quadrilateral mesh revisited several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent the effect of the bi section condition on the degenerate mesh conditions is also checked

delay independent stability test for systems with multiple time delays delay independent stability dis of a general class of linear time invariant lti multiple time delay system mtds is investigated in the entire delay parameter space stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays we build an analytical approach that requires the inspection of the roots of finite number of single variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays excluding infinite delays the approach enables to test the necessary and sufficient conditions of dis of lti mtds technically known as weak dis as well as the robust stability of single delay systems against all variations in delay ratios the proposed approach which does not require any parameter sweeping and graphical display becomes possible by establishing a link between the infinite spectrum and algebraic geometry case studies are provided to show the effectiveness of the approach

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

superconvergence of fully discrete rectangular mixed finite element methods of parabolic control problems in this paper we investigate the superconvergence property of the numerical solution of a quadratic parabolic optimal control problem by using fully discrete mixed finite element methods the space discretization of the state variable is done using usual mixed finite elements whereas the time discretization is based on difference methods the state and the co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions we derive the superconvergence results for the control and the state approximation some numerical examples are presented to confirm the theoretical investigations

a weak galerkin finite element method with polynomial reduction the weak galerkin wg is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions in the implementation the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom the accuracy and the computational complexity of the corresponding wg scheme is significantly impacted by the selection of such polynomials this paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation for illustrative purpose the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction consequently a new weak galerkin finite element method is proposed and analyzed 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 presents some numerical results to demonstrate the power of the wg method in dealing with finite element partitions with arbitrary polygons in 2d or polyhedra in 3d the numerical examples include various finite element partitions such as triangular mesh quadrilateral mesh honeycomb mesh in 2d and mesh with deformed cubes in 3d

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

brief paper all stabilizing pid controllers for time delay systems this paper presents a method to compute the entire set of stabilizing pid controller parameters for an arbitrary including unstable linear time delay system the main contribution is to handle the infinite number of stability boundaries in the k d k i plane for a fixed proportional gain k p for retarded open loops it is shown that the stable region in the k d k i plane consists of convex polygons concerning neutral loops a new phenomenon is introduced for certain systems and certain k p the exact stable region in the k d k i plane can be described by the limit of a sequence of polygons with an infinite number of vertices this sequence may be well approximated by convex polygons moreover the paper describes a necessary condition for k p intervals potentially having a stable region in the k d k i plane thus the set of stabilizing controller parameters can be calculated after gridding k p in these intervals a matlab tool implementing the presented method is available

superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems in this paper we investigate superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems the presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables moreover the matching relation i e lbb condition between the mixed element spaces v h and wh is not necessary thus we can choose the approximation spaces more flexibly in order to derive the superconvergence we will use classical mixed finite element spaces the state and co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions at first we derive some superconvergence properties for the control variable the state variables and the adjoint state variables then using a recovery operator we obtain a superconvergence result for the control variable next combining energy approach with postprocessing method we derive a posteriori error estimates for optimal control problems we will show that the method does not involve the jump residuals which make the analysis simpler finally a numerical example is given to demonstrate the theoretical results on superconvergence

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

blossom quad a non uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm a new indirect way of producing all quad meshes is presented the method takes advantage of a wellknown algorithm of the graph theory namely the blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time the new blossom quad algorithm is compared with standard indirect procedures meshes produced by the new approach are better both in terms of element shape and in terms of size eld ef ciency

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

superconvergence of mixed covolume method for elliptic problems on triangular grids in this paper we consider the superconvergence of a mixed covolume method on the quasi uniform triangular grids for the variable coefficient matrix poisson equations the superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained with these superconvergence estimates we establish the superconvergence estimates and the l error estimates for the mixed covolume method for the elliptic problems based on the superconvergence of the mixed covolume method under the condition that the triangulation is uniform we construct a post processing method for the approximate velocity which improves the order of approximation of the approximate velocity

