multi level monte carlo weak galerkin method with nested meshes for stochastic brinkman problem abstract this paper is devoted to the numerical analysis of a multi level monte carlo weak galerkin mlmcwg approximation with nested meshes for solving stochastic brinkman equations with two dimensional spatial domain with weak gradient operator and a stabilizer at hand the weak galerkin wg technique is a high order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries flows with jump fluid viscosity coefficients or high contrast permeability fields given by each sample the multi level monte carlo mlmc technique with nested meshes balances the sampling error and the spatial approximation error where the computational cost can be sharply reduced to log linear complexity with respect to the degree of freedom in spatial direction the nested meshes requirement is introduced here in order to simplify the analysis which can be generalized to mlmc with non nested meshes error estimates are derived in terms of the spatial meshsize and the number of samples the numerical tests are provided to illustrate the behavior of the mlmcwg method and verify our theoretical results regarding optimal convergence of the approximate solutions

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 monte carlo study of ant traffic in a uni directional trail abstract in this study we aim to investigate ant traffic in the uni directional ant trail we consider two types of ants moving in the trail one of which smells well and the other does not the theoretical base of the study is similar to that of the nagel schreckenberg nasch model but we do not use the exclusion rule the asymmetrical exclusion rule is employed instead ants are placed on the trail as mixed by keeping the number of poor smelling ants constant the traffic in the trail is studied as a function of the number of good smelling ants and the evaporation rate probability of pheromone f the fundamental physical quantities i e mean speed v and flux f interestingly show non monotonic density dependence for some values of f at some densities

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

modeling of soil water retention from saturation to oven dryness most analytical formulas used to model moisture retention in unsaturated porous media have been developed for the wet range and are unsuitable for applications in which low water contents are important we have developed two models that fit the entire range from saturation to oven dryness in a practical and physically realistic way with smooth continuous functions that have few parameters both models incorporate a power law and a logarithmic dependence of water content on suction differing in how these two components are combined in one model functions are added together model sum in the other they are joined smoothly together at a discrete point model junction both models also incorporate recent developments that assure a continuous derivative and force the function to reach zero water content at a finite value of suction that corresponds to oven dryness the models have been tested with seven sets of water retention data that each cover nearly the entire range the three parameter sum model fits all data well and is useful for extrapolation into the dry range when data for it are unavailable the two parameter junction model fits most data sets almost as well as the sum model and has the advantage of being analytically integrable for convenient use with capillary bundle models to obtain the unsaturated hydraulic conductivity

polynomial preserving recovery for a class of weak galerkin finite element methods abstract this paper presents the polynomial preserving recover ppr postprocessing technique for the weak galerkin wg finite element methods on triangular meshes the proposed technique involves a fine tuning parameter in a stabilizer that improves the convergence order of the finite element methods the supercloseness between the lagrangian interpolation and the wg solution is analyzed which leads to the main result about the superconvergence in the gradient of the wg solution numerical results are presented to illustrate the theoretical analysis

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

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

a locking free weak galerkin finite element method for elasticity problems in the primal formulation this paper presents an arbitrary order locking free numerical scheme for linear elasticity on general polygonal polyhedral partitions by using weak galerkin wg finite element methods like other wg methods the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element such local problems are derived from weak formulations of the corresponding differential operators through integration by parts locking free error estimates of optimal order are derived in a discrete h 1 norm and the usual l 2 norm for the approximate displacement when the exact solution is smooth numerical results are presented to demonstrate the efficiency accuracy and the locking free property of the weak galerkin finite element method

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

a locking free numerical approximation for quasilinear poroelasticity problems abstract this paper is devoted to a new formulation which couples the weak galerkin method and the finite element method for approximating solutions of the equations of quasi static poroelasticity which model flow through elastic porous media it is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium the steady state version of the system is recast in terms of displacement pressure and volumetric stress and the well posedness of both the continuous and discrete three field formulations is proved error estimates for the proposed numerical method are obtained these show that the method is locking free numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method

