Pre-Publicación 2026-13
Jessika Camaño, Ricardo Oyarzúa, Katherine Rojo, Segundo Villa-Fuentes:
A mixed finite element method based on pseudostress and stream-function for the Navier–Stokes problem in 2D
Abstract:
In this work, we propose and analyze a new pseudostress–stream function dual-mixed varia- tional formulation for the stationary Navier–Stokes problem in two dimensions with nonho- mogeneous Dirichlet boundary conditions. More precisely, after rewriting the Navier–Stokes system in terms of the pseudostress tensor and the velocity field, the latter is decomposed by means of a Helmholtz–Weyl decomposition in L^p(Ω), which leads to a three-field formu- lation in which the pseudostress, the stream function, and a Lagrange multiplier associated with momentum conservation are the primary unknowns. The associated Galerkin scheme is defined using lowest-order Raviart–Thomas elements for the pseudostress tensor, contin- uous piecewise linear elements for the stream function, and Crouzeix–Raviart elements for the Lagrange multiplier. We prove the well-posedness of both the continuous and discrete formulations by means of the Banach–Nečas-Babuška theorem and Banach’s fixed-point theorem, under a sufficiently small data assumption. In addition, we derive a priori error estimates and establish the optimal rate of convergence. Finally, numerical experiments are provided to validate the theoretical results and to demonstrate the efficiency and accuracy of the proposed method. The present contribution retains the main advantages of classical pseudostress-based formulations, such as the capability to recover additional variables of interest and to ensure momentum conservation, while introducing new features, including exactly divergence-free velocity approximations and a direct approximation of the stream function.
