15:45
MS23: Recent Advances in Numerical Simulation of Incompressible Flows (Part 1)
Chair: Christoph Lehrenfeld
15:45
25 mins
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Virtual Elements for the Stokes and Navier-Stokes equations in 2D
Lourenco Beirao da Veiga, Carlo Lovadina, Giuseppe Vacca
Abstract: The Virtual Element Method (in short VEM, introduced in [3, 4]) is recent
generalization of the Finite Element Method that enjoys also a connection with
modern Mimetic schemes. By avoiding the explicit integration of the shape functions that span the discrete Galerkin space and introducing a novel construction of the associated sti ness matrix, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/ polyhedral meshes (even non-conforming) also with non-convex elements and still yields a conforming solution with (possibly) high order accuracy.
In the present talk we introduce the Virtual Element Method in the framework
of fluid dynamics, more specifically the Stokes and Navier-Stokes equations
in two dimensions [1, 2]. We present a method of general order of accuracy that
(in addition to enjoying the important advantage of handling general polytopal
meshes) exploits the flexibility of Virtual Elements in order to obtain an exactly
divergence-free solution. This is well known to yield a set of advantages,
when compared to more traditional inf-sup stable methods, that we explore both theoretically and numerically. After a detailed introduction to the method, we present the main theoretical results and close the presentation by showing numerical tests. In a companion talk in the same Minisymposium a co-author will also present the 3D case.
References
[1] L. Beir~ao da Veiga, C. Lovadina, and G. Vacca. Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model.
Numer. Anal., 51(2):509-535, 2017.
[2] L. Beir~ao da Veiga, C. Lovadina, and G. Vacca. Virtual elements for
the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal.,
56(3):1210-1242, 2018.
[3] L. Beir~ao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and
A. Russo. Basic principles of virtual element methods. Math. Models Methods
Appl. Sci., 23(1):199-214, 2013.
[4] L. Beir~ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The Hitchhiker's
Guide to the Virtual Element Method. Math. Models Methods Appl. Sci.,
24(8):1541-1573, 2014.
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16:10
25 mins
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Virtual Elements for the Stokes and Navier-Stokes equations in 3D
Lourenco Beirao da Veiga, Franco Dassi, Giuseppe Vacca
Abstract: In the present talk we develop the Virtual Element Method (VEM) for the Stokes and Navier--Stokes equation in three dimensions for general polynomial order $k \geq 2$.
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In addition to the important flexibility of dealing with general polyhedral meshes, the presented scheme leads to an exactly divergence-free discrete velocity solution.
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We highlight that this feature is not shared by most of the standard mixed Finite Element methods, where the divergence-free constraint is imposed only in a weak (relaxed) sense.
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We analyse the theoretical property of the method and then show specific tests that underline the divergence free nature of the discrete solutions.
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The proposed Virual Elements is the natural extension to the 3D case of the corresponding 2D divergence-free VEM elements introduced in \cite{Stokes:divfree, Bdv-Lovadina-Vacca:2018} and presented by a co-author in a companion talk in the same Minisymposium.
@article{Bdv-Lovadina-Vacca:2018,
title={Virtual elements for the {N}avier--{S}tokes problem on polygonal meshes},
author={Beir{\~a}o da Veiga, L. and Lovadina, C. and Vacca, G.},
journal={SIAM J. Numer. Anal.},
VOLUME = {56},
NUMBER = {3},
PAGES = {1210--1242},
year={2018}
}
@article {Stokes:divfree,
author = {Beir{\~a}o da Veiga, L. and Lovadina, C. and Vacca, G.},
title = {Divergence free virtual elements for the {S}tokes problem on polygonal meshes},
JOURNAL = {ESAIM Math. Model. Numer. Anal.},
FJOURNAL = {ESAIM. Mathematical Modelling and Numerical Analysis},
year = {2017},
pages = {509--535},
volume = {51},
number = {2},
}
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16:35
25 mins
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The unfitted HHO method for the Stokes problem on curved domains
Erik Burman, Guillaume Delay, Alexandre Ern
Abstract: We design a hybrid high-order (HHO) method to approximate
the Stokes problem on curved domains using unfitted meshes.
We prove inf-sup stability and a priori estimates with
optimal convergence rates. Moreover, we provide numerical
simulations that corroborate the theoretical convergence rates.
A cell-agglomeration procedure is used to prevent the appearance of small cut cells.
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17:00
25 mins
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A gradient-robust, well-balanced discretisation for the compressible barotropic Stokes problem
Christian Merdon, Alexander Linke
Abstract: This talk suggests a novel well-balanced discretisation of the stationary compressible Stokes problem that has a number of interesting properties. An upwind-stabilised finite volume discretisation of the continuity equation and a pseudo time integration for the density ensures existence of a discrete solution and non-negativity of the discrete density. Second, a reconstruction operator is used in the right-hand side of a Bernardi--Raugel finite element discretisation of the momentum balance that maps discretely divergence-free testfunctions to divergence-free ones. This ensures a certain well-balanced property in the sense that arbitrary gradient forces are balanced by the discrete pressure if there is enough mass to compensate them. Moreover, if the Mach number converges to zero, the scheme converges to a pressure-robust discretisation of the incompressible Stokes problem. All properties are demonstrated in numerical examples.
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