## Preprint 2023-19

## Isaac Bermudez, Claudio I. Correa, Gabriel N. Gatica, Juan P. Silva:

### A perturbed twofold saddle point-based mixed finite element method for the Navier-Stokes equations with variable viscosity

### Abstract:

This paper proposes and analyzes a new mixed variational formulation for the Navier-Stokes equations with constant density and variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, the present approach employs a technique previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms and the pressure. Secondly, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the nonlinear viscosity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We analyze the continuous and discrete solvability of this problem using a relevant abstract theory developed specifically for this purpose, by linearizing the perturbation and applying the classical Banach fixed point theorem. In particular, given an integer $\ell \, \geq \, 0$, feasible choices of finite element subspaces include piecewise polynomials of degree $\, \leq \, \ell$ for the gradient of velocity, Raviart-Thomas spaces of order $\ell$ for the pseudostress, and piecewise polynomials of degree $\, \leq \, \ell$ for the velocity. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence, are reported.