Preprint 2026-34
Gonzalo A. Benavides, Sergio Caucao, Gabriel N. Gatica:
New skew-symmetry-based mixed finite element methods for the Navier-Stokes equations with variable viscosity
Abstract:
In this paper, we introduce and analyze a new skew-symmetry-based mixed formulation for the Navier--Stokes equations with variable viscosity, which leads to the derivation of two families of stable mixed finite element methods for their numerical approximation. Motivated by the need to employ the symmetric stress tensor in this setting, our approach introduces the strain rate and vorticity tensors as independent unknowns. Their average, acting on the velocity, provides one half of the convective term, while the remaining contribution is accounted for in the momentum equation by introducing an incomplete symmetric stress tensor containing half of the tensor product of the velocity with itself. The resulting variational system exhibits a typical saddle-point structure in Banach spaces, but with a nonlinear main operator. A key step in its derivation is the introduction of a null expression given by the tensor inner product of suitably chosen symmetric and skew-symmetric tensors. We then show that the variational system is equivalent to a reduced problem posed on the kernel of the coupling bilinear form. Existence and uniqueness of a solution to the latter are obtained by means of a fixed-point argument that linearizes the nonlinear part of the aforementioned operator, yielding a skew-symmetric bilinear form in its place, and by applying the Schauder and Banach fixed-point theorems. In contrast, the solvability analysis of the Galerkin scheme is carried out directly in the original saddle-point setting. Under suitable hypotheses on the finite element subspaces, existence and uniqueness of a discrete solution are proved through a similar fixed-point strategy, using the Brouwer and Banach fixed-point theorems. A priori error estimates are then derived, including a corresponding bound for the pressure approximation, with this unknown having been eliminated from the continuous formulation and subsequently recovered by postprocessing. PEERS- and AFW-based finite element subspaces of order $k \ge 0$, defined on shape-regular simplicial meshes, are shown to satisfy the required stability conditions, and the associated convergence rates are established. Finally, several numerical experiments illustrating the performance of the methods and confirming the theoretical findings are reported.
