on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

a modified weak galerkin finite element method for a class of parabolic problems in this paper we consider the solution of parabolic equation using the modified weak galerkin finite element procedure which is named as mwg fem based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties in which the classical gradient operator is replaced by a modified weak gradient operator optimal order error estimates in a discrete l2 norm and h1 norm are established for the corresponding modified weak galerkin finite element solutions finally we numerically verify the convergence theory for the mwg fem through some examples

a multigrid method for the pseudostress formulation of stokes problems the purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator mathcal a gamma mathbf grad mathbf div over 2 times 2 matrix valued functions this system is derived from the pseudostress velocity formulation z cai b lee and p wang siam j numer anal 42 2004 pp 843 859 of two dimensional stokes problems through the penalty method or natural time discretization for the respective steady or unsteady state problems here gamma 0 is a constant associated with either the penalty parameter or the time step size and mathcal a is a singular map in this paper we develop a multigrid solver for the discrete problem using both the raviart thomas rt and the brezzi douglas marini bdm finite elements we show that the multigrid convergence rate is o frac 1 gamma 2 1 gamma 2 m where m is the number of smoothings this convergence rate is independent of the mesh size and the number of levels used in multigrid moreover numerical results presented in this paper show that the multigrid convergence rate for the bdm elements does not depend on the parameter gamma

stabilized finite element methods for the velocity pressure stress formulation of incompressible flows nonlinear model formulated in terms of velocity pressure and the extra stress tensor the incompressible navier stokes is discretized by stabilized finite element method the stabilized method proposed is analyzed for the full nonlinear model and is applicable to various combinations of interpolation functions including the simplest equal order linear and bilinear elements

fully discrete stabilized multiphysics finite element method for the polymer gel model abstract in this paper we propose a fully discrete stabilized multiphysics finite element method by the pressure projection method to solve the polymer gel model to reveal the multi physical process of the polymer gel model we reformulate the polymer gel model by introducing a new variable to get a new multi field problem then we propose the fully discrete stabilized multiphysics finite element method using the simplest equal order element of p 1 p 1 p 1 without the discrete inf sup condition to solve the new reformulated system and we prove that the new method not only has no locking phenomenon but also preserves some energy conservation laws also we prove that the stabilized multiphysics finite element method has optimal convergence order finally we give the numerical example to verify the theoretical results

on inf sup stabilized finite element methods for transient problems we consider the behavior of inf sup stabilization in the context of transient problems with multiple time scales our motivation for studying this setting is provided by reacting flows problems for which small time steps are necessary in the integration process we show that for algorithms defined through a process wherein spatial and temporal discretizations are separated the coupling of implicit time integration with spatial inf sup stabilization may lead to anomalous pressure behavior including the onset of spurious oscillations for very small time steps effectively this coupling introduces a stability criterion resulting in a dependence between the spatial grid size and the time step we illustrate our theoretical results by numerical examples that demonstrate the stability criterion 2004 elsevier b v all rights reserved

incompressible flow past a circular cylinder dependence of the computed flow field on the location of the lateral boundaries the influence of the location of the lateral boundaries on 2d computation of unsteady incompressible flow past a circular cylinder is investigated the case of reynolds number 100 is used as a benchmark and several quantities characterizing the unsteady flow are obtained for a range of lateral boundary locations the computations are performed with two distinct finite element formulations space time velocity pressure formulation and velocity pressure stress formulation we conclude that the distance between the cylinder and the lateral boundaries can have a significant effect on the strouhal number and other flow quantities the minimum distance at which this influence vanishes has been found to be larger than what is commonly assumed

weak galerkin method for the biot s consolidation model abstract we develop a weak galerkin wg finite element method for the biot s consolidation model in the classical displacement pressure two field formulation weak galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations backward euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme we study the well posedness of the linear system at each time step and also derive the overall optimal order convergence of the wg formulation such wg scheme is designed on general shape regular polytopal meshes and provides stable and oscillation free approximation for the pressure without special treatment numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak galerkin finite element method