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

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

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

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

superconvergence and extrapolation for mixed finite element methods on rectangular domains asymptotic expansions for the rt raviart thomas mixed finite element approximation by the lowest order rectangular element associated with a second order elliptic equation on a rectangular domain are derived superconvergence for the vector field along the gauss lines is obtained as a result of the expansion a procedure of postprocessed extrapolation is presented for the scalar field as well as procedures of pure richardson extrapolation for both the vector and the scalar fields

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

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

on some mixed finite element methods with numerical integration in this paper a new family of mixed finite volume methods is analyzed for the approximation of a reaction diffusion problem all the methods are obtained starting from the dual mixed formulation of the problem and then employing the lowest order raviart thomas finite element spaces plus a suitable quadrature formula for the mass matrix this allows for the use of different averages of the inverse diffusion coefficient to enforce the constitutive law for the fluxes at the interelement boundary in a finite volume fashion the soundness of the methods is supported by an error analysis which shows optimal cal o h convergence rate with respect to the standard mixed finite element norm in the case of both smooth and piecewise smooth coefficients numerical results on test problems with both smooth and nonsmooth coefficients support the theoretical estimates

mixed finite element methods for general quadrilateral grids abstract we study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the h div norm this new element is designed so that the h div projection h satisfies h p h div a rigorous optimal order error estimate is carried out by proving a modified version of the bramble hilbert lemma for vector variables we show that a local h div projection reproducing certain polynomials suffices to yield an optimal l 2 error estimate for the velocity and hence our approach also provides an improved error estimate for original raviart thomas element of lowest order numerical experiments are presented to verify our theory

mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems we construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order pdes in the state variable e g pressure and its flux e g darcy velocity an important point is that no staggered grids or covolumes are used to stabilize the system only a single primary grid system is adopted and the degrees of freedom are imposed on the interfaces the approximate flux is sought in the lowest order raviart thomas space and the pressure field in the rotated q1 nonconforming space furthermore we demonstrate that the present finite volume method can be interpreted as a rotated q1 nonconforming finite element method for the pressure with a simple local recovery of flux numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media the mixed hybrid finite element mhfe and the multipoint flux approximation mpfa methods are well suited for anisotropic heterogeneous domains since both are locally conservative and can handle general irregular grids in this work behaviours and performances of mhfe and mpfa methods are studied numerically for different heterogeneities and anisotropy factors on parallelograms and then on a more general quadrilateral grid the superiority of mpfa in terms of accuracy and efficiency is clearly demonstrated for parallelogram grids in the case of more general quadrilateral grids mpfa becomes more central processing unit time consuming than mhfe for high anisotropy factors both methods give results with significant non physical oscillations copyright 2008 john wiley sons ltd

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

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

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

superconvergence of h div finite element approximations for the stokes problem by l2 projection methods a general superconvergence result of h div finite element approximations for the stokes equations is established by using l 2 projection method regularity assumptions for the stokes problem with regular partitions is required numerical experiments are given to verify the theoretical results

optimal biquadratic finite volume element methods on quadrilateral meshes in this paper an optimal biquadratic finite volume element scheme is established and analyzed for elliptic equations on quadrilateral meshes it is proved that the new scheme has optimal o h 2 convergence rate in h 1 norm and o h 3 convergence rate in l 2 norm with the assumption that each element is an h 2 parallelogram

parameter space stability analysis of lti time delay systems with parametric uncertainties a novel approach to stability analysis of linear time invariant lti time delay systems of retarded type with incommensurate time delays and polynomial dependence on parametric uncertainties is presented using a branch and bound algorithm which relies on taylor models and polynomials in bernstein form we first determine the stability crossing set in the delay parameter space and then evaluate stability for each disjoint region the novel approach can be used to either nonconservatively check stability of time delay systems with interval parameters or to map stable regions to a low dimensional parameter space while taking additional interval parameters into account the algorithm is applied to several examples with parametric uncertainties and or incommensurate time delays

robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion this paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals two algorithms are presented both rely on the expansion of a multivariate polynomial into bernstein polynomials the first one is an improvement of the so called bernstein algorithm and checks the hurwitz determinant for positivity over the parameter set the second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the bernstein polynomials numerical results to real world control problems are presented showing the efficiency of both algorithms

weighted poincare type inequalities for differential forms inls μ averaging domains abstract we first introduce a new class of weighted functions and obtain some basic properties of this class then as applications of this class we prove local weighted integral inequalities for differential forms finally we obtain the global weighted integral inequalities for differential forms inls averaging domains which can be considered as generalizations of the poincare inequality for sobolev functions

parallel mixed fem simulation of a class of single phase models with non local operators anomalous sub diffusion processes governed by non local operators have been used in various applications these are recently adapted to model complex single and multi phase fluid flow in unconventional reservoirs to understand the long time behavior of various quantities of interest such processes requiring long time simulation are modeled by a non local in time fractional derivative based darcy s law and a conservation of mass equation leading to a coupled system of first order fractional partial differential equations fpdes standard time stepping discretization of these continuous models leads to computational bottleneck for long time simulation memory limitations in each node of high performance computing hpc clusters dictate computational constraints for resolving fine scale spatial structures in the models we describe and implement a parallel in time and parallel in space mixed finite element fem based discretization computer model to simulate the fpdes our hybrid parallel in time and space framework facilitates efficient long time simulation of the first order computer model to overcome the time stepping computational bottleneck and within the node memory constraints approximate the continuous models with large degrees of freedom dof

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

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

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

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

interior superconvergence in mortar and non mortar mixed finite element methods on non matching grids we establish interior velocity superconvergence estimates for mixed finite element approximations of second order elliptic problems on non matching rectangular and quadrilateral grids both mortar and non mortar methods for imposing the interface conditions are considered in both cases it is shown that a discrete l 2 error in the velocity in a compactly contained subdomain away from the interfaces con1 2 higher than the error in the whole domain for the non mortar method we also establish pressure superconvergence which is needed in the velocity analysis numerical results are presented in confirmation of the theory

poincare inequalities with the radon measure for differential forms we establish the local and global poincare inequalities with the radon measure for the solutions to the nonlinear elliptic partial differential equation for differential forms

lipschitz and bmo norm inequalities for operators abstract in this paper we obtain lipschitz norm inequalities and bmo norm inequalities for operators applied to differential forms satisfying the nonhomogeneous a harmonic equation

inequalities for green s operator with lipschitz and bmo norms in this paper we obtain lipschitz norm and bmo norm inequalities for green s operator applied to differential forms we also establish norm comparison theorems for the bmo norm and the lipschitz norm of green s operator

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

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

poincare type inequalities for the homotopy operator with lφ ω norms abstract we obtain poincare type inequalities for the homotopy operator applied to differential forms in l m averaging domains which can be used to estimate the norms of operators and the integral properties of differential forms

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

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

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

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

on the application of the continuous galerkin finite element method for conservation problems one major drawback that prevents the use of the standard continuous galerkin finite element method in solving conservation problems is its lack of a locally conservative flux our present work has developed a simple postprocessing for the continuous galerkin finite element method resulting in a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh the postprocessing requires an auxiliary fully neumann problem to be solved on each finite element these local problems are independent of each other and in two dimensions involve solving only a 3 by 3 system in the case of triangular elements and a 4 by 4 system for quadrilateral elements a convergence analysis for the method is provided and its performance is demonstrated through numerical examples of multiphase flow with triangular and quadrilateral elements along with a description of its parallel implementation

