An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier-Stokes flows

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DOI:10.1051/m2an/2014013 www.esaim-m2an.org

AN EFFICIENT COMPUTATIONAL FRAMEWORK FOR REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION

OF PARAMETRIZED NAVIER–STOKES FLOWS

Andrea Manzoni

¹

Abstract.We present the current Reduced Basis framework for the eﬃcient numerical approximation of parametrized steady Navier–Stokes equations. We have extended the existing setting developed in the last decade (seee.g.[S. Deparis,SIAM J. Numer. Anal.46(2008) 2039–2067; A. Quarteroni and G. Rozza, Numer. Methods Partial Diﬀer. Equ.23 (2007) 923–948; K. Veroy and A.T. Patera, Int.

J. Numer. Methods Fluids47(2005) 773–788]) to more general affine and nonaffine parametrizations (such as volume-based techniques), to a simultaneous velocity-pressure error estimates and to a fully decoupled Offline/Online procedure in order to speedup the solution of the reduced-order problem.

This is particularly suitable for real-time and many-query contexts, which are both part of our final goal. Furthermore, we present an efficient numerical implementation for treating nonlinear advection terms in a convenient way. A residual-based a posteriori error estimation with respect to a truth, full-order Finite Element approximation is provided for joint pressure/velocity errors, according to the Brezzi–Rappaz–Raviart stability theory. To do this, we take advantage of an extension of the Successive Constraint Method for the estimation of stability factors and of a suitable fixed-point algorithm for the approximation of Sobolev embedding constants. Finally, we present some numerical test cases, in order to show both the approximation properties and the computational efficiency of the derived framework.

Mathematics Subject Classiﬁcation. 65M15, 65M60, 65N12, 76D07, 78M34.

Received July 23, 2013. Revised February 5, 2014.

Published online July 15, 2014.

1. Introduction

In this paper we describe some new contributions to the Reduced Basis (RB) approximation and a pos- teriori error estimation of parametrized steady incompressible Navier–Stokes equations. We consider a mixed velocity/pressure formulation of the problem, by addressing the case of (both affine and nonaffine) physical and geometrical parametrizations, and including for the first time the case of external flows around nonaffinely parametrized profiles. Our last goal is to solve real-time numerical simulations and optimization or many-query problems in a very efficient way. Although very strong efforts have been carried out in the last decade to achieve this goal, classical, full-order approximation techniques – such as the Finite Element (FE) method – still entail strong computational costs, so that a further enhancement of the computational efficiency is needed.

Keywords and phrases.Reduced Basis Method, parametrized Navier–Stokes equations, steady incompressible ﬂuids,a posteriori error estimation, approximation stability.

1 SISSA Mathlab – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

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This is the main reason whyReduced-Order Models(ROMs), such as the RB method, have gained increasing popularity in recent years. The goal of a RB method is to approximate the manifold of solutions –i.e., the set of solutions obtained by varying the parameter vector – through a Galerkin projection onto a space spanned by few precomputed snapshots (that is, solutions of the full-order problem at some selected parameters values).

In this work we restrict ourselves to the case of parametrized steady incompressible Navier–Stokes equations. In fact, this problem arises in several fluid flows applications; its rapid and reliable solution represents a remarkable challenge for a ROM, because of nonlinearities, stability and efficiency issues [16].

First of all, the nonlinear nature of Navier–Stokes equations yields serious concerns from both a physical and a computational standpoint. For instance, ﬂow can undergo strong changes (bifurcation phenomena) when the Reynolds number reaches critical values, or even loss of symmetry. Thus, in order to obtain a reliable RB approximation, we require that the manifold of the solutions is smooth and low-dimensional, and that it consists of a branch of nonsingular solutions²– the latter is a standard assumption also in the FE method [4,5].

Moreover, we need to face the stability issue related to the calculation of pressure ﬁelds. As already in the Stokes case, we can recover the pressure stability at the reduced order level by enriching the reduced velocity space with the so-calledsupremizer solutions. In this way, we manage to fulﬁll a Brezzi inf-sup condition for the reduced spaces as well. Concerning instead the stability of the approximation, we rely on the Brezzi–Rappaz–

Raviart(BRR) theory [4,5] for the approximation of branches of nonsingular solutions. We remind that stability factors aﬀect not only the continuous dependence of the solution on data [32], but also the a posteriori error estimation ensuring the reliability of the RB method.