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

a modified weak galerkin finite element method in this paper we introduce a new discrete weak gradient operator and a new weak galerkin wg finite element method for second order poisson equations based on this new operator this newly defined discrete weak gradient operator allows us to use a single stabilizer which is similar to the one used in the discontinuous galerkin dg methods without having to worry about choosing a sufficiently large parameter in addition we will establish the optimal convergence rates and validate the results with numerical examples

weak galerkin finite element methods for darcy flow anisotropy and heterogeneity this paper presents a family of weak galerkin finite element methods wgfems for darcy flow computation the wgfems are new numerical methods that rely on the novel concept of discrete weak gradients the wgfems solve for pressure unknowns both in element interiors and on the mesh skeleton the numerical velocity is then obtained from the discrete weak gradient of the numerical pressure the new methods are quite different than many existing numerical methods in that they are locally conservative by design the resulting discrete linear systems are symmetric and positive definite and there is no need for tuning problem dependent penalty factors we test the wgfems on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in darcy flow

an a posteriori error estimator for the weak galerkin least squares finite element method abstract in this paper we derive an a posteriori error estimator for the weak galerkin least squares wg ls method applied to the reaction diffusion equation we show that this estimator is both reliable and efficient allowing it to be used for adaptive refinement due to the flexibility of the wg ls discretization we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh finally we present numerical experiments that confirm the effectiveness of the estimator and demonstrate the robustness and efficiency of the proposed adaptive wg ls approach

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

least squares galerkin methods for parabolic problems i semidiscretization in time this is the first part of a series of papers on least squares galerkin methods for parabolic initial boundary value problems these methods are based on the minimization of a least squares functional for an equivalent first order system over space and time with respect to suitable discrete spaces this paper presents the derivation and analysis of one step methods for semi discretization in time from least squares principles for linear parabolic problems one of the most important features of the least squares methodology is the built in a posteriori estimate for the approximation error this is a consequence of the equivalence of the least squares functional to the consistency error associated with a time step measured in an appropriate norm for the presentation in this paper we focus our attention on the specific combination of piecewise linear not necessarily continuous functions with continuous piecewise linear functions for the flux and scalar variable respectively for the resulting method a convergence result is shown for the scalar variable and the associated flux

an analysis of a weak galerkin finite element method for stationary navier stokes problems abstract in this article we present and analyze a weak galerkin finite element method for stationary navier stokes problems this weak galerkin finite element scheme is based on a shape regular partition consisting of arbitrary polygons polyhedra we first establish a discrete embedding inequality that is useful in weak galerkin finite element analysis for nonlinear problems then the stability and unique existence are proved for the discrete velocity and pressure by means of a discrete inf sup condition furthermore we derive the optimal error estimates for velocity approximation in the discrete h 1 norm and pressure approximation in the l 2 norm respectively numerical examples are provided that corroborate the optimal convergence of the proposed method

stabilized finite element methods for the velocity pressure stress formulation of incompressible flows abstract formulated in terms of velocity pressure and the extra stress tensor the incompressible navier stokes equations are discretized by stabilized finite element methods the stabilized methods proposed are analyzed for a linear model and extended to the navier stokes equations the numerical tests performed confirm the good stability characteristics of the methods these methods are applicable to various combinations of interpolation functions including the simplest equal order linear and bilinear elements

a taxonomy of consistently stabilized finite element methods for the stokes problem stabilized mixed methods can circumvent the restrictive inf sup condition without introducing penalty errors for properly chosen stabilization parameters these methods are well posed for all conforming velocity pressure pairs however their variational forms have widely varying properties first stabilization offers a choice between weakly or strongly coercive bilinear forms that give rise to linear systems with identical solutions but very different matrix properties second coercivity may be conditional upon a proper choice of a stabilizing parameter here we focus on how these two aspects of stabilized methods affect their accuracy and efficient iterative solution we present results that indicate a preference of krylov subspace solvers for strongly coercive formulations stability criteria obtained by finite element and algebraic analyses are compared with numerical experiments while for two popular classes of stabilized methods sufficient stability bounds correlate well with numerical stability our experiments indicate the intriguing possibility that the pressure stabilized galerkin method is unconditionally stable

all principal congruence link groups abstract we enumerate all the principal congruence link complements in s 3 thereby answering a question of w thurston

the arbitrary order mimetic finite difference method for a diffusion equation with a non symmetric diffusion tensor abstract we present the arbitrary order mimetic finite difference mfd discretization for the diffusion equation with non symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes the diffusion tensor is assumed to be positive definite the asymmetry of the diffusion tensor requires changes to the standard mfd construction we present new approach for the construction that guarantees positive definiteness of the non symmetric mass matrix in the space of discrete velocities the numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme for higher orders schemes we observed super convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor the new scheme was also tested on a time dependent problem modeling the hall effect in the resistive magnetohydrodynamics