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 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

superconvergence of weak galerkin finite element approximation for second order elliptic problems by l2 projections the weak galerkin finite element method wg fem is a novel numerical method that was first proposed and analyzed by wang and ye 2013 12 for general second order elliptic problems on triangular meshes based on a discrete weak gradient in general the weak galerkin finite element formulations for partial differential equations can be derived naturally by replacing usual derivatives by weakly defined derivatives in the corresponding variational forms the superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate wang 2000 11 proposed and analyzed superconvergence of the standard galerkin finite element method by l 2 projections the main idea behind the l 2 projections is to project the finite element solution to another finite element space with a coarse mesh and a higher order of polynomials the objective of this paper is to establish a general superconvergence result for the weak galerkin finite element approximations for second order elliptic problem by l 2 projection methods the results of numerical experiments show great promise for the robustness reliability flexibility and accuracy of superconvergence in wg fem by l 2 projection methods

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

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

superconvergence of mixed finite element methods for optimal control problems t in this paper we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods the state and co state variables are approximated by the lowest order raviart thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique we derive l 2 superconvergence properties for the flux functions along the gauss lines and for the scalar functions at the gauss points via mixed projections moreover global l 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique thus based on these superconvergence estimates some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods finally some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence

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

an economical cascadic multigrid method for the weak galerkin finite element approximation of second order elliptic problems abstract in this paper we present both error and computational complexity estimates of economical cascadic multigrid solver for the linear system of equations arising from weak galerkin finite element approximation of second order elliptic problems on triangular meshes our analysis shows that the proposed economical cascadic multigrid method with the conjugate gradient smoother is optimal in both accuracy and computational complexity for two dimensional problems in addition an elimination technique is utilized to further improve the computation efficiency the interior degrees of freedom on each element are removed from the resulting linear system so that the size of the problem becomes much smaller we demonstrate the accuracy and efficiency of our proposed economical cascadic multigrid methods through ample numerical experiments the numerical results show that compared with the usual cascadic multigrid method the elimination economical cascadic multigrid method saves computational cost significantly

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

superconvergence of finite volume methods for the second order elliptic problem abstract this paper derives a superconvergence result for finite volume approximations of the second order elliptic problem by using a l2 projection post processing technique the superconvergence result is applicable to different kind of finite volume methods and to general quasi uniform meshes

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

on the regularity and uniformness conditions on quadrilateral grids in the literature there are various regularity conditions as well as many different so called almost parallelogram conditions on the subdivisions of quadrilateral grids in this paper we investigate and clarify their relations

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational 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

imbedding theorems in orlicz sobolev space of differential forms abstract in this paper we prove imbedding inequalities with l norms in the orlicz sobolev space of the forms and establish l norm inequalities for the related operators applied to differential forms we also obtain the global imbedding theorems in l m averaging domains

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

estimates of weighted integrals for differential forms in this paper we obtain some weighted integral estimates for differential forms which are generalizations of the poincare inequality the caccioppoli type estimate and the weak reverse holder inequality respectively these results can be used to study the integrability of differential forms and to estimate the integrals for differential forms

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

arλ ω weighted imbedding inequalities for a harmonic tensors abstract we prove new versions of a r weighted imbedding inequalities for a harmonic tensors locally and globally the results are used to estimate the integrals of a homotopy operator t from the banach space l s l to the sobolev space w 1 s l 1 l 1 2 n and sobolev poincare imbedding inequality with a r weight

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

two families of h div mixed finite elements on quadrilaterals of minimal dimension we develop two families of mixed finite elements on quadrilateral meshes for approximating mathbf u p solving a second order elliptic equation in mixed form standard raviart thomas rt and brezzi douglas marini bdm elements are defined on rectangles and extended to quadrilaterals using the piola transform which are well known to lose optimal approximation of nabla cdot mathbf u arnold boffi falk spaces rectify the problem by increasing the dimension of rt so that approximation is maintained after piola mapping our two families of finite elements are uniformly inf sup stable achieve optimal rates of convergence and have minimal dimension the elements for mathbf u are constructed from vector polynomials defined directly on the quadrilaterals rather than being transformed from a reference rectangle by the piola mapping and then supplemented by two one for the lowest order basis functions that are piola mapped one family has full h div approximation mathbf u p

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

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

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

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type