## Preprint 2021-22

## Ricardo Oyarzúa, Manuel Solano, Paulo Zuñiga:

### Analysis of an unfitted mixed finite element method for a class of quasi-Newtonian Stokes flow

### Abstract:

We propose and analyze an unfitted method for a dual-dual mixed formulation of a class of Stokes models with variable viscosity depending on the velocity gradient, in which the pseudoestress, the velocity and its gradient are the main unknowns. On a fluid domain Ω with curved boundary Γ we consider a Dirichlet boundary condition and employ an approach previously applied to the Stokes equations with constant viscosity, which consists of approximating Ω by a polyhedral computational subdomain Ω_h, not necessarily fitting Ω, where a Galerkin method is applied to compute solution. Furthermore, to approximate the Dirichlet data on the computational boundary Γ_h, we make use of a transferring technique based on integrating the discrete velocity gradient. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas of order k ≥ 0 for the pseudostress, and discontinuous polynomials of degree k for the velocity and its gradient. For the a priori error analysis we provide suitable assumptions on the mesh near the boundary Γ ensuring that the associated Galerkin scheme is well-posed and optimally convergent with O(h^(k+1)). Next, for the case when Γ_h is taken as a piecewise linear interpolation of Γ, we develop a reliable and quasi-efficient residual-based a posteriori error estimator. Numerical experiments verify our analysis and illustrate the performance of the associated a posteriori error indicator.