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In this paper a time-explicit weak Galerkin finite element method is introduced and analyzed for parabolic equations. The main idea relies on the inclusion of a stabilization term in the temporal direction in addition to the usual static stabilization in the weak Galerkin framework. Both semi-discrete and fully-discrete schemes in time are presented, as well as their stability and error analysis. Numerical results are reported for this new explicit weak Galerkin finite element method. [ABSTRACT FROM AUTHOR]

This article introduces a particular weak Galerkin (WG) element on rectangular/cuboid partitions that uses k$$ k $$th order polynomial for weak finite element functions and (k+1)$$ \left(k+1\right) $$th order polynomials for weak derivatives. This WG element is highly accurate with convergence two orders higher than the optimal order in an energy norm and the L2$$ {L}^2 $$ norm. The superconvergence is verified analytically and numerically. Furthermore, the usual stabilizer in the standard weak Galerkin formulation is no longer needed for this element. [ABSTRACT FROM AUTHOR]

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5‐ and 7‐point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme. [ABSTRACT FROM AUTHOR]

The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L 2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper. [ABSTRACT FROM AUTHOR]

A weak Galerkin finite element method is devised for the Poisson equation with Dirichlet boundary value when curved elements are employed in the numerical scheme. Optimal order error estimates are derived for the weak Galerkin solution in both the H 1 -norm and the L 2 -norm. Numerical results are produced to demonstrate the performance of the weak Galerkin method on general curved partitions. [ABSTRACT FROM AUTHOR]

This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h -acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP. [ABSTRACT FROM AUTHOR]

A discrete divergence free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in [17] . Discrete divergence free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence free basis derived, the discrete divergence free WG scheme can eliminate pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence free WG method. [ABSTRACT FROM AUTHOR]

The weak Galerkin (WG) methods have been introduced in Mu et al. (2013, 2014), [14] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique. [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]

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H 1 and L 2 norms are established for the present WG finite element solutions. Extensive numerical experiments have been conducted to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L ∞ norm for both C 1 and H 2 continuous solutions. [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]

In this paper, we construct an auxiliary space multigrid preconditioner for the weak Galerkin method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a “coarse” space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In our construction, we conveniently use the H 1 conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the H 1 conforming piecewise linear finite element space. We successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as the reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments further support the theoretical results. [ABSTRACT FROM AUTHOR]

A finite element method, called generalized Weak Galerkin (gWG), is introduced for the Stokes equation by using a new weak gradient notion in the usual weak Galerkin approach. The gWG method allows polynomial elements of arbitrary order in any combination for the velocity variable, but with proper constraints on the pressure element due to the inf-sup condition. Error estimates are derived for the velocity approximation in a mesh-dependent energy norm, as well as in the L 2 norm for both the velocity and the pressure approximations. Some numerical examples are presented to verify the accuracy, theoretical convergence order, and robustness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete H2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L2 norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003-1029, 2014 [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]

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal-order error estimates in both H1 and L2 norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

Abstract: Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both and norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order to for the solution itself in norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order to in the norm for or Lipschitz continuous interfaces associated with a or continuous solution. [Copyright &y& Elsevier]

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye () for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established in Wang and Ye (). The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing. [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]

This paper presents a new numerical method for div-curl systems with the normal boundary condition by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low H α -regularity (α > 0) assumption for the true solution, but also an effective approximation of the normal harmonic vector fields on domains with complex topology. Seven numerical experiments are conducted and the results are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields. [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 numerical scheme is devised by the primal-dual weak Galerkin(PDWG) framework. • The PDWG method can be interpreted as a constraint optimization process. • The PDWG scheme conserves mass locally on each element. • Optimal order error estimates are derived for the PDWG solution. This article devises a new numerical method for first-order transport problems by using the primal-dual weak Galerkin (PD-WG) finite element method recently developed in scientific computing. The PD-WG method is based on a variational formulation of the modeling equation for which the differential operator is applied to the test function so that low regularity for the exact solution of the original equation is sufficient for computation. The PD-WG finite element method indeed yields a symmetric system involving both the original equation for the primal variable and its dual for the dual variable (also known as Lagrangian multiplier). For the linear transport problem, it is shown that the PD-WG method offers numerical solutions that conserve mass locally on each element. Optimal order error estimates in various norms are derived for the numerical solutions arising from the PD-WG method with weak regularity assumptions on the modelling equations. A variety of numerical results are presented to demonstrate the accuracy and stability of the new 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]

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 article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O (h r) , 1.5 ≤ r ≤ 2 , for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O (h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory. [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 introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions on the exact solution and the interface and boundary data. It is proved that the PDWG method is stable and accurate with optimal order of error estimates in discrete and Sobolev norms. In particular, the error estimates are derived under the low regularity assumption of u ∈ H δ (Ω) for δ > 1 2 for the exact solution u. Extensive numerical experiments are conducted to provide numerical solutions that verify the efficiency and accuracy of the new PDWG method. • A PDWG method is proposed for second order elliptic interface problems. • PDWG is stable and accurate with optimal order error estimates in various norms. • Error estimates are derived under low regularity assumption for exact solution. • Numerical experiments are conducted to verify the efficiency and accuracy of PDWG. [ABSTRACT FROM AUTHOR]