Furthermore, for the sake of computational eﬃciency, we need to properly extend the usual Oﬄine/Online splitting [24,27] also to nonlinear terms, in order to make both the reduced-order approximation and the error estimation independent of the full-order approximation, and thus very cheap.

In this paper we show how to deal with these issues. For the first time, we address the multifaceted components behind the analysis of the complete steady framework. This includes nonaffine geometrical parametrizations, error bound estimations for velocity/pressure fields jointly, an improved algorithm for the estimation of parametrized lower bounds of stability factors and a modified fixed-point algorithm for the estimation of Sobolev embedding constants, together with rigorous mathematical proofs. In particular, this framework takes advantage of a full uncoupling between the estimation of stability factors and the construction of a reduced space through a greedy selection of velocity-pressure snapshots. This option has been made possible thanks to a recent extension of the natural-norm Successive Constraint Method (SCM) [13,29] to the case of nonlinear parametrized operators, addressed in [18].

More in detail, we present a general parametrized formulation of Navier–Stokes equations (Sect.2), capable of managing with both (physical and geometrical) affine and nonaffine parametrizations, by relying in the latter case on the Empirical Interpolation Method (EIM) [1]. We also recall the main assumptions required to ensure the well-posedness of both the parametrized formulation and the full-order approximation (Sects.3–4), pointing out which features must be enforced at the reduced-order level (Sect.5). Another original contribution of this paper is the setting ofa posteriorierror estimation for velocity and pressure fields jointly (Sect.6), obtained by exploiting the BRR theory. Moreover, we sketch the main points of the extended SCM [18] for computing lower bounds of stability factors (Sect.7), and provide a detailed proof of a fixed-point approximation of the Sobolev embedding constants entering in the expression of the error bound (Sect. 8). Finally, we show some numerical tests dealing with low and moderate Reynolds flows in parametrized geometries (Sect. 9). Technical proofs of the stated results are detailed in Appendix A.

We refer to [10,28] for former contributions on the RB approximation of Stokes problems, and to the recent work [26], for both a stability anda posteriorierror analysis based on the Brezzi’s and the Babuˇska’s inf-sup the- ories, respectively. In the Navier–Stokes case, after the pioneering works by Peterson [22], Ito and Ravindran [14], a general framework for both RB approximation anda posteriorierror estimation has been established by Patera,

2RB methods might provide reliable approximations also when bifurcation points are included in the parameter space, if the latter is properly sampled; seee.g.[12]. Further investigation is ongoing on this issue.

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Veroy [31] and Nguyen [21]. This has been further developed in more recent years [7,8] by considering a complete natural norm framework, whereas a ﬁrst case of simple nonaﬃne geometrical parametrizations can be found in [23].

2. Parametrized formulation of steady Navier–Stokes equations

The Navier–Stokes (NS) equations provide a model for the ﬂow motion of a viscous Newtonian incompressible ﬂuid. In the steady case, on a spatial domain ˜Ω⊂R^d,d= 2,3 (which we assume to be piecewiseC²with convex corners), they read as follows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−νΔv+δ(v· ∇)v+∇p=f in ˜Ω

∇ ·v= 0 in ˜Ω

v=0 onΓD₀

v=g_D onΓDg

−pn+ν∂v

∂n =g_N onΓN,

(2.1)

where (v, p) = (v(μ), p(μ)) are the velocity and the pressure fields defined on ˜Ω, for some givenf ∈(L²( ˜Ω))^d, gD∈(H¹^/²(ΓD))^d,gN ∈(H¹^/²(ΓN))^d. We denote byΓD=ΓD₀∪ΓD_g the Dirichlet portion of∂Ω, and by˜ nthe normal unit vector to∂Ω. We also define the Reynolds number as Re =˜ L|v¯|/ν, whereLis a characteristic length of the domain, ¯v a typical velocity of the flow andν the kinematic viscosity. We concentrate on laminar flows, withRe∈[1,10³], in the cased= 2, although the whole framework keeps holding for the cased= 3 as well. Here μ= (μp,μg)^T ∈ D ⊂R^P is a vector of parameters which may characterize either the geometrical configuration – so that ˜Ω= ˜Ω(μg) – or physical properties, such asν=ν(μp), boundary datagD=gD(μp),gN =gN(μp) or source terms f = f(μp). For the sake of notation, we shall distinguish between np physical parameters μp∈ Dp ⊂Rⁿ^p andng =P −np physical parametersμg ∈ Dg ⊂Rⁿ^g. Hereon, we omit the dependence onμ wherever understood, and we adopt the convention that repeated indices are implicitly summed over.

