Research Outputs
Analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem
In this paper, we propose and analyze a new momentum conservative mixed finite element method for the Navier–Stokes problem posed in nonstandard Banach spaces. Our approach is based on the introduction of a pseudostress tensor relating the velocity gradient with the convective term, leading to a mixed formulation where the aforementioned pseudostress tensor and the velocity are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree K for the pseudostress tensor and discontinuous piece–wise polynomial elements of degree K for the velocity. With this choice of spaces, the equilibrium equation is exactly satisfied if the external force belongs to the velocity discrete space, thus the method conserves momentum, which constitutes one of the main feature of our approach. For both, the continuous and discrete problems, the Banach–Nečas–Babuška and Banach's fixed-point theorems are employed to prove unique solvability. We also provide the convergence analysis and particularly prove that the error decay with optimal rate of convergence. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Finally, several numerical results illustrating the performance of the method are provided.
Divergence-free finite elements for the numerical solution of a hydroelastic vibration problem
In this paper, we analyze a divergence-free finite element method to solve a fluid–structure interaction spectral problem in the three-dimensional case. The unknowns of the resulting formulation are the fluid and solid displacements and the fluid pressure on the interface separating both media. The resulting mixed eigenvalue problem is approximated by using appropriate basis of the divergence-free lowest order Raviart–Thomas elements for the fluid, piecewise linear elements for the solid and piecewise constant elements for the interface pressure. It is proved that eigenvalues and eigenfunctions are correctly approximated and some numerical results are reported in order to assess the performance of the method.
A five-field mixed formulation for stationary magnetohydrodynamic flows in porous media
We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman–Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier associated to the divergence-free condition of the magnetic field, a convenient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. The resulting mixed scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor, discontinuous piecewise polynomial elements of degree for the velocity and the translation of the velocity gradient, Nédélec elements of degree for the magnetic field and Lagrange elements of degree for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.
Weights for moments’ geometrical localization: A canonical isomorphism
This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.
Basis for high order divergence-free finite element spaces
A method classically used in the lower polynomial degree for the construction of a finite element basis of the space of divergence-free functions is here extended to any polynomial degree for a bounded domain without topological restrictions. The method uses graphs associated with two differential operators: the gradient and the divergence, and selects the basis using a spanning tree of the first graph. It can be applied for the two main families of degrees of freedom, weights and moments, used to express finite element differential forms.
A posteriori error analysis of a momentum conservative Banach spaces based mixed-FEM for the Navier-Stokes problem
In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.
A graph-based algorithm for the approximation of the spectrum of the curl operator
We analyze a new algorithm for the finite element approximation of a family of eigenvalue problems for the curl operator that includes, in particular, the approximation of the helicity of a bounded domain. It exploits a tree-cotree decomposition of the graph relating the degrees of freedom of the Lagrangian finite elements and those of the first family of Nédélec finite elements to reduce significantly the dimension of the algebraic eigenvalue problem to be solved. The algorithm is well adapted to domains of general topology. Numerical experiments, including a not simply connected domain with a not connected boundary, are presented in order to assess the performance and generality of the method.
Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics
In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided.