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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. [ABSTRACT FROM AUTHOR]

This article introduces a weak Galerkin (WG) finite element method for quad-curl problems in three dimensions. It is proved that the proposed WG method is stable and accurate in an optimal order of error estimates for the exact solution in discrete norms. In addition, an L 2 error estimate in an optimal order except the lowest order k = 2 is derived for the WG solution. Some numerical experiments are conducted to verify the efficiency and accuracy of our WG method and furthermore a superconvergence has been observed from the numerical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. Some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general framework for weak Galerkin generalized multiscale (WG-GMS) finite element method for the elliptic problems with rapidly oscillating or high contrast coefficients. This general WG-GMS method features in high order accuracy on general meshes and can work with multiscale basis derived by different numerical schemes. A special case is studied under this WG-GMS framework in which the multiscale basis functions are obtained by solving local problem with the weak Galerkin finite element method. Convergence analysis and numerical experiments are obtained for the special case. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a theoretical foundation for interior penalty discontinuous Galerkin methods for second-order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra that satisfy certain shape regularity conditions characterized in a recent paper by two of the authors, Wang and Ye (2012) [11]. The usual -conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. The interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. Results with such general meshes have important application in computational sciences. [Copyright &y& Elsevier]

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element methods on general polyhedral elements. One of the sequences uses polynomials of equal order for all the finite element spaces involved in the sequence and the other one uses polynomials of naturally descending orders. It is shown that the diagrams in both de Rham complexes commute for general polyhedral elements. The exactness of one of the complexes is established for the lowest order element. [ABSTRACT FROM AUTHOR]

Abstract: 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 and norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation. [Copyright &y& Elsevier]

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]

An extended P 1 -nonconforming finite element method is developed in this article for the Dirichlet boundary value problem of convection–diffusion–reaction equations on general polytopal partitions. This new method was motivated by the simplified weak Galerkin method, and makes use of only the degrees of freedom on the boundary of each element and, hence, has reduced computational complexity. Numerical stability and optimal order of error estimates in H 1 and L 2 norms are established for the corresponding numerical solutions. Some numerical results are presented to computationally verify the mathematical convergence theory. A superconvergence phenomenon on rectangular partitions is noted and illustrated through various numerical experiments. [ABSTRACT FROM AUTHOR]

This article proposes and analyzes the generalized weak Galerkin (gWG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the gWG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual L 2 norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed gWG method. [ABSTRACT FROM AUTHOR]

In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the evolutionary Oseen equation. The gWG method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical schemes are developed and analyzed for their convergence, stability, and error estimates. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples. [ABSTRACT FROM AUTHOR]

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H 2 norm and a L 2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation. [ABSTRACT FROM AUTHOR]

A new L p -primal–dual weak Galerkin method (L p -PDWG) with p > 1 is proposed for the first-order transport problems. The existence and uniqueness of the L p -PDWG numerical solution is established. In addition, the L p -PDWG method offers a numerical solution which retains mass conservation locally on each element. An optimal order error estimate is established for the primal variable. A series of numerical results are presented to verify the efficiency and accuracy of the proposed L p -PDWG scheme. [ABSTRACT FROM AUTHOR]

This paper presents a new L p -primal–dual weak Galerkin (PDWG) finite element method for the div–curl system with the normal boundary condition for p > 1. Two crucial features for the proposed L p -PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div–curl system under the low W α , p -regularity (α > 0) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the L q -norm for the primal variable where 1 p + 1 q = 1. A series of numerical experiments are presented to demonstrate the performance of the proposed L p -PDWG algorithm. [ABSTRACT FROM AUTHOR]

In this article, the authors present a new L p -primal–dual weak Galerkin method (L p -PDWG) for convection–diffusion equations. Comparing with the standard L 2 -PDWG method, the solution calculated from the L p -PDWG may exhibit some important advantages and features (e.g., less jumps cross the element interface when p → 1 , or sparsity by using p = 1 and wavelet basis approximation). The existence and uniqueness of the numerical solution is discussed, and an optimal-order error estimate is derived in the L q -norm for the primal variable, where 1 p + 1 q = 1 with p > 1. Furthermore, error estimates are established for the numerical approximation of the dual variable in the standard W m , p norm, 0 ≤ m ≤ 2. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed L p -PDWG method. [ABSTRACT FROM AUTHOR]

This article presents a new primal–dual weak Galerkin (PDWG) finite element method for transport equations in non-divergence form. The PDWG method employs locally reconstructed differential operators and stabilizers in the weak Galerkin framework, and yields a symmetric discrete linear system involving the primal variable and the dual variable (known as the Lagrangian multiplier) for the adjoint equation. Optimal order error estimates are established in various discrete Sobolev norms for the corresponding numerical solutions. Numerical results are reported to illustrate the accuracy and efficiency of the new PDWG method. [ABSTRACT FROM AUTHOR]