effective implementation of the weak galerkin finite element methods for the biharmonic equation abstract the weak galerkin wg methods have been introduced in mu et al 2013 2014 14 for solving the biharmonic equation the purpose of this paper is to develop an algorithm to implement the wg methods effectively this can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size in fact this reduced global system is equivalent to the schur complements of the wg methods the unknowns of the schur complement of the wg method are those defined on the element boundaries the equivalence of the wg method and its schur complement is established the numerical results demonstrate the effectiveness of this new implementation technique

a weak galerkin finite element scheme with boundary continuity for second order elliptic problems abstract a new weak galerkin wg finite element method for solving the second order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed the wg method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces different combination gives rise to different weak galerkin finite element methods which makes wg methods highly flexible and efficient in practical computation this paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme yet without losing the accuracy of the numerical approximation error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper also presents some numerical experiments to demonstrate the power of the wg method the numerical results show that the wg method achieves the optimal convergence order with a relatively low computational cost which reveals a great promise of the flexibility and accuracy of the wg method

a hybrid discontinuous galerkin method for elliptic problems a new family of hybrid discontinuous galerkin methods is studied for second order elliptic equations our proposed method is a generalization of the cell boundary element cbe method y jeon and e j park appl numer math 58 2008 pp 800 814 which allows high order polynomial approximations our method can be viewed as a hybridizable discontinuous galerkin method b cockburn j gopalakrishnan and r lazarov siam j numer anal 47 2009 pp 1319 1365 using a bauman oden type local solver the method conserves the mass in each element and the average flux is continuous across the interelement boundary for even degree polynomial approximations optimal order error estimates measured in the energy norm are proved numerical examples are presented to show the performance of the method

discontinuous galerkin methods for the stokes equations with nonlinear damping term on general meshes abstract this paper presents the interior penalty discontinuous galerkin ipdg methods to solve the stationary stokes equations with nonlinear damping term the ipdg discrete schemes are established on general meshes the corresponding consistency and stabilization of those schemes are proved subsequently we analyze the existence boundedness and uniqueness of the discrete solutions then the optimal error estimates are derived in the l 2 norm and h 1 like dg norm for the velocity variable and l 2 norm for the pressure variable respectively finally some numerical experiments in two dimensional are reported to demonstrate the theoretical results and show the robustness of our discrete schemes

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

a symmetric weak galerkin method for solving non divergence form elliptic equations abstract we reformulate the second order non divergence form elliptic equations in a symmetric variation form i e a least squares form we design a weak galerkin finite element method for this high regularity formulation the optimal order of convergence is proved numerical results verify the theory in addition numerical results compared to the existing weak galerkin method for the non symmetric form and the new methods show advantage on the accuracy and the degree of freedoms

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

an immersed finite volume element method for 2d pdes with discontinuous coefficients and non homogeneous jump conditions an immersed finite volume element method is developed to solve 2d elliptic interface problems with a variable coefficient that has a finite jump across an interface the solution and the flux may also have a finite jump across the interface using the source removal technique an equivalent elliptic interface problem with homogeneous jump conditions is obtained the nodal basis functions are constructed to satisfy the homogeneous jump conditions near the interface and the usual finite element nodal basis functions are applied away from the interface the resulting linear problem is simple and easy to solve a proof of the error estimate in the energy norm is given numerical experiments demonstrate the convergence rates of the proposed method with the usual o h 2 in the l 2 the l norms and o h in the h 1 norm

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

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

an improved fractal equation for the soil water retention curve the water retention curve is important for predicting soil physical properties and processes until recently equations for the were empirical advances in fractal geometry have led to the derivation of physical models for the however both existing fractal equations have only two parameters and thus are relatively inflexible we derived a new three parameter fractal model for the this equation was fitted to 36 s for a silt loam soil with a wide range of structural conditions the new equation fitted these data much better than the existing equations the parameters of the new equation a d and d are physical entities corresponding to the air entry value tension draining the smallest pores and fractal dimension respectively estimates of the a d and d were physically reasonable with median values of 2 9 10 1 kpa 1 6 104 kpa and 2 87 respectively in contrast the existing equations yielded anomalous estimates of either a or d the new equation was able to fit for a variety of porous media including sandstone glass beads sands sieved soil and undisturbed soils ranging from very fine sandy loam to heavy clay the a and d were more sensitive to structural and textural variation than d the new equation represents an improvement over existing models in terms of both goodness of fit and the physical significance of its parameters