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

a new modified weak galerkin finite element scheme for solving the stationary stokes equations abstract in this paper a modified weak galerkin method is proposed for the stokes problem the numerical scheme is based on a novel variational form of the stokes problem the degree of freedoms in the modified weak galerkin method is less than that in the original weak galerkin method while the accuracy stays the same in this paper the optimal convergence orders are given and some numerical experiments are presented to verify the theory

a weak galerkin finite element method for the navier stokes equations abstract this paper introduces a weak galerkin wg finite element method for the navier stokes equations in the primal velocity pressure formulation optimal order error estimates are established for the corresponding numerical approximations it must be emphasized that the wg finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular numerical experiments are presented to support the theoretical results

a weak galerkin finite element method for burgers equation abstract we propose a weak galerkin wg finite element method for solving the one dimensional burgers equation based on a new weak variational form both semi discrete and fully discrete wg finite element schemes are established and analyzed we prove the existence of the discrete solution and derive the optimal order error estimates in a discrete h 1 norm and the standard l 2 norm respectively numerical experiments are presented to illustrate our theoretical analysis

accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities the accuracy of the darcy velocity flux and stream function computed from lowest order triangle based control volume and mixed finite element approximations to the two dimensional pressure equation is considered the control volume finite element method similar to integrated finite difference methods and analogous to the interpolation of galerkin finite element results over control volumes is shown to yield a conservative velocity field and smooth streamlines the streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method which approximates the pressure and velocity variables separately it is shown that for systems with only moderate degrees of heterogeneity the control volume finite element method is the more computationally efficient alternative i e it provides more accurate flow results for a given number of unknowns for more variable or discontinuous permeability fields by contrast such as sand shale systems the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns

free surface flow simulations in the presence of inclined walls difficulties associated with free surface finite element flow simulations are related to a nonlinear and advective nature of most hydrodynamic flows b requirements for compatibility between velocity and pressure interpolation c maintaining a valid computational mesh in the presence of moving boundaries and d enforcement of the kinematic conditions at the free surface focusing on the last issue we present an extension of the free surface elevation equation to cases where the prescribed direction of the surface node motion is not uniformly vertical the resulting hyperbolic generalized elevation equation is discretized using a galerkin least squares formulation applied on the surface mesh the elevation field so obtained is then used to impose displacement boundary conditions on the elastic mesh update scheme that governs the movement of interior mesh nodes the proposed method is used to solve a two dimensional problem of sloshing in a trapezoidal tank and a three dimensional application involving flow in a trapezoidal channel with bridge supports

the weak galerkin finite element method for the transport reaction equation abstract we present and analyze a weak galerkin finite element method for solving the transport reaction equation in d space dimensions this method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon polyhedra we derive the l 2 error estimate of o h k 1 2 order for the discrete solution when the kth order polynomials are used for k 0 moreover for a special class of meshes we also obtain the optimal error estimate of o h k 1 order in the l 2 norm a derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given numerical examples on compatible and non compatible meshes are provided to show the effectiveness of this weak galerkin method

an effective implementation for stokes equation by the weak galerkin finite element method abstract in this paper we introduce and analyze the schur complement technique to a weak galerkin wg for short finite element method for solving stokes equation due to the special structure of the weak functions the main idea of schur decomposition method is to express the interior velocity functions defined inside partition elements in terms of the velocity functions defined on the partition boundary and the pressure functions this can largely reduce the degree of freedom of the derived discrete linear system we investigate the well posedness of the solution generated from schur complement and its coincidence with the standard wg method finally we exhibit some numerical experiments to illustrate the theoretical analysis

