A mass-conserving stress-yielding formulation for the Stokes equation with exact stress symmetry
We introduce a new finite element discretization for a mixed formulation of the two-dimensional incompressible Stokes equations with symmetric viscous stresses. The method is based on a mass-conserving stress-yielding formulation (Gopalakrishnan, Lederer, Schöberl; A mass conserving mixed stress formulation for the Stokes equations; IMA J. Numer. Anal., Vol 40, 2020) with symmetric stresses, with the key novelty that the symmetry of the stress tensor is enforced exactly as an intrinsic property of the discrete space. This space is constructed on the Clough--Tocher macroelement split and consists of matrix-valued functions whose normal-tangential components across macroelement interfaces are continuous. Since the exact symmetry is enforced via local polynomial bubble functions on the subelements, which can be eliminated via static condensation, the number of globally coupled degrees of freedom coincide with those of the original mass-conserving stress-yielding formulation. The resulting pressure-robust method is stable without requiring any additional enrichment. We establish optimal convergence rates for all the variables and present numerical experiments that confirm the theoretical results.