Enriched finite element subspaces for dualdual mixed formulations in fluid mechanics and elasticity ?
In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dualdual ones), which are analyzed by means of a slight generalization of the well known BabukaBrezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, RaviartThomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces.
twofold saddle point raviartthomas peers enriched subspaces