In this paper,we introduce the weak Galerkin(WG)method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains.We define the WG spaces ...In this paper,we introduce the weak Galerkin(WG)method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains.We define the WG spaces in the polygonal meshes and construct corresponding discrete schemes.We prove the existence and uniqueness of the WG scheme by the discrete inf-sup condition and monotone operator theory.Then,we derive the optimal error estimates for the velocity and pressure.Numerical experiments are presented to verify the efficiency of the WG method.展开更多
This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interio...This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interior of elements, respectively, and piecewise polynomials of degrees k and k + 1 for the boundary parts of the velocity and pressure, respectively. Wellposedness of the discrete scheme is established. The method yields a globally divergence-free velocity approximation. Optimal priori error estimates are derived for the velocity gradient and pressure approximations. Numerical results are provided to confirm the theoretical results.展开更多
A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theo...A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute(a constant-order away from 90 degrees).To avoid this difficulty,a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions.The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges.Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation of the maximum principle.Numerical examples are presented.展开更多
The weak Galerkin(WG)method is a nonconforming numerical method for solving partial differential equations.In this paper,we introduce the WG method for elliptic equations with Newton boundary condition in bounded doma...The weak Galerkin(WG)method is a nonconforming numerical method for solving partial differential equations.In this paper,we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains.The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications.We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions.The error estimates are derived.Numerical experiments are presented to verify the theoretical analysis.展开更多
This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system...This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.展开更多
This article is devoted to studying the application of the weak Galerkin(WG)finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds.The WG method uses discontinuous polynomi...This article is devoted to studying the application of the weak Galerkin(WG)finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds.The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions.The non-conforming finite element space of the WG method is the key of the lower bound property.It also makes the WG method more robust and flexible in solving eigenvalue problems.We demonstrate that the WG method can achieve arbitrary high convergence order.This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements.Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.展开更多
This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique.A high order lower bound can be obtained at a relatively low cost via the proposed me...This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique.A high order lower bound can be obtained at a relatively low cost via the proposed method.The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions.Numerical examples are presented to validate the theoretical analysis.展开更多
The linear hyperbolic equation is of great interest inmany branches of physics and industry.In this paper,we use theweak Galerkinmethod to solve the linear hyperbolic equation.Since the weak Galerkin finite element sp...The linear hyperbolic equation is of great interest inmany branches of physics and industry.In this paper,we use theweak Galerkinmethod to solve the linear hyperbolic equation.Since the weak Galerkin finite element space consists of discontinuous polynomials,the discontinuous feature of the equation can be maintained.The optimal error estimates are proved.Some numerical experiments are provided to verify the efficiency of the method.展开更多
The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the correspond...The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.展开更多
A modified weak Galerkin(MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the...A modified weak Galerkin(MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree k and k-1 are used to approximate the velocity u and the pressure p, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than k-1. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity.Optimal order error estimates are established for the velocity and pressure approximations in H^1 and L^2 norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.展开更多
The weak Galerkin （WG） finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend...The weak Galerkin （WG） finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin （DG） methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak flmctions defined on the same edge. We introduce an over-penalized weak Galerkin （OPWG） method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in H1 and L2 norms are established for the finite element （Pk（K）, Pk（e）, RTk（K））. In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CO, and BICGSTAB iterative methods.展开更多
In this paper,we propose a pressure-robust weak Galerkin(WG)finite element scheme to solve the Stokes-Darcy problem.To construct the pressure-robust numerical scheme,we use the divergence-free velocity reconstruction ...In this paper,we propose a pressure-robust weak Galerkin(WG)finite element scheme to solve the Stokes-Darcy problem.To construct the pressure-robust numerical scheme,we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme.This numerical scheme is easy to implement because it only need to modify the right side.We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient.Moreover,the errors of the velocity function reach the optimal convergence orders under the energy norm,as validated by both theoretical analysis and numerical results.展开更多
This study proposes a class of augmented subspace schemes for the weak Galerkin(WG)finite element method used to solve eigenvalue problems.The augmented subspace is built with the conforming linear finite element spac...This study proposes a class of augmented subspace schemes for the weak Galerkin(WG)finite element method used to solve eigenvalue problems.The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigen-function approximations in the WG finite element space defined on the fine mesh.Based on this augmented subspace,solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented sub-space.The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size,as demonstrated by the accompanying error esti-mates.Finally,a few numerical examples are provided to validate the proposed numerical techniques.展开更多
In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential a...In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed.The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error.The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle,quadrilateral and polygonal meshes with hanging nodes.Finally,numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.展开更多
The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution varia...The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.展开更多
We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is rem...We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and L2 norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.展开更多
基金supported by the National Natural Science Foundation of China(NSFC)(Grant No.12301519)supported by the National Natural Science Foundation of China(Grant Nos.12271208,11901015)。
文摘In this paper,we introduce the weak Galerkin(WG)method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains.We define the WG spaces in the polygonal meshes and construct corresponding discrete schemes.We prove the existence and uniqueness of the WG scheme by the discrete inf-sup condition and monotone operator theory.Then,we derive the optimal error estimates for the velocity and pressure.Numerical experiments are presented to verify the efficiency of the WG method.
基金supported by Major Research Plan of National Natural Science Foundation of China (Grant No. 91430105)
文摘This paper proposes a weak Galerkin finite element method to solve incompressible quasi-Newtonian Stokes equations. We use piecewise polynomials of degrees k + 1(k 0) and k for the velocity and pressure in the interior of elements, respectively, and piecewise polynomials of degrees k and k + 1 for the boundary parts of the velocity and pressure, respectively. Wellposedness of the discrete scheme is established. The method yields a globally divergence-free velocity approximation. Optimal priori error estimates are derived for the velocity gradient and pressure approximations. Numerical results are provided to confirm the theoretical results.
