classical choices for the decomposition in the hp finite element framework to very general decomposition of the domain a new stabilization for the discrete bilinear form with explicit bounds in h and p is introduced numerical experiments validate the theoretical results we also exhibit a numerical comparison between hp virtual elements and hp finite elements

a virtual element method for stochastic stokes equations abstract in this paper a virtual element method for the stochastic stokes equations driven by an additive white noise is proposed and analyzed the velocity is approximated by the lowest order virtual element which is originally designed for the poisson equation and the projection is also taken as the one originally for the poisson equation while the pressure is approximated by the traditional discontinuous piecewise constant element for stable approximations we adopt a stabilization associating with the pressure jumps we show the inf sup condition and derive the stability we moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the green function numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained

a robust numerical method for stokes equations based on divergence free h div finite element methods a computational method based on a divergence free h div approach is presented for the stokes equations in this article this method is designed to find velocity approximation in an exact divergence free subspace of the corresponding h div finite element space that is the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation a strength of this approach is that the saddle point problem for stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available the resulting discrete system can then be solved by using existing sophisticated solvers the aim of this article is to demonstrate the efficiency and robustness of h div finite element methods for stokes equations the results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions

numerical approximation of mindlin reissner plates we consider a finite element approximation of the so called mindlin reissner formulation for moderately thick elastic plates we show that stability and optimal error bounds hold independently of the value of the thickness 1 introduction the so called mindlin reissner model for moderately thin plates is often used by engineers in connection with plate and shell problems it is well known that many numerical schemes for this model are satisfactory only when the thickness parameter t is not too small for a very small t some bad behavior such as the locking phenomenon might occur here we present a method which is uniformly good as t goes to zero and we prove optimal error estimates for transversal displacement rotations and shear stresses with constants independent of t

mimetic finite difference methods for diffusion equations on non orthogonal non conformal meshes mimetic discretizations based on the support operators methodology are derived for non orthogonal locally refined quadrilateral meshes the second order convergence rate on non smooth meshes is verified with numerical examples

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity