A reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations

Shafqat Ali; Francesco Ballarin; Gianluigi Rozza

doi:10.3934/acse.2023008

Article Contents

2023, Volume 1, Issue 2: 180-201. Doi: 10.3934/acse.2023008

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A reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations

1.
mathLab, Mathematics area, SISSA, Trieste, Italy
2.
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
3.
Department of Mathematics and Physics, Università Cattolica del Sacro Cuore, Brescia, Italy

^*Corresponding author: Gianluigi Rozza
^*Corresponding author: Gianluigi Rozza

Received: May 2023

Revised: June 2023

Early access: June 2023

Published: June 2023

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Abstract

In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserve the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours [1]. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.

Keywords:

Mathematics Subject Classification: Primary: 65M22, 65M60; Secondary: 35Q30, 76D05.

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Figure 1. Parametrized domain

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Figure 2. Stokes problem: Franca-Hughes stabilization with $ \mathbb{P}_2/\mathbb{P}_2 $ FE pair; RB solutions for Velocity field (left) and Pressure field (right) at different time step from top to bottom; $ t = 0.02, 0.04, 0.06, 0.1, 0.12 $, $ N_u = N_p = 30 $

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Figure 3. Stokes problem: Franca-Hughes stabilization with $ \mathbb{P}_2/\mathbb{P}_2 $ on cavity flow; $ L^2 $-error in time for velocity (left) and pressure (right) with stabilization coefficient $ \delta = 0.05 $ and $ \Delta{t} = 0.02 $

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Figure 4. Stokes problem: $ L^2 $-error in time for velocity (left) and pressure (right) with stabilization coefficient $ \delta = 0.05 $ and $ \Delta{t} = 0.02. $ using $ \mathbb{P}_1/{\mathbb{P}_0} $

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Figure 5. Stokes problem: Franca-Hughes stabilization; $ L^2 $-error in time for Velocity (left) and pressure (right) using $ \mathbb{P}_2/\mathbb{P}_2 $ and $ \delta = 0.05 $, $ \Delta{t} = 0.02, 0.002, 0.0002 $. offline-online stabilization without supremizer

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Figure 6. Navier-Stokes problem with SUPG stabilization; physical parametrization on cavity flow; Error between FE and RB solution for velocity (left) and pressure (right) using $ \mathbb{P}_1/{\mathbb{P}_1} $

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Figure 7. Navier-Stokes problem with SUPG stabilization; physical parametrization on cavity flow; Error between FE and RB solution for velocity (left) and pressure (right) using $ \mathbb{P}_2/{\mathbb{P}_2} $

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Figure 8. Navier-Stokes problem with SUPG stabilization using $ \mathbb{P}_1/{\mathbb{P}_1} $: Velocity (left) and pressure (right) error for physical and geometrical parameters on cavity flow

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Table 1. Stokes problem: Computational details of unsteady Stokes problem (3)

Number of Parameters	2: $ \mu_1 $(viscosity), $ \mu_2 $(domain's length)
$ \mu_1 $ range offline	[0.25, 0.75]
$ \mu_2 $ range offline	[1, 2]
$ \mu_1 $ value online	0.57
$ \mu_2 $ value online	1.78
Final time	0.2
Time step $ \Delta t $	0.02
$ N_{train} $	25
$ N_{max} $	25
Stabilization coefficient $ \delta $	0.05
FE degrees of freedom ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 6222 $ $ 18300 $ ($ \mathbb{P}_2/{\mathbb{P}_2} $)
RB dimension	$ N_u=N_s=N_p=30 $
Computation time ($ \mathbb{P}_2/{\mathbb{P}_1} $)	$ 1780s $ (offline), $ 300s $ (online) with supremizer
Offline time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 1046s $ (offline-online stabilization with supremizer) $ 738s $ (offline-online stabilization without supremizer) $ 980s $ (offline-only stabilization with supremizer)
Offline time ($ \mathbb{P}_2/{\mathbb{P}_2} $)	$ 2260s $ (offline-online stabilization with supremizer) $ 1945s $ (offline-online stabilization without supremizer) $ 1730s $ (offline-only stabilization with supremizer)
Online time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 103s $ (offline-online stabilization with supremizer) $ 82s $ (offline-online stabilization without supremizer) $ 81s $ (offline-only stabilization with supremizer)
Online time ($ \mathbb{P}_2/{\mathbb{P}_2} $)	$ 242s $ (offline-online stabilization with supremizer) $ 180s $ (offline-online stabilization without supremizer) $ 90s $ (offline-only stabilization with supremizer)

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Table 2. Navier-Stokes problem with physical parameter only: Computational details of unsteady Navier-Stokes problem without Empirical Interpolation

Physical parameter	$ \mu $ (Reynolds number)
Range of $ \mu $	[100,200]
Online $ \mu $ (example)	130
FE degrees of freedom	5934 ($ \mathbb{P}_1/{\mathbb{P}_1} $)
RB dimension	$ N_u=N_s=N_p=30 $
Offline time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 40612s $ (offline-online stabilization with supremizer) $ 38781s $ (offline-online stabilization without supremizer)
Online time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 4640s $ (offline-online stabilization with supremizer) $ 4040s $ (offline-online stabilization without supremizer)
Time step	$ 0.02 $
Final time	$ 0.5 $

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Table 3. Computational details for unsteady Navier-Stokes problem with physical and geometrical parameters: stabilization and computational reduction

Physical parameter	$ \mu_1 $ (Reynolds number)
Geometrical parameter	$ \mu_2 $ (horizontal length of domain)
Range of $ \mu_1 $	[100,200]
Range of $ \mu_2 $	[1.5, 3]
$ \mu_1 $ online (example)	130
$ \mu_2 $ online (example)	2
FE degrees of freedom	6222 ($ \mathbb{P}_1/{\mathbb{P}_1} $)
RB dimension	$ N_u=N_s=N_p=30 $
Offline time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 44693s $ (offline-online stabilization with supremizer) $ 40153s $ (offline-online stabilization without supremizer)
Online time ($ \mathbb{P}_1/{\mathbb{P}_1} $)	$ 5169s $ (offline-online stabilization with supremizer) $ 4724s $ (offline-online stabilization without supremizer)
Time step	$ 0.02 $
Final time	$ 0.5 $

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