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This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new 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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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]

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]

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]