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

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

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

new discretization schemes for time harmonic maxwell equations by weak galerkin finite element methods abstract this paper introduces new discretization schemes for time harmonic maxwell equations in a connected domain by using the weak galerkin wg finite element method the corresponding wg algorithms are analyzed for their stability and convergence error estimates of optimal order in various discrete sobolev norms are established for the resulting finite element approximations

an efficient numerical scheme for the biharmonic equation by weak galerkin finite element methods on polygonal or polyhedral meshes this paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak galerkin wg finite element methods the wg finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual h 2 semi norm weak partial derivatives and their approximations called discrete weak partial derivatives are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra the discrete weak partial derivatives serve as building blocks for the wg finite element method the resulting matrix from the wg method is symmetric positive definite and parameter free an error estimate of optimal order is derived in an h 2 equivalent norm for the wg finite element solutions error estimates in the usual l 2 norm are established yielding optimal order of convergence for all the wg finite element algorithms except the one corresponding to the lowest order i e piecewise quadratic elements some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme

traffic flow cellular automaton order parameter and its conjugated field we use a cellular automaton traffic model in order to study a nonequilibrium phase transition we define an order parameter and show that its conjugated field is a parameter of randomness of the model we analyze the symmetries of the free unbroken and of the jammed broken phases our results are consistent with a second order phase transition at p 0 nontrivial critical exponents have also been obtained

mixed virtual element methods for elastodynamics with weak symmetry abstract we propose and analyze a mixed virtual element method for linear elastodynamics in velocity stress formulation with weak symmetry in this formulation the symmetry of the stress is relaxed by the rotation of the displacement and the system of second order differential equation in time is reduced to first order differential equations in time by introducing velocity the proposed method uses h div conforming virtual element space of order k k 1 for the stress and discontinuous piecewise polynomial spaces of degree k for the velocity and rotation for time discretization we use the crank nicolson scheme both semidiscrete and fully discrete error estimates are robust for nearly incompressible materials numerical experiments confirm our theoretical predictions

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

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

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

vibration control for a nonlinear three dimensional flexible manipulator trajectory tracking abstractthis paper proposes an innovative approach to the trajectory tracking in three dimensional space and vibration control problems in the presence of a nonlinear three dimensional flexible manipulator based on the partial differential equation model unlike two dimensional plane we select spherical coordinates to describe the position of the end point in three dimensional space this novel approach makes it possible to realise the trajectory tracking by controlling the two angles in spherical coordinates meanwhile a vibration control scheme is proposed to restrain vibrations in addition the existence and uniqueness of solutions are demonstrated finally the performance of the desired trajectory tracking the proposed vibration control scheme and their convergence properties are demonstrated by numerical simulations

three parameter lognormal distribution model for soil water retention many models for soil water retention have been proposed however most of these models are curve fitting equations and do not emphasize the physical significance of their empirical parameters a new retention model that exhibits increased flexibility was developed by applying three parameter lognormal distribution laws to the pore radius distribution function r and to the water capacity function which was taken to be the pore capillary pressure distribution function this model contains three parameters that are closely related to the statistics of the bubbling pressure c the mode 0 of and the standard deviation of transformed by comparison of this model with three existing models the van genuchten model the brooks corey model and the modified tani model it was shown that c 0 and are all essential for a general retention model

modeling of steep layers in singularly perturbed diffusion reaction equation via flexible fine scale basis abstract we present a new stabilized method for the diffusion reaction equation which develops sharp boundary and or internal layers for the reaction dominated case i e singularly perturbed case the method relies on an improved expression for the stabilization parameter that is derived via the fine scale variational formulation facilitated by the variational multiscale vms framework the proposed fine scale basis consists of enrichment functions which may be nonzero at element edges the derived stabilization parameter enjoys spatial variation over element interiors and along inter element boundaries that helps model the rapid variation of the residual of the euler lagrange equations over the domain this feature also facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects the method is able to better satisfy the maximum principle as compared to other existing methods we present a priori mathematical analysis of the stability and convergence of the method for the diffusion reaction equation optimal convergence on meshes comprised of linear triangles and bilinear quadrilateral elements are presented for smooth problems as well as for problems with steep boundary layers stability and accuracy features of the method for problems with discontinuous forcing function internal layers and boundary layers are shown and its performance on unstructured and distorted meshes comprised of quadrilateral and triangular elements is highlighted