If parameters affect the geometrical configuration (i.e. ng ≥ 1), we assume that the parameterized configuration ˜Ω(μg) can be obtained as the image of a reference domain Ω through a parametrized map T(·;μg) :R²×D →R². In general, we can deal with original domains made up ofK_dommutually nonoverlapping open subdomains{Ω^k}^K_k₌₁^dom, so that original and reference subdomains can be linked via either an affine or a nonaffine mapT^k(·;μg) :R²× D →R²,i.e.Ω˜^k(μ) =T^k(Ω^k;μg), 1≤k≤K_dom. Seee.g.[27] for the regularity assumptions required to the mapsT^k(·;μg) in order to ensure the well-posedness of (2.2).

Theparametrized variational formulation of (2.1) can be obtained by standard integration by parts. When dealing with parametrized geometries, we trace the usual variational formulation, written on ˜Ω(μg), back on the reference domainΩ, so that: givenμ∈ D ⊂R^P, we seek for (v, p) = (v(μ), p(μ))∈V ×Qsuch that

˜a(v,w;μ) +b(p,w;μ) +c(v,v,w;μ) =F(w;μ) ∀w∈V

b(q,v;μ) =G(q;μ) ∀q∈Q. (2.2)

The parametrized forms appearing in (2.2) are deﬁned as follows [23,28]:

a(v,w;μ) =

K_dom

k=1

Ω^k

∂v

∂xi

κ^k_ij(·;μ)∂w

∂xj

, b(q,w;μ) =−

K_dom

k=1

Ω^k

qχ^k_ij(·;μ)∂wj

∂xi

, (2.3)

c(v,w,z;μ) =

K_dom

k=1

Ω^k

viχ^k_ji(·;μ)∂wm

∂xj

zm, (2.4)

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whereas the transformation tensors appearing in (2.3)-(2.4) are given by:

κ^k(x;μ) =ν(μp) J_T^k ·;μg

₋T

J_T^k x;μg

₋₁

|JT^k x;μg

|, 1≤k≤K_dom (2.5)

χ^k(x;μ) = J_T^k x;μg

₋T

|JT^k(x;μg)|, 1≤k≤K_dom; (2.6) hereJ_T^k :R²× Dg→R^2×2 is the Jacobian matrix of the mapT^k(·;μg), and|JT^k|denotes its determinant.

Instead, the linear form is given by Fs(w;μ) =

K_dom

k=1

Ω^k

f(μp)·w |JT^k(·;μg)|dΩ+

K¯_dom

k=1

Γ_N^k

g^N(μp)·w|JT^k(·;μg)t|dΓ,

where f(μp) ∈ (L²(Ω))² is a forcing term per unit mass, t is the tangential unit vector to the boundary, Γ_N^k =∂Ω^k∩ΓN and ¯K_dom ≤K_dom is the number of subdomains sharing at least one Neumann side. From now on, we assume thatg^N =0. Moreover, we consider a lifting approach to deal with essential (Dirichlet) boundary conditions. Thus, we introduce a lift functionvD∈(H¹(Ω))²such thatvD|Γ_Dg =gDandvD|Γ_D

0 =0, to extend non-homogeneous boundary conditions to the interior of the domain, and denote by

˜a(v(μ),w;μ) =a(v(μ),w;μ) + d(v(μ),w;μ), F(w;μ) =Fs(w;μ) +Fd(w;μ), where

d(v,w;μ) =c(vD,v,w;μ) +c(v,vD,w;μ) (2.7) and

Fd(w;μ) =−a(vD,w)−c(vD,vD,w;μ), G(q;μ) =−b(q,vD;μ) (2.8) are the terms resulting from the lifting of non-homogeneous Dirichlet conditions.

We denote by V = (H₀¹_,Γ_D(Ω))², Q = L²(Ω) the functional spaces for velocity and pressure, where H₀¹_,Γ_D(Ω) ={v∈H¹(Ω) :v|Γ_D = 0}. We also equipV andQwith the following notions of inner products and norms:

· V = (·,·)¹V^/², (v,w)V = (v,w)₍H¹(Ω))² ∀v,w∈V, · Q= (·,·)¹Q^/², (p, q)Q=ν( ¯μp)(p, q)L²(Ω) ∀p, q∈Q,

(2.9) respectively, being (v,w)₍H¹(Ω))² = (∇v,∇w)₍L²(Ω))²+ (v,w)₍L²(Ω))² andν( ¯μp) the kinematic viscosity corresponding to a selected value ¯μp. Moreover, we recall that the following Poincar´e inequality holds:

1

1 +C_p² ≤ ∇v²_L2(Ω)

v²_H1(Ω)

≤1 ∀v∈V, H₀¹(Ω)⊂V ⊂H¹(Ω), (2.10) whereCp=Cp(Ω)>0 denotes the Poincar´e constant.

For the sake of the analysis, we also provide a more compact notation. Let us denote by X the product space given byX =V ×Q, by Y(μ) = (v(μ), p(μ))∈ X and W = (w, q). Thus, the parametrized abstract formulation (2.2) can be rewritten in the following form: ﬁndY(μ) = (v(μ), p(μ))∈X(Ω) such that

A(Y(μ), W;μ) +C(Y(μ), Y(μ), W;μ) = ˜F(W;μ) ∀W ∈X (2.11) where

A(Y, W;μ) = ˜a(v,w;μ) +b(p,w;μ) +b(q,v;μ), C(Y, Y, W;μ) =c(v,v,w;μ),

F˜(W;μ) =F(w;μ) +G(q;μ).

(2.12)

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3. Well posedness of parametrized formulation

We now recall the assumptions required to ensure the well-posedness of the parametrized formulation (2.11) (or (2.2)); some of these will be automatically inherited also by the full-order and the reduced-order approximations. Moreover, we introduce theaffinity assumptionto be fulfilled by the parametrized forms, in order to ensure an efficient Offline/Online computational splitting.

3.1. Well-posedness analysis

The well-posedness of parametrized NS equations can be analyzed either by extending the theory for linear saddle-point problems to account for the nonlinear terms, or by applying the BRR theory [4,5], valid for a wider class of nonlinear equations. These two approaches can be regarded as generalizations of theBrezzi theoryand theBabuˇska theory, respectively, to the nonlinear NS case. Here we recall the main results necessary to frame this analysis, which are also useful in view of the RB approximation and thea posteriorierror estimation.

Recalling our parametrized formulation (2.2), we assume that the bilinear formsa(·,·;μ) :V ×V →Rand b(·,·;μ) :V ×V →Rare continuous: for anyμ∈ D,

γa(μ) = sup

v∈V w∈supV

a(v,w;μ) vVwV

<+∞, γb(μ) = sup

q∈Q w∈supV

b(q,w;μ) wVqQ

<+∞ (3.1)

and satisfy the following stability assumptions:a(·,·;μ) is coercive overV,i.e.

∃α^LB(μ)>0 : α(μ) = inf

w∈V

a(w,w;μ)

w²_V ≥α^LB(μ) ∀μ∈ D (3.2)

andb(·,·;μ) is inf-sup stable overV ×Q,i.e.

∃β^LB_Br(μ)>0 : βBr(μ) = inf

q∈Q sup

w∈X

b(q,w;μ)

wVqQ ≥β^LB_Br(μ) ∀μ∈ D. (3.3) Moreover, thanks to H¨olderinequality and Sobolev embedding theorems (seee.g.[30], Chapt. 2, Sect. 1.1), the trilinear formc(·,·,·;μ) :V ×V ×V →Ris continuous:

γc(μ) = sup

v∈V w∈supV

supz∈V

c(v,w,z;μ) vVwVzV

<+∞ ∀μ∈ D, (3.4)

where the continuity constantγc(μ) is expressed, in the parametrized case, as

γc(μ) = ˜ρ²Mc(μ)≤ρ²Mc(μ); (3.5)

Mc(μ) is a function depending on the parametrization; see equation (3.15) for its deﬁnition. Hereρ=ρ(Ω) and

˜

ρ= ˜ρ(Ω) denote the Sobolev embedding constants ρ²= sup

v∈V

v²_L4(Ω)

(∇v,∇v)L²(Ω)

, ρ˜²= sup

v∈V

v²_L4(Ω)

(v, v)H¹(Ω)

(3.6) respectively, beingwL^p(Ω)= _Ω|w|^p₁/p

. In particular, ˜ρ²≤ρ² thanks to (2.10).