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

applications of a computer implementation of poincare s theorem on fundamental polyhedra poincarcs theorem asserts that a group f of isometries of hyperbolic space h is discrete if its generators act suitably on the boundary of some polyhedron in h and when this happens a presentation of f can be derived from this action we explain methods for deducing the precise hypotheses of the theorem from calculation in f when f is algorithmi cally defined and we describe a file of fortran programs that use these methods for groups f acting on the upper half space model of hyperbolic 3 space h we exhibit one modest example of the application of these programs and we summarize computations of repesenta tions of groups psl 2 0 where 6 is an order in a complex quadratic number field in the early 1880 s h poincare discovered a general theorem allowing one to deduce the discreteness of and a presentation for a group g of isometries of hyperbolic space from its action on a hyperbolic polyhedron under certain condi tions theorems of this sort are part of the foundations of his theories of fuchsian and kleinian groups that have become very popular again and h seifert has recently given us a modern proof of a fairly general version of poincare s theorem in 12 see also 7 this theorem has been little used over the past century perhaps partly because its hypotheses have seemed very difficult to verify for a given group g except in very special circumstances one reason for doubting that poincare s theorem is unreasonably difficult to apply to fairly general discrete groups is that no general alternative method for accomplishing its tasks has been proposed the present paper is devoted to demonstrating that poincare s theorem can indeed be applied to given groups in apparently difficult cases and that much of the work can be done by a computer our experience suggests that the theorem is really very helpful in guiding the user to an understanding of the details of the action of g starting from a state of near ignorance an outline of this paper is as follows in section 1 we begin by stating seifert s version of poincare s theorem and explaining how we would apply it to an algorithmically defined group g of isometries of hyperbolic space h o in section 2 we specialize to the situation 00 for which we wrote our file poincare of fortran

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

singular integrals on self similar sets and removability for lipschitz harmonic functions in heisenberg groups in this paper we study singular integrals on small that is measure zero and lower than full dimensional subsets of metric groups the main examples of the groups we have in mind are euclidean spaces and heisenberg groups in addition to obtaining results in a very general setting the purpose of this work is twofold we shall extend some results in euclidean spaces to more general kernels than previously considered and we shall obtain in heisenberg groups some applications to harmonic in the heisenberg sense functions of some results known earlier in euclidean spaces

weak galerkin finite element method for viscoelastic wave equations abstract in this article we consider a weak galerkin finite element method wg fem for solving one type of viscoelastic wave equation a discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme we introduce both the semi and fully discrete wg fems and establish the corresponding stability estimates optimal order error estimate in h 1 norm is derived numerical experiments are preformed to verify the theoretical results

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

stabilized finite element method for incompressible flows with high reynolds number in the following paper we discuss the exhaustive use and implementation of stabilization finite element methods for the resolution of the 3d time dependent incompressible navier stokes equations the proposed method starts by the use of a finite element variational multiscale vms method which consists in here of a decomposition for both the velocity and the pressure fields into coarse resolved scales and fine unresolved scales this choice of decomposition is shown to be favorable for simulating flows at high reynolds number we explore the behaviour and accuracy of the proposed approximation on three test cases first the lid driven square cavity at reynolds number up to 50 000 is compared with the highly resolved numerical simulations and second the lid driven cubic cavity up to re 12 000 is compared with the experimental data finally we study the flow over a 2d backward facing step at re 42 000 results show that the present implementation is able to exhibit good stability and accuracy properties for high reynolds number flows with unstructured meshes

the weak galerkin finite element method for incompressible flow abstract we study the weak galerkin finite element method for stationary navier stokes problem we propose a weak finite element velocity pressure space pair that satisfies the discrete inf sup condition this space pair is then employed to construct a stable weak galerkin finite element scheme without adding any stabilizing term or penalty term we prove a discrete embedding inequality on the weak finite element space which together with the discrete inf sup condition enables us to establish the unique existence and stability estimates of the discrete velocity and pressure then we derive the optimal error estimates for velocity and pressure approximations in the h 1 norm and l 2 norm respectively numerical experiments are provided to illustrate the theoretical analysis

the weak galerkin finite element method for the symmetric hyperbolic systems abstract in this paper we present and analyze a weak galerkin finite element wg method for solving the symmetric hyperbolic systems this method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other by introducing special weak derivative we construct a stable weak galerkin scheme and derive the optimal l 2 error estimate of o h k 1 2 order for the discrete solution when the k order polynomials are used for k 0 as application we discuss this wg method for solving the singularly perturbed convection diffusion reaction equation and derive an e uniform error estimate of order k 1 2 numerical examples are provided to show the effectiveness of the proposed wg method

