Divergence‐free tangential finite element methods for incompressible flows on surfaces

Abstract

In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H 1 -conformity allows us to construct finite elements which are-due to an application of the Piola transformation-exactly tangential. To reintroduce continuity (in a weak sense)we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, H(div Γ )-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy. K E Y W O R D Sdivergence-conforming finite elements, incompressible Navier-Stokes equations, Piola transformation, surface PDEs, tangential vector field Int J Numer Methods Eng. 2020;121:2503-2533.wileyonlinelibrary.com/journal/nme

A numerical approach for fluid deformable surfaces

“…The computational examples can provide benchmark problems for other numerical approaches, which can be extended to the considered model, e.g. Nitschke et al (2017), Olshanskii et al (2018), Torres-Sanchez, Santos-Olivan & Arroyo (2020), Lederer et al (2019). They also form the basis for more complex models, which include coupling with concentration fields for proteins and dependency of H 0 on concentration in lipid bilayers, or coupling with liquid crystal theory as in Nitschke et al (2019a) for Erickson-Leslie type models or with Landau-de Gennes theory on surfaces (Nitschke et al 2019b) for Beris-Edwards type models, which also can be extended by active contributions to model, e.g.…”

“…The growing interest in these phenomena in biology is in contrast with the available tools to numerically solve the governing equations. Even for surface fluids on stationary surfaces, where the governing equations have been known since the pioneering work of Scriven (1960), numerical tools have only been developed recently (see Nitschke, Voigt & Wensch (2012), Gross & Atzberger (2018) for simply-connected surfaces and Nitschke, Praetorius & Voigt (2017), Reuther & Voigt (2018b), Fries (2018), Lederer, Lehrenfeld & Schöberl (2019) for general surfaces). The governing equations for fluid deformable surfaces have been more recently derived in a different context (see Arroyo & DeSimone (2009), Salbreux & Jülicher (2017), Miura (2018)), but have never been solved in a general setting.…”

A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow

“…To design this experiment, we follow what done in [26,24,34]. The initial velocity field is given by the counter-rotating upper and lower hemispheres with speed approximately equal 1 closer to equator and vanishing at the poles.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Stemming from the work in [117], different HHO formulations of the incompressible Navier-Stokes equations were proposed, incorporating a skew-symmetric form of the convection term [31] and a globally divergence-free velocity approximation to achieve robustness in presence of large irrotational body forces [45]. Moreover, special attention was devoted in recent years to the development of hybridised DG schemes [126] with pointwise divergence-free velocity [171,181,223] and with relaxed H(div)-conformity [177,178], as well as divergence-conforming hybrid DG discretisations for incompressible flows on surfaces [179]. It is worth noting that all the above mentioned references focus on viscous laminar flows and preliminary promising results on the incompressible Reynolds averaged Navier-Stokes (RANS) equations coupled with the Spalart-Allmaras turbulence model were recently presented in [210].…”