A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy
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A new mixed‑FEM for steady‑state natural convection models allowing conservation of momentum and thermal energy Sergio Caucao1,2 · Ricardo Oyarzúa3,4 · Segundo Villa‑Fuentes3 Received: 14 November 2019 / Revised: 10 August 2020 / Accepted: 26 September 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this work we present a new mixed finite element method for a class of steadystate natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, 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 the vector unknown, and discontinuous piece-wise polynomial elements of degree k for the velocity and temperature. With this choice of spaces, both, momentum and thermal energy, are conserved if the external forces belong to the velocity and temperature discrete spaces, respectively, which constitutes one of the main feature of our approach. We prove unique solvability for both, the continuous and discrete problems and provide the corresponding convergence analysis. Further variables of interest, such as the fluid pressure, the fluid vorticity, the fluid velocity gradient, and the heat-flux 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. Keywords Stationary Boussinesq equations · Divergence-conforming · Stressvelocity formulation · Mixed finite element method · Conservation of momentum · Conservation of thermal energy This work was partially supported by CONICYT-Chile through project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, project Fondecyt 1161325, Becas de Doctorado Nacional año académico 2018 and BecasChile para postdoctorado en el extranjero convocatoria 2018; by Department of Mathematics, University of Pittsburgh; and by Universidad del Bío-Bío through VRIP-UBB project 194608 GI/C. Extended author information available on the last page of the article
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Mathematics Subject Classification 65N30 · 65N12 · 65N15 · 35Q79 · 80A20 · 76R05 · 76D07
1 Introduction The motion of a liquid or gas, generated by some parts of the fluid being heavier than other parts, or in other words, produced by density differences as, for example, when a liquid in a vessel is heated from below, is a process known as nat
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