New mixed finite elements for plane elasticity and Stokes equations

Abstract

We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element spaces consist respectively of piecewise quadratic polynomials and piecewise cubic polynomials such that the divergence of each space restricted to a single simplex is contained in the corresponding displacement approximation space. We derive stability and optimal order approximation for the elements. We also give some numerical results to verify the theoretical results.

For the Stokes equation, introducing the symmetric part of the gradient tensor of the velocity as a stress variable, we present a stress-velocity-pressure field Stokes system. We use some plane elasticity mixed finite elements, including the two elements we proposed, to approximate the stress and velocity fields, and use continuous piecewise polynomial functions to approximate the pressure with the gradient of the pressure approximation being in the corresponding velocity finite element spaces. We derive stability and convergence for these methods.