Under the above assumptions, problem (2.2) admits a solution (see e.g. [30], Chapt. 2, Thm. 1.2) in the casevD =0; the same conclusion holds in the casevD =0 by replacinga(·,·;μ) with ˜a(·,·;μ) (see e.g. [30], Chapt. 2, Thm. 1.6). Moreover, the solution is also unique under a suitablesmall dataassumption. In this way, we can analyze the well-posedness of parametrized steady Navier–Stokes equations by extending the framework developed for parametrized Stokes equations [26], based on the Brezzi theory[3].

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On the other hand, the more general BRR theory [4,5] can be applied to state the well-posedness of the parametrized formulation (2.11). As the Babuˇska theory for mixed variational problems, it is based on the global operator ˜A(·,·;μ) :X×X→R, deﬁned in our case by

A(U, W˜ ;μ) =A(U, W;μ) +C(U, U, W;μ). (3.7) Here A(·,·;μ) andC(·,·,·;μ) denote the operators deﬁned in (2.12). Moreover, we denote by

d ˜A(Y;μ)(U, W) =A(U, W;μ) +C(Y, U, W;μ) +C(U, Y, W;μ) (3.8) the Fr´echet derivative of ˜A(·,·;μ) with respect to the ﬁrst variable. According to the BRR theory, problem (2.11) admits a solution if and only if the following continuity condition:

γ(μ) = sup

U∈X

sup

W∈X

d ˜A(Y(μ);μ)(U, W) UXWX

<+∞, ∀ μ∈ D, (3.9)

and the following (Babuˇska) inf-sup condition:

∃β^LB(μ)>0 : β(μ) = inf

U∈X sup

W∈X

d ˜A(Y(μ);μ)(U, W)

UXWX ≥β^LB(μ) ∀μ∈ D (3.10)

hold. Moreover, the solution in a neighborhood of Y(μ) is unique. Besides the well-posedness analysis, this framework will be exploited in Section6 also to derive ana posteriorierror bound for the RB approximation.

3.2. Aﬃne and nonaﬃne parametric dependence

In order to develop an Offline/Online computational procedure, we require that the forms (2.3)–(2.4) fulfill the assumption of affine parametric dependence. In our case, provided the physical parametrizations depend just onμp but not on the spatial coordinatesx, if the transformation mapsT^k(·;μg) are affine, the parametrized tensor (2.5)–(2.6) depend just onμ. In this way the bilinear forms (2.3) can be written, for some Qa, Qb >0, as:

a(v,w;μ) =

Q_a

q=1

Θ^q_a(μ)a^q(v,w), b(q,w;μ) =

Q_b

q=1

Θ_b^q(μ)b^q(q,w); (3.11) whereqis a condensed index of i, j, k quantities and, for 1≤k≤K_dom, 1≤i, j≤2,

Θ^q_a⁽^i,j,k⁾(μ) =κ^k_ij(μ), a^q⁽^i,j,k⁾(v,w) =

Ω^k

∂v

∂xi

∂w

∂xj

dΩ, (3.12)

Θ^q_b⁽^i,j,k⁾(μ) =χ^kij(μ), b^q⁽^i,j,k⁾(q,w) =−

Ω^k

q∂wi

∂xj

dΩ. (3.13)

For the trilinear form and the source term (e.g.in the caseg^N =0), we have instead:

c(v,w,z;μ) =

Q_c

q=1

Θ^qc(μ)c^q(v,w,z), Fs(w;μ) =

Q_s

q=1

Θ^qs(μ)Fs^q(w), (3.14)

for someQc, Qs>0, whereqandq are condensed indexes of (i, j, k) andk quantities, respectively, and Θ_c^q⁽^i,j,k⁾(μ) =χ^k_ji(μ), c^q⁽^i,j,k⁾(v,w,z) =

Ω^k

vi

∂wk

∂xj

zk dΩ, Θ^q_s⁽^k⁾(μ) =|JT^k(·;μ)|, F_s^q⁽^k⁾(w) =

Ω^k

f ·wdΩ.