machine learning and transport simulations for groundwater anomaly detection abstract this paper presents studies on modeling and algorithms for groundwater anomaly detection specifically conductivity along with four other surrogates are used for identifying anomaly in groundwater the one class support vector machine 1 svm technique is utilized for model training and real data from colorado water watch is used for testing the model and algorithms design of code modules in python is briefly discussed since groundwater contamination rarely happens in reality we also use synthetic data from numerical simulations of flow and transport in porous media to test this model for anomaly detection

a stable weak galerkin finite element method for stokes problem abstract we study the weak galerkin finite element method for stokes problem a new weak galerkin finite element velocity pressure space pair is presented which satisfies the discrete inf sup condition based on this space pair we establish a stable weak galerkin approximation scheme without adding any stability term or penalty term then we further derive the optimal error estimates for velocity and pressure approximations respectively numerical experiments are provided to illustrate the theoretical analysis

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a locally conservative finite element method based on piecewise constant enrichment of the continuous galerkin method this paper presents a locally conservative finite element method based on enriching the approximation space of the continuous galerkin method with elementwise constant functions the proposed method has a smaller number of degrees of freedom than the discontinuous galerkin method numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method we also present a theoretical analysis of the method and establish optimal convergence of numerical solutions

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

a new weak galerkin finite element scheme for general second order elliptic problems abstract a new weak galerkin wg finite element scheme is presented for general second order elliptic problems in this paper in this new scheme a skew symmetric form has been used for handling the convection term the advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well posedness of the system in this scheme the wg elements are designed to have the form of p k t p k 1 e that is we choose the polynomials of degree k 1 on each element and the polynomials of degree k 1 on the edge face of each element as a result fewer degrees of freedom are generated in the new wg finite element scheme it is also worth pointing out that the wg finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons polyhedra which are shape regular optimal order error estimates are presented for the corresponding numerical approximation in various norms some numerical results are reported to confirm the theory

a weak finite element method for elliptic problems in one space dimension we present a weak finite element method for elliptic problems in one space dimension our analysis shows that this method has more advantages than the known weak galerkin method proposed for multi dimensional problems for example it has higher accuracy and the derived discrete equations can be solved locally element by element we derive the optimal error estimates in the discrete h1 norm the l2 norm and l norm respectively moreover some superconvergence results are also given finally numerical examples are provided to illustrate our theoretical analysis

artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows this note investigates the so called arti cial added mass e ect which is responsible for devastating instabilities within sequentially staggered fluid structure interaction fsi simulations where incompressible fluids are considered a discrete representation of the added mass operatorma is given and instability conditions are evaluated for different temporal discretisation schemes it is proven that for every sequentially staggered scheme and given spatial discretisation of a problem a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable the analysis is quite general and does not depend upon the particular spatial discretisation schemes used however here special attention is put on stabilised finite elements employed on the fluid partition numerical investigations further highlight the results

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

superconvergence of a modified weak galerkin approximation for second order elliptic problems by l2 projection method abstract this paper derives a superconvergence result for the modified weak galerkin mwg finite element method of the second order elliptic problem the convergence rate of the mwg approximation is improved by 30 after applying a low cost l 2 projection post processing technique these superconvergence phenomena are proved theoretically and confirmed numerically

split least squares finite element methods for linear and nonlinear parabolic problems in this paper we propose some least squares finite element procedures for linear and nonlinear parabolic equations based on first order systems by selecting the least squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub procedures one of which is for the primary unknown variable u and the other is for the expanded flux unknown variable s optimal order error estimates are developed finally we give some numerical examples which are in good agreement with the theoretical analysis

computing arithmetic kleinian groups arithmetic kleinian groups are arithmetic lattices in psl 2 c we present an algorithm which given such a group gamma returns a fundamental domain and a finite presentation for gamma with a computable isomorphism

