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Analysis of an augmented mixed-FEM for the Navier-Stokes problem

Authors: Jessika Camaño, Ricardo Oyarzúa and Giordano Tierra
Journal: Math. Comp.
MSC (2010): Primary 65N15, 65N30, 76D05, 76M10
Published electronically: June 20, 2016
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Abstract: In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a ``nonlinear-pseudostress'' tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. 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. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach fixed point theorem and the Lax-Milgram lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree $ k$ for the nonlinear-pseudostress tensor and continuous piecewise polynomial elements of degree $ k+1$ for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.

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Additional Information

Jessika Camaño
Affiliation: Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile – and – CI$^{2}$MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Ricardo Oyarzúa
Affiliation: Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile – and – CI$^{2}$MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Giordano Tierra
Affiliation: Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Prague 8, 186 75, Czech Republic
Address at time of publication: Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, Pennsylvania 19122

Keywords: Navier-Stokes, mixed finite element method, augmented formulation, Raviart-Thomas elements
Received by editor(s): November 19, 2014
Received by editor(s) in revised form: July 29, 2015, and September 12, 2015
Published electronically: June 20, 2016
Additional Notes: This research was partially supported by CONICYT-Chile through project Inserción de Capital Humano Avanzado en la Academia 79130048; project Fondecyt 11140691, project Fondecyt 11121347, project Anillo ACT1118 (ANANUM); by DIUBB project 120808 GI/EF; and by Ministry of Education, Youth and Sports of the Czech Republic through the ERC-CZ project LL1202.
Article copyright: © Copyright 2016 American Mathematical Society