With Prof. Xu, Dr. Wu, Dr. Hong, we present a unified framework ([9]) for the design of finite element methods (FEM) of different kinds, including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin (DG) methods, hybrid DG methods and weak Galerkin methods.  We illustrate the main idea using a model second order elliptic boundary problem. This problem admits two main variational formulations, namely the primal and mixed formulation respectively.  The primal formulation requires certain continuity for the primal variable u, namely u belongs to H¹$, while the mixed formulation requires certain continuity for the (mixed) flux variable p, namely p belongs to H(div). The design of finite element methods then amounts to an appropriate approximation of the aforementioned continuity requirements for either u or p.  There are roughly four different ways to approximately impose these continuity requirements: (1) strongly (conforming); (2) weakly (nonconforming); (3) by Lagrangian multiplier (hybrid or stabilized hybrid), and (4) by penalization (discontinuous Galerkin).  Using the notation of the DG-gradient and DG-divergence (which are reduced to standard weak derivatives when the underlying finite element spaces are conforming), every existing finite element method considered in this paper can be expressed in terms of one of the following: (1) DG-gradient and DG-gradient^* (primal), (2) DG-divergence and DG-divergence^* (mixed), and (3) DG-divergence^* and DG-gradient^* (DG). 

In light of this general framework, a new mixed DG method is proposed, and we apply it to solve linear elasticity problem with arbitrary order discontinuous finite element spaces. For the mixed methods for linear elasticity problem, it is very challenging to develop the stable mixed finite element methods because the stress tensor needs to be symmetric according to the principle of conservation of angular momentum. In [10], we study the mixed LDG method for solving linear elasticity by discontinuous P_k+1-P_k finite element pairs for the stress and displacement with k ≥ 0 for any spatial dimension in a unified fashion. We note that the stress is discretized in the DG space with strongly imposed symmetry. Our contributions are twofold. First, by introducing a mesh dependent norm for the stress, we give a prior error analysis, which shows that optimal L²-error estimate for displacement and optimal H_h(div) error estimate for stress. Second, when the P_k+2−P_k+1 Stokes pair is stable and k ≥ d, we prove the optimal L²-error estimate for the stress by the BDM projection and a symmetrization technique.

[9] Qingguo Hong, Fei Wang, Shuonan Wu and Jinchao Xu*, A Unified Study of Continuous and Discontinuous Galerkin Methods, SCIENCE CHINA Mathematics, 62 (2019), 1-32.

[10] Fei Wang, Shuonan Wu and Jinchao Xu, A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry, Journal of Scientific Computing, 82:2 (2020).

[11] Yanxia Qian, Shuonan Wu and Fei Wang, A Mixed Discontinuous Galerkin Method with Symmetric Stress for Brinkman Problem Based on the Velocity-Pseudostress Formulation, Computer Methods in Applied Mechanics and Engineering, 368 (2020), 113177.