Verónica Anaya

In this work, we present two Nitsche-based mixed finite element methods. The first one concerns the scalar Poisson problem on a domain whose boundary is partitioned into Dirichlet and Neumann (linear) conditions. For this problem, we introduce a new discrete variational formulation that enables the weak imposition of mixed boundary conditions. The proposed method is analyzed within the Babuška-Brezzi framework, ensuring the well-posedness of the discrete problem. We derive optimal a priori error estimates and present numerical experiments that confirm the theoretical convergence rates and demonstrate the robustness of the proposed scheme. The second method is a mixed Nitsche-type discretization for the unilateral (non-linear) Signorini conditions in the scalar case, as a simple prototype for contact and friction problems or other variational inequalities associated with boundary conditions. We present the full variational formulation. The performance of the method is assessed by numerical experiments that illustrate the accurate enforcement of the boundary conditions and the expected behavior under mesh refinement.

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use fixed--point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.

We develop the a posteriori error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The disc...

A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L²-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an a posteriori error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and also confirm the theoretical findings.