In this article, a modified weak Galerkin method based on weak Laplacian is presented and analyzed for the biharmonic equation. A modified stabilizer is designed to ensure the solution uniqueness and existence for the numerical scheme. This new scheme is based on the use of local finite element spaces in the form of (Pk(T), Pk-2(dT), Pk-2(dT)), k >= 2, in which polynomials of degree k-2 are employed to approximate both the trace of the unknown function and its normal derivative on each element boundary. Optimal order of error estimates is established in various Sobolev norms. Numerical examples are presented to validate the effectiveness and feasibility of the new weak Galerkin scheme.

This paper introduces a new primal-dual weak Galerkin finite element method for the div-curl system subject to tangential boundary conditions, under minimal regularity assumptions on the exact solution. The proposed numerical scheme is formulated through a weak variational form that avoids the use of classical derivatives of the exact solution, complemented by a dual problem within the general framework of the weak Galerkin method. Optimal order error estimates in L-2 are established for solution vector fields belonging to H-theta(Omega) with theta >1/2. The existence and uniqueness of the discrete solution are proved for connected domains with general topological characteristics, including general first and second Betti numbers. Numerical results are presented to validate the theoretical convergence results.

This paper presents and analyzes a parallelizable iterative procedure based on domain decomposition for primal-dual weak Galerkin (PDWG) finite element methods applied to the Poisson equation. The existence and uniqueness of the PDWG solution are established. Optimal order of error estimates are derived in both a discrete norm and the L^2 norm. The convergence analysis is conducted for domain decompositions into individual elements associated with the PDWG methods, which can be extended to larger subdomains without any difficulty.

This paper introduces a generalized weak Galerkin (gWG) finite element method for linear elasticity problems on general polygonal and polyhedral meshes, providing a flexible and efficient framework that accommodates non-polynomial approximation spaces. The generalized weak differential operators are constructed as the sum of classical differential operators and solutions of simple element-level correction problems that account for boundary discontinuity, thereby reducing computational cost and yielding increased flexibility compared to standard weak Galerkin formulations. The proposed gWG framework naturally admits arbitrary finite-dimensional approximation spaces, including non-polynomial activation-based spaces with random parameters. Error equations and optimal-order error estimates are established for the proposed method. Numerical experiments demonstrate that the method is locking-free, robust under varing mesh geometries, and effective on general polygonal and polyhedral partitions. In particular, activation-based interior approximation spaces often yield convergence behavior comparable to classical polynomial spaces.

This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact differential identities that link a function and its derivatives through polynomial weight factors. This formulation expresses smooth functions via divergence-based relations connecting derivatives of all orders with systematically scaled polynomial coefficients. This framework provides a natural foundation for constructing high-order numerical quadrature formulas, particularly for multi-dimensional domains. By exploiting the divergence structure, volume integrals are systematically transformed into boundary integrals using the Divergence Theorem, recursively reducing the integration domain from an n-dimensional body to its (n-1)-dimensional facets, and ultimately to its vertices. The article further enhances the framework's accuracy by introducing a complex-shift technique. It is demonstrated that by positioning the expansion center at specific roots of unity in the complex plane, lower-order error terms are annihilated, yielding high-order real-valued quadrature rules with minimal function evaluations. Additionally, a rigorous geometric analysis of the affine transformations required for surface integration is provided, deriving explicit formulas for the transformation of normal vectors and surface measures. The proposed method offers a robust, systematic, and computationally efficient alternative to tessellation-based quadrature for arbitrary flat-faced polytopes.

This paper presents a generalized weak Galerkin (gWG) finite element method for linear elasticity problems on general polygonal and polyhedral meshes. The proposed framework is flexible and efficient, allowing for the use of nonpolynomial approximating functions. The generalized weak differential operators are defined as an element-level correction of the classical differential operators accounting for boundary discontinuities. This construction reduces computational cost and provides greater flexibility than standard weak Galerkin formulations. The gWG framework naturally accommodates arbitrary finite-dimensional approximation spaces, including nonpolynomial activation-based spaces with randomly selected parameters. Error equations and error estimates are established for the proposed method. Numerical experiments demonstrate that the method is locking-free, robust with respect to mesh geometry, and effective on general polygonal and polyhedral partitions. In particular, activation-based interior approximation spaces exhibit convergence behavior comparable to that of classical polynomial spaces.

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.

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 H1-norm and the L2-norm. Numerical results are produced to demonstrate the performance of the weak Galerkin method on general curved partitions.

An upwind weak Galerkin finite element scheme was devised and analyzed in this article for convection-dominated Oseen equations. The numerical algorithm was based on the weak Galerkin method enhanced by upwind stabilization. The resulting finite element scheme uses equal-order, say k, polynomial spaces on each element for the velocity and the pressure unknowns. With finite elements of order k⩾ 1 , the numerical solutions are proved to converge at the rate of O(h^k+1/2) in an energy-like norm for convection-dominated Oseen equations. Numerical results are presented to demonstrate the accuracy and effectiveness of the upwind weak Galerkin scheme.

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete $H^1$ and $L^2$ norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

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.

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the $L^2$ norm. A series of numerical experiments are conducted to validate the theoretical developments.

A parallelizable iterative procedure based on domain decomposition is presented and analyzed for weak Galerkin finite element methods for second order elliptic equations. The convergence analysis is established for the decomposition of the domain into individual elements associated to the weak Galerkin methods or into larger subdomains. A series of numerical tests are illustrated to verify the theory developed in this paper.

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.

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.

In this article, the authors present a new Lp-primal–dual weak Galerkin method (Lp-PDWG) for convection–diffusion equations. Comparing with the standard L2-PDWG method, the solution calculated from the Lp-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 Lq-norm for the primal variable, where 1p+1q=1 with p>1. Furthermore, error estimates are established for the numerical approximation of the dual variable in the standard Wm,p norm, 0≤m≤2. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed Lp-PDWG method.

A new Lp-primal–dual weak Galerkin method (Lp-PDWG) with p>1 is proposed for the first-order transport problems. The existence and uniqueness of the Lp-PDWG numerical solution is established. In addition, the Lp-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 Lp-PDWG scheme.

This article extends a recently developed superconvergence result for weak Galerkin (WG) approximations for modeling partial differential equations from constant coefficients to variable coefficients. This superconvergence features a rate that is two-order higher than the optimal-order error estimates in the usual energy and L 2 norms. The extension from constant to variable coefficients for the modeling equations is highly non-trivial. The underlying technical analysis is based on the use of a sequence of projections and decompositions. Numerical results are presented to confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

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 L2 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.

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.