a lowest order weak galerkin finite element method for stokes flow on polygonal meshes abstract this paper presents a lowest order weak galerkin wg finite element method for solving the stokes equations on convex polygonal meshes constant vectors are used separately in element interiors and on edges to approximate fluid velocity whereas constant scalars are used on elements to approximate the pressure for the constant vector basis functions their discrete weak gradients are established in a matrix space that is based on the c w 0 space chen and wang 2017 whereas their discrete weak divergences are calculated as elementwise constants to circumvent the saddle point problem a reduced scheme for velocity is established by using three types of basis functions for the discretely divergence free subspace a procedure for subsequent pressure recovery is also developed error analysis along with numerical experiments on benchmarks are presented to demonstrate accuracy and efficiency of the proposed new method

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

gls and evss methods for a three field stokes problem arising from viscoelastic flows in this paper order one finite elements together with galerkin least squares gls methods are used for solving a three held stokes problem arising from the numerical study of viscoelastic hows stability and convergence results are established even when the solvent viscosity is small compared to the viscosity due to the polymer chains an iterative algorithm decoupling velocity pressure and stress calculations is proposed the link with the modified elastic viscous split stress evss method studied in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 r guenette m fortin j non newtonian fluid mech 60 1995 27 52 is presented numerical results are in agreement with theoretical predictions and with those presented in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 c 2001 elsevier science b v all rights reserved

analyse numerique des ecoulements quasi newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau we prove abstract error estimates for the approximation of the velocity and the pressure by a mixed fem of quasi newtonian flows whose viscosity obeys the power law or the carreau law these estimates are optimal in some cases they can be applied to most finite elements used for the solution of stokes s problem on prouve des estimations d erreur abstraites pour l approximation de la vitesse et la pression par une mef mixtes d ecoulements quasi newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau ces estimations sont optimales dans certains cas elles peuvent etre appliquees a la plupart des elements finis utilises pour la resolution du probleme de stokes

an h 1 galerkin mixed finite element method for parabolic partial differential equations in this paper an h1 galerkin mixed finite element method is proposed and analyzed for parabolic partial differential equations with nonselfadjoint elliptic parts compared to the standard h1 galerkin procedure c1 continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method moreover it is shown that the finite element approximations have the same rates of convergence as in the classical mixed method but without lbb consistency condition and quasiuniformity requirement on the finite element mesh finally a better rate of convergence for the flux in l2 norm is derived using a modified h1 galerkin mixed method in two and three space dimensions which confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed method under extra regularity assumptions on the exact solution

a modified weak galerkin finite element method for the stokes equations a modified weak galerkin mwg finite element method is introduced for the stokes equations in the primary velocity pressure formulation this method is based on the weak galerkin wg finite element method developed recently in wang and ye 0000 the unknowns associated with element boundaries in the wg method have been eliminated in the mwg method as the result the mwg method has fewer degrees of freedom optimal order error estimates are established for the corresponding numerical approximation in various norms numerical results are presented to confirm the theory

a finite volume method for stokes problems on quadrilateral meshes abstract we present a finite volume method for stokes problems using the isoparametric q 1 q 0 element pair on quadrilateral meshes to offset the lack of the i n f s u p condition a jump term of discrete pressure stabilizing term is added to the continuity approximation equation thus we establish a stabilized finite volume scheme on quadrilateral meshes then based on some superclose estimates we derive the optimal error estimates in the h 1 and l 2 norms for velocity and in the l 2 norm for pressure respectively numerical examples are provided to illustrate our theoretical analysis we emphasize that our work is the first time to propose and analyze a finite volume method for stoke problems using isoparametric elements on quadrilateral meshes

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

a coupling of weak galerkin and mixed finite element methods for poroelasticity abstract in this paper we present a coupling of a weak galerkin method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase in poroelasticity equation because our method provides a stable element combination for the mixed linear elasticity problem it can avoid the poroelasticity locking mathematically indeed uniform in time error estimates of all the unknowns are obtained for both semidiscrete scheme and fully discrete scheme without assuming that the constrained specific storage coefficient is uniformly positive compared with the method proposed by jeonghun j lee our method can get the arbitrary convergence order by changing degree of the polynomial as for the computational cost we can use the modified weak galerkin method to eliminate the boundary term and reduce the number of unknowns in addition we do not use gronwall inequality in the error analysis finally the numerical experiments verify the theoretical analysis and show the effectiveness to overcome spurious pressure oscillations