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On the other hand, if the parametric mapsT^k(·;μg) are nonaffine, parametrized tensors are function of both spatial coordinatesxand parameterμ. In this case we rely on the EIM (seee.g.[1,17]) in order to recover the affinity assumption. If we assume, e.g., to deal with a global map (K_dom = 1), each component of the tensors κandχhas to be approximated by an affine expansion:

κij(x,μ) =

K^a_ij

k=1

β_kî,j(μ)ξ_kî,j(x) +εâ_i,j(x;μ), χij(x,μ) =

K^b_ij

k=1

γî,j_k (μ)η_kî,j(x) +ε^b_i,j(x;μ), (3.15) for 1≤i, j≤2, whereK_ijâ andK_ij^b represent the total number of terms obtained for each tensorial component through EIM. In particular, we denote by Mc(μ) = √

2 maxq=1,...,QcΘ^cq(μ)L^∞(D)maxq=1,...,Qcη^qL^∞(Ω), whereη^q is deﬁned in (3.15) –q is a condensed index ofi, j, k – and it is such thatη^q = 1 in the aﬃne case.

The same expansion (made now byK^s terms) is set up for the tensor appearing at the right-hand-side of velocity equation:

|det(JT(x;μ))|=

K^s

k=1

δk(μ)ψk(x) +ε^s(x;μ). (3.16)

All the coefficients β_kî,j’s, γ_kî,j’s, δk’s, ξ_kî,j’s, η_kî,j’s andψk’s are efficiently computable scalar functions and the error terms are guaranteed to be under some prescribed tolerance:

ε⁽i,jâ,b⁾(·;μ)_∞≤εÊIM_tol , ε^s(·;μ)_∞≤εÊIM_tol ∀μ∈ D.

In this way, we can recover the aﬃne expansions (3.11)–(3.14) by setting a^q⁽^i,j,k⁾(v,w) =

Ω

ξ_k^i,j(x)∂v

∂xi

∂w

∂xj

, b^q⁽^i,j,k⁾(p,w) =−

Ω

η_k^i,j(x)p∂wi

∂xj

, c^q⁽^i,j,k⁾(v,w,z) =

Ω^k

viη_k^i,j(x)∂wk

∂xj

zk, F_s^q⁽^k⁾(w) =

Ω^k

ψk(x)f·w,

with Θ_a^q(μ) = β_k^i,j(μ), Θ_b^q(μ) = Θ^q_c(μ) = γ^i,j_k (μ), Θ^q_s(μ) = δk(μ). Thus, we can express all the operators appearing in (2.2) as linear combinations of parameter-independent forms through some weights given by real functions of the parameters.

4. Truth approximation: Stability and algebraic formulation

We brieﬂy recall the main features related to the FE truth approximation, over which the RB approximation is built, as well the conditions ensuring its stability. We also build the algebraic version of the truth approximation, since it is required later on to set the corresponding structures for the RB approximation.

4.1. Formulation and stability

Let V^N ⊂ V and Q^N ⊂ Q be two subspaces of V and Q, of dimension NV,NQ < +∞, respectively, and let us denote byv^N(μ)∈V^N andp^N(μ)∈Q^N the FE approximations for the velocity and the pressure fields [25]. We suppose that the dimension of the FE spaces is large enough that the differencesv^N(μ)−v(μ)V, p^N(μ)−p(μ)Qcan be neglected – in other words, it can be effectively considered as a “truth” approximation.

Moreover, (2.9) deﬁnes our inner product and norm for members ofV^N ⊂V andQ^N ⊂Q, respectively.

We can now introduce the Galerkin-FE approximation of the parametrized problem (2.11) (or (2.2)): given μ∈ D, we seek forY^N(μ) = (v^N(μ), p^N(μ))∈X^N :=V^N×Q^N such that

A Y^N(μ), W^N;μ

+C Y^N(μ), Y^N(μ), W^N;μ

= ˜F W^N;μ

∀W^N ∈X^N (4.1)

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or, equivalently,

˜a v^N(μ),w^N;μ

+b p^N(μ),w^N;μ

+c v^N(μ),v^N(μ),w^N;μ

=F w^N;μ

∀w^N ∈V^N b q^N,v^N(μ);μ

=G q^N;μ

∀q^N ∈Q^N. (4.2) All the forms are continuous over the discrete spacesV^N andQ^N. In particular, a discrete version of the Sobolev embedding result can be stated (seee.g.[30]), where

ρ²_N = sup

v∈VN

v²_L4(Ω)

(∇v,∇v)L²(Ω)

, ρ²_V,_N = sup

v∈VN

v²_L4(Ω)

(v, v)H¹(Ω)

(4.3) are the discrete versions of the Sobolev constants in (3.6). Conversely, the approximation stability is ensured by imposing that the coercivity and inf-sup conditions are still valid at the discrete level. In particular,a(·,·;μ) is automatically coercive overV^N:

∃α^LB_N (μ)>0 : α^N(μ) = inf

v∈VN

a(v,v;μ) v²V

≥α^LB_N (μ) ∀μ∈ D. (4.4)

On the other hand, we require thatb(·,·;μ) is inf-sup stable overV^N×Q^N, so that the following discrete Brezzi inf-sup condition [3]:

∃ β_Br,^LB_N(μ)>0 : βBr,N(μ) = inf

q∈Q^N sup

w∈V^N

b(q,w;μ)

wVqQ ≥β_Br,^LB_N(μ) ∀μ∈ D (4.5) holds. This last property is ensurede.g.by choosingV^N×Q^Nas the space of Taylor–HoodP2−P1elements [11];

however, this choice is not restrictive – the whole construction keeps holding for other spaces combinations as well. We also introduce the (inner, pressure) supremizer operatorT_p^µ:Q^N →V^N, deﬁned as

(T_p^µq,w)V =b(q,w;μ) ∀w∈V^N, (4.6)

so that we can express the inf-sup condition (4.5) as (seee.g.[26,28]) T_p^µq= arg sup

w∈V^N

b(q,w;μ) wV

and (βBr,N(μ))²= inf

q∈QN

(T_p^µq, T_p^µq)V

q²Q

· (4.7)

This operator will play a fundamental role in the stability of the RB approximation, as we will see in Section5.

Furthermore, (4.2) admits a solution in the case of homogeneous Dirichlet conditions if (4.4) and (4.5) hold. As before, in the case of inhomogeneous Dirichlet conditions, we need to replacea(·,·;μ) with ˜a(·,·;μ). Moreover, by deﬁningγ_c^N(μ) =ρ²_V,_NMc(μ)≤ρ²_NMc(μ), uniqueness is ensured by the followingsmall dataassumption:

4γ_c^N(μ)

α^LB_N (μ)₂F(·;μ)(H−1(Ω))² <1. (4.8) In the same way as for the continuous level, the BRR theory provides a more general framework to state the well-posedness of the ﬁnite dimensional approximation (2.11). We remark that the discrete version of continuity and inf-sup stability conditions (3.9)–(3.10) reads as follows: for allμ∈ D,

γ_N(μ) = sup

U∈X^N

sup

W∈X^N

d ˜A Y^N(μ);μ (U, W) UXWX

<+∞ (4.9)

∃β^LB_N (μ)>0 : β_N(μ) = inf

U∈X^N sup

W∈X^N

d ˜A Y^N(μ);μ (U, W) UXWX

≥β^LB_N (μ). (4.10)

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4.2. Algebraic formulation of the FE truth approximation

We can now derive the matrix formulation corresponding to the Galerkin-FE approximation (4.2); this consti- tutes an essential ingredient in order to build our RB approximation in the following. Let us denote by{φ^vi}^Ni=1^V

and {φ^pi}^Ni=1^Q the Lagrangian basis of the FE spacesV^N andQ^N, respectively, so that we can express the FE velocity and pressure as

v^N(μ) =

NV

i=1

u^N_i (μ)φ^vi, p^N(μ) =

NQ

i=1

p^N_i (μ)φ^pi. (4.11)

We remark that the solution to (4.1) is vanishing on the whole Dirichlet boundary, so that the corresponding velocity approximation fulﬁlling the boundary conditions is given byv^N(μ) +v^N_D, beingv_D^N∈V^N a discrete function interpolating Dirichlet data. Thus, by denoting asv_N(μ)∈R^N^V andp_N(μ)∈R^N^Q the vectors of the degrees of freedom appearing in (4.11), problem (4.2) can be