Alternative approaches for the derivation of discontinuous Galerkin methods for nonlinear mechanics

Discontinuous Galerkin methods are commonly derived by seeking a weak statement of the governing differential equations via a weighted-average approach allowing for discontinuous fields at the element interfaces of the discretization. In order to ensure consistency and stability of the formulation, this approach requires the definition of a numerical flux and a stabilization term. Discontinuous Galerkin methods may also be formulated from a linear combination of the governing and compatibility equations weighted by suitable operators. A third approach based on a variational statement of a generalized energy functional has been
proposed recently for finite elasticity. This alternative approach naturally leads to an expression of the numerical flux and the stabilization terms in the context of large deformation mechanics problems. This paper compares these three approaches and establishes the conditions under which identical formulations are obtained.