H(div) conforming methods for the rotation form of the incompressible fluid equations
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H(div) conforming methods for the rotation form of the incompressible fluid equations Xi Chen1 · Corina Drapaca2 Received: 22 January 2020 / Revised: 20 August 2020 / Accepted: 29 August 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzmán et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the ) ( incompressible Navier-Stokes equation with the full version of the stress tensor 𝜈 ∇u + ∇uT − 32 (∇ ⋅ u)𝕀 , instead
of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzmán et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order p ∈ {2, 3} , (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization. Keywords Rotation form · Bernoulli function · Full stress tensor · Incompressible Navier-Stokes · Finite element methods · H(div) conforming
* Xi Chen [email protected] Corina Drapaca [email protected] 1
Department of Mechanical Engineering, The Pennsylvania State University, State College, PA 16802, USA
2
Department of Engineering Science and Mechanics, The Pennsylvania State University, State College, PA 16802, USA
13
Vol.:(0123456789)
32
Page 2 of 26
X. Chen, C. Drapaca
Mathematics Subject Classification 65N30
1 Introduction One of the most important properties that numerical discretizations for incompressible viscous flows should have is pressure robustness. Classical mixed finite element methods lack pressure robustness in the sense that the pressure approximation scaled by the inverse of the viscosity arises [19] in the error estimate of the velocity, which leads to the pressure induced locking seen especially in convection dominated flows [26]. From a physical point of view, this means that the mass is not conserved. Even with the help of grad-div stabilization [1, 5, 14, 18, 29], the mass conservation is still not satisfied exactly [19]. The way to resolve this problem is to approximate the Helmholtz-Hodge decomposition exactly. This can be achieved by making the weakly divergence free subspace pointwise divergence free.
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