a multiscale finite element method for the incompressible navier stokes equations this paper presents a new multiscale finite element method for the incompressible navier stokes equations the proposed method arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales modeling of the unresolved scales corrects the lack of stability of the standard galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields specifically the equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine scale problem a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed convergence studies for the method on uniform skewed as well as composite meshes are presented numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method 2005 elsevier b v all rights reserved

analysis of piecewise linear approximations to the generalized stokes problem in the velocity stress pressure formulation in this paper we study continuous piecewise linear polynomial approximations to the generalized stokes equations in the velocity stress pressure first order system formulation by using a cell vertex finite volume least squares scheme this method is composed of a direct cell vertex finite volume discretization step and an algebraic least squares step where the least squares procedure is applied after the discretization process is accomplished this combined approach has the advantages of both finite volume and least squares approaches an error estimate in the h1 product norm for continuous piecewise linear approximating functions is derived it is shown that with respect to the order of approximation for h2 regular exact solutions the method exhibits an optimal rate of convergence in the h1 norm for all unknowns velocity stress and pressure

a weak galerkin finite element method for solving nonlinear convection diffusion problems in two dimensions abstract we study weak galerkin wg finite element method fem for solving nonlinear convection diffusion problems a wg finite element scheme is presented based on a new variational form we prove the energy conservation law and stability of the continuous time wg fem in particular optimal order error estimates are established for the wg fem approximation in both a discrete h1 norm and l2 norm numerical experiments are performed to confirm the theoretical results

boundedness of singular integrals on c1 α intrinsic graphs in the heisenberg group abstract we study singular integral operators induced by 3 dimensional calderon zygmund kernels in the heisenberg group we show that if such an operator is l 2 bounded on vertical planes with uniform constants then it is also l 2 bounded on all intrinsic graphs of compactly supported c 1 functions over vertical planes in particular the result applies to the operator r induced by the kernel k z h z 2 z h 0 the horizontal gradient of the fundamental solution of the sub laplacian the l 2 boundedness of r is connected with the question of removability for lipschitz harmonic functions as a corollary of our result we infer that the intrinsic graphs mentioned above are non removable apart from subsets of vertical planes these are the first known examples of non removable sets with positive and locally finite 3 dimensional measure

a comparative study on the weak galerkin discontinuous galerkin and mixed finite element methods this paper presents a comparative study on the newly introduced weak galerkin finite element methods wgfems with the widely accepted discontinuous galerkin finite element methods dgfems and the classical mixed finite element methods mfems for solving second order elliptic boundary value problems we examine the differences similarities and connection among these methods in scheme formulations implementation strategies accuracy and computational cost the comparison and numerical experiments demonstrate that wgfems are viable alternatives to mfems and hold some advantages over dgfems due to their properties of local conservation normal flux continuity no need for penalty factor and definiteness of discrete linear systems

an absolutely stable pressure poisson stabilized finite element method for the stokes equations the pressure poisson stabilized galerkin method for the stokes equation requires the choice of a positive parameter existing theoretical predictions for the range of parameter values that yield stable discretizations seem to be very pessimistic when compared to the computational evidence motivated by this wide gap we first examine a continuous prototype for this class of schemes we show that the prototype is absolutely stable i e it is stable for all parameter values and is optimally accurate we then define a new practical variant of the well known pressure poisson stabilized scheme we prove that the new method is absolutely stable just like its continuous prototype and that it achieves optimal convergence rates with respect to the same mesh independent norms the new method differs from the standard pressure poisson stabilized method in several important aspects first its definition does not degrade to a penalty formulation for the lowest order nodal spaces second the method is absolutely stable with respect to the natural norm for the problem while the standard pressure poisson stabilized method is stable with respect to a mesh dependent norm

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

on the uniform rectifiability of ad regular measures with bounded riesz transform operator the case of codimension 1 we prove that if is a d dimensional ahlfors david regular measure in mathbb r d 1 then the boundedness of the d dimensional riesz transform in l2 implies that the non baup david semmes cells form a carleson family combined with earlier results of david and semmes this yields the uniform rectifiability of

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties