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Stabilization techniques for pressure recovery applied to POD-Galerkin methods for the incompressible Navier-Stokes equations G. Stabile, G. Rozza SISSA, International School for Advanced Studies, MathLab, Mathematics Area, Trieste, Italy Quantification of Uncertainty Improving Efficiency and Technology QUIET 2017 – Trieste 18-21/07/2017

Framework and motivations • In order to efficiently apply Uncertainty Quantification in computational fluid dynamics problems one needs inexpensive computational models to solve the forward problem . In this direction the development of efficient and reliable reduced order models (ROMs) would be a great advantage. • It is well known that Galerkin based ROMs of the incompressible Navier-Stokes equations suffer from stability issues for what concern the pressure term . The considered system of PDEs consists in the unsteady parametrized incompressible Navier Stokes Equations.  ∂ u ∂ t + ( u · ∇ ) u − ∇ · ν ∇ u = − ∇ p in Ω    ∇ · u = 0 in Ω    u = u ( µ ) on ∂ Ω , in (1)  u = 0 on ∂ Ω , 0     ( µ ∇ u − p I ) n = 0 on ∂ Ω , out  The offline stage is performed using a Finite Volume Method (OpenFOAM) while the projection and online stage are based on the in-house package ITHACA-FV . G. Stabile Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations 1/ 3

Methods Reduced Order Modelling Most of the problems require high dimensional parametrized simulations. OFFLINE STAGE Reduced Basis Approximation of the Full order model: fields Generation of the high dimensional system of POD reduced PDEs: Solved with a N u a i ϕ i u≈u r = ∑ i = 1 Basis spaces Finite Volume technique V =[ϕ 1 , ϕ 2 , ... , ϕ N u ] N p b i χ i p≈ p r = ∑ i = 1 U =[ u ( t 1 ) ,u ( t 2 ) , ... ,u ( t n )] Q =[χ 1 , χ 2 , ... , χ N p ] P =[ p ( t 1 ) , p ( t 2 ) , ... , p ( t n )] Galerkin projection of the governing equation onto the reduced basis spaces STAB. METHODS ONLINE STAGE Poisson equation Lower Dimensional System for Of ODEs pressure T + a' C a M ˙ a = aB + b K K a = 0 Enrichment of the velocity space to ensure the Inf-Sup Stability issues condition G. Stabile Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations 2/ 3

Conclusions References [1] G. Stabile, S. Hijazi, A. Mola, S. Lorenzi, and G. Rozza. POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder. In Press, 2017 [2] G. Stabile and G. Rozza, Stabilized Reduced order POD-Galerkin techniques for finite volume approximation of the parametrized Navier–Stokes equations, submitted, 2017. [3] F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier–Stokes equations, International Journal for Numerical Methods in Engineering, vol. 102, no. 5, pp. 1136–1161, 2015. G. Stabile Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations 3/ 3