基金This work was supported in part by the NSF under Grant DMS-1115118.
文摘A weak Galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak Galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute(a constant-order away from 90 degrees).To avoid this difficulty,a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions.The discrete maximum principle is then established for the full weak Galerkin approximation using the relations between the degrees of freedom located on elements and edges.Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation of the maximum principle.Numerical examples are presented.
基金China Postdoctoral Science Foundation through grant 2019M661199 and Postdoctoral Innovative Talent Support Program(BX20190142)Q.Zhai was partially supported by National Natural Science Foundation of China(12271208,11901015)+1 种基金R.Zhang was supported in part by National Natural Science Foundation of China(grant 11971198,11871245,11771179,11826101)the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education of China(housed at Jilin University).
文摘The weak Galerkin(WG)method is a nonconforming numerical method for solving partial differential equations.In this paper,we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains.The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications.We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions.The error estimates are derived.Numerical experiments are presented to verify the theoretical analysis.
基金supported by Zhejiang Provincial Natural Science Foundation of China(LY19A010008).
文摘This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.
基金supported in part by China Natural National Science Foundation(91630201,U1530116,11771179)by the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University,Changchun,130012,P.R.China+3 种基金supported in part by the National Natural Science Foundation of China(NSFC 11471031,91430216)and the U.S.National Science Foundation(DMS–1419040)supported by Science Challenge Project(No.TZ2016002)National Natural Science Foundations of China(NSFC 11771434,91330202,11371026,91430108,11771322,11626033,11601368)the National Center for Mathematics and Interdisciplinary Science,CAS.
文摘This article is devoted to studying the application of the weak Galerkin(WG)finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds.The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions.The non-conforming finite element space of the WG method is the key of the lower bound property.It also makes the WG method more robust and flexible in solving eigenvalue problems.We demonstrate that the WG method can achieve arbitrary high convergence order.This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements.Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.
基金The work of Q.Zhai was partially supported by China Postdoc total Science Foundation(2018M640013,2019T120008)The work of X.Hu was partially supported by NSF grant(DMS-1620063)+1 种基金The work of R.Zhang was supported in part by China Natural National Science Foundation(91630201,11871245,11771179)by the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University,Changchun,130012,P.R.China.
文摘This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique.A high order lower bound can be obtained at a relatively low cost via the proposed method.The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions.Numerical examples are presented to validate the theoretical analysis.
基金The research of R.Zhangwas supported in part by China Natural National Science Foundation(U1530116,91630201,11471141)by the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University,Changchun,130012,P.R.China.
文摘The linear hyperbolic equation is of great interest inmany branches of physics and industry.In this paper,we use theweak Galerkinmethod to solve the linear hyperbolic equation.Since the weak Galerkin finite element space consists of discontinuous polynomials,the discontinuous feature of the equation can be maintained.The optimal error estimates are proved.Some numerical experiments are provided to verify the efficiency of the method.
文摘The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.
基金Supported by the Natural National Science Foundation of China(Grant Nos.91630201 U1530116+6 种基金 1172610211771179 93K172018Z01 11701210 JJKH20180113KJ 20190103029JH)the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University
文摘A modified weak Galerkin(MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree k and k-1 are used to approximate the velocity u and the pressure p, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than k-1. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity.Optimal order error estimates are established for the velocity and pressure approximations in H^1 and L^2 norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.
基金The research of the second author is supported in part by the Natural Science Foundation of Gansu Province, China （Grant 18JR3RA290）, and Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund （the second phase）.
文摘The weak Galerkin （WG） finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin （DG） methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak flmctions defined on the same edge. We introduce an over-penalized weak Galerkin （OPWG） method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in H1 and L2 norms are established for the finite element （Pk（K）, Pk（e）, RTk（K））. In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CO, and BICGSTAB iterative methods.
基金supported by the National Natural Science Foundation of China(Grant Nos.11901015,12271208,12001232,12201246,22341302)the National Key Research and Development Program of China(Grant Nos.2020YFA0713602,2023YFA1008803)+1 种基金the Fundamental Research Funds for the Central Universities housed at Jilin University(Grant No.93Z172023Z05)the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education of China housed at Jilin University.
文摘In this paper,we propose a pressure-robust weak Galerkin(WG)finite element scheme to solve the Stokes-Darcy problem.To construct the pressure-robust numerical scheme,we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme.This numerical scheme is easy to implement because it only need to modify the right side.We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient.Moreover,the errors of the velocity function reach the optimal convergence orders under the energy norm,as validated by both theoretical analysis and numerical results.
基金partly supported by the Beijing Natural Science Foundation(Grant No.Z200003)by the National Natural Science Foundation of China(Grant Nos.12331015,12301475,12301465)+1 种基金by the National Center for Mathematics and Interdisciplinary Science,Chinese Academy of Sciencesby the Research Foundation for the Beijing University of Technology New Faculty(Grant No.006000514122516).
文摘This study proposes a class of augmented subspace schemes for the weak Galerkin(WG)finite element method used to solve eigenvalue problems.The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigen-function approximations in the WG finite element space defined on the fine mesh.Based on this augmented subspace,solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented sub-space.The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size,as demonstrated by the accompanying error esti-mates.Finally,a few numerical examples are provided to validate the proposed numerical techniques.
基金supported by the National Natural Science Foundation of China(Grant No.12471363).
文摘In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed.The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error.The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle,quadrilateral and polygonal meshes with hanging nodes.Finally,numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.
文摘The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.
基金supported by the National Natural Science Foundation of China(Grant Nos.1901015,12271208,11971198,91630201,11871245,11771179,11826101)by the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University.
文摘We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and L2 norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.