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

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

a lowest order weak galerkin method for linear elasticity abstract the lowest order weak galerkin wg method is considered for linear elasticity based on the displacement formulation the new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution a priori error estimates of optimal order in the discrete h 1 and l 2 norms for the displacement are proved when the solution is smooth the error estimates are independent of the lame constant thus the performance of the new method does not deteriorate as the elastic material becomes incompressible further a simple post processing technique to obtain a numerical approximation of the stress is presented a careful error analysis reveals that the l 2 norm error in the stress is also optimal and independent of several numerical experiments confirm the locking free property and optimal convergence rates of the new method

traffic model with an absorbing state phase transition we consider a modified nagel schreckenberg ns model in which drivers do not decelerate if their speed is smaller than the headway number of empty sites to the car ahead in the original ns model such a reduction in speed occurs with probability p independent of the headway as long as the current speed is greater than zero in the modified model the free flow state with all vehicles traveling at the maximum speed v max is absorbing for densities smaller than a critical value c 1 v max 2 the phase diagram in the p plane is reentrant for densities in the range c c both small and large values of p favor free flow while for intermediate values a nonzero fraction of vehicles have speeds v max in addition to representing a more realistic description of driving behavior this change leads to a better understanding of the phase transition in the original model our results suggest an unexpected connection between traffic models and stochastic sandpiles

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

phase coexistence induced by a defensive reaction in a cellular automaton traffic flow model abstract traffic flow modeling is an elusive example for the emergence of complexity in dynamical systems of interacting objects in this work we introduce an extension of the nagel schreckenberg nasch model of vehicle traffic flow that takes into account a defensive driver s reaction such a mechanism acts as an additional nearest neighbor coupling the defensive reaction dynamical rule consists in reducing the driver s velocity in response to deceleration of the vehicle immediately in front of it whenever the distance is smaller than a security minimum this new mechanism when associated with the random deceleration rule due to fluctuations considerably reduces the mean velocity by adjusting the distance between the vehicles it also produces the emergence of bottlenecks along the road on which the velocity is much lower than the road mean velocity besides the two standard phases of the nasch model corresponding to the free flow and jammed flow the present model also exhibits an intermediate phase on which these two flow regimes coexist as it indeed occurs in real traffics these findings are consistent with empirical results as well as with the general three phase traffic theory

a stabilizer free weak galerkin finite element method on polytopal meshes abstract a stabilizing penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries removing stabilizers from discontinuous galerkin finite element methods will simplify formulations and reduce programming complexity significantly the goal of this paper is to introduce a stabilizer free weak galerkin wg finite element method for second order elliptic equations on polytopal meshes this new wg method keeps a simple symmetric positive definite form and can work on polygonal polyhedral meshes optimal order error estimates are established for the corresponding wg approximations in both a discrete h 1 norm and the l 2 norm numerical results are presented verifying the theorem

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

a primal discontinuous galerkin method with static condensation on very general meshes we propose an efficient variant of a primal discontinuous galerkin method with interior penalty for the second order elliptic equations on very general meshes polytopes with eventually curved boundaries efficiency especially when higher order polynomials are used is achieved by static condensation i e a local elimination of certain degrees of freedom cell by cell this alters the original method in a way that preserves the optimal error estimates numerical experiments confirm that the solutions produced by the new method are indeed very close to that produced by the classical one

simplified weak galerkin and new finite difference schemes for the stokes equation abstract this article presents a simplified formulation for the weak galerkin finite element method for the stokes equations without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak galerkin algorithm the simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions a particular application of the simplified weak galerkin on rectangular partitions yields a new class of 5 and 7 point finite difference schemes for the stokes equations an explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements error estimates of optimal order are established for the simplified weak galerkin finite element method in the h 1 and l 2 norms furthermore a superconvergence of order o h 3 2 is established on rectangular partitions for the velocity approximation in the h 1 norm at cell centers and a similar superconvergence is derived for the pressure approximation in the l 2 norm at cell centers some numerical results are reported to confirm the convergence and superconvergence theory