Fei Wang

Fei Wang, Yuanming Xiao, Jinchao Xu

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise $H^1$-norm and in the $L^2$-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no "sufficiently large" parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are given to support the theoretical results.

Qingguo Hong, Fei Wang, Shuonan Wu et al.

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

Fei Wang, Shuonan Wu, Jinchao Xu

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the stress and of degree $k$ for the displacement ($k\geq 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k \geq 0$, the $H({\rm div})$-like error estimate for the stress and $L^2$ error estimate for the displacement are optimal. We further establish the optimal $L^2$ error estimate for the stress provided that the $\mathcal{P}_{k+2}-\mathcal{P}_{k+1}^{-1}$ Stokes pair is stable and $k \geq d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.

Fei Wang, Tianyi Zhang, Weimin Han

Discontinuous Galerkin (DG) methods are considered for solving a plate contact problem, which is a 4th-order elliptic variational inequality of second kind. Numerous $C^0$ DG schemes for the Kirchhoff plate bending problem are extended to the variational inequality. Properties of the DG methods, such as consistency and stability, are studied, and optimal order error estimates are derived. A numerical example is presented to show the performance of the DG methods; the numerical convergence orders confirm the theoretical prediction.