Robust error estimates for weak Galerkin finite element method for singularly perturbed 2D reaction-diffusion elliptic boundary-value problems on various layer-adapted meshes

A modified weak Galerkin finite element method for the linear elasticity problem in mixed form

A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

An absolutely stable weak Galerkin finite element method for the Darcy–Stokes problem

A weak Galerkin finite element method for second-order elliptic problems

A new weak Galerkin finite element method for elliptic interface problems

A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation

A weak Galerkin finite element method for 1D semiconductor device simulation models

A modified weak Galerkin finite element method for the Stokes equations

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple WG finite element method is introduced for second‐order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory.

This paper concerns the numerical approximation of elliptic interface problems via least-squares-based weak Galerkin (WG) finite element method. This method allows the use of totally discontinuous functions on finite element partitions consisting of arbitrary polygon shape. Further, the method is capable of solving the unknown and the flux simultaneously with optimal order convergence rates.

A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems

An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes

A stabilizer-free weak Galerkin finite element method on polytopal meshes

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete \begin{document}$ H^1 $\end{document} norm and the standard \begin{document}$ L^2 $\end{document} norm. The numerical examples are tested on various meshes and confirm the theory.

The weak Galerkin finite element method for the symmetric hyperbolic systems