A conforming and divergence-free approximation for the pseudostress-based formulation of the Stokes problem

Abstract: In this work we propose and analyze a conforming and mass-conservative pseudostress-based numerical scheme for the Stokes problem. More precisely, we extend previous results on pseduostress-velocity formulations for the Stokes problem, with unknowns originally in $H(\div)$ and $L^2$, respectively, and apply a Helmholtz decomposition to the velocity to derive a new dual-mixed formulation for the fluid flow problem, which considers now the velocity in $H(\div)$.

Consequently, we obtain a three-field mixed variational formulation where the pseudostress and the velocity, both in $H(\div)$ and a further unknown in $H^1$, representing the null function, are the main unknowns of the system.

Then the associated Galerkin scheme can be defined by employing Raviart--Thomas elements of degree $k$ for the pseudostress, divergence-free Raviart--Thomas elements of degree $k$ for the velocity, and Lagrange elements of degree $k+1$ for the aforementioned additional unknown.

For both, the continuous and discrete problems, we employ the Babu\v ska--Brezzi theory to prove unique solvability.

We also provide the convergence analysis and derive the corresponding theoretical rate of convergence. In addition, we propose a reliable and efficient residual-based a posteriori error estimator for the resulting Galerkin scheme.

Finally, numerical results illustrating the performance of the method are provided.