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Following Castillo et al. (2000) and Cockburn (2003), a general framework of constructing discontinuous Galerkin (DG) methods is developed for solving the linear elasticity problem. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method discussed by Castillo et al. (2000) and Cockburn (2003). The error bounds inL2-norm,H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established.

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Two nonconforming finite element Stokes complexes ended with the nonconforming P 1 -P 0 element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(grad curl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(grad curl)-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem is decoupled into two mixed methods of the Maxwell equation and the nonconforming P 1 -P 0 element method for the Stokes equation, based on which a fast solver is developed.