A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

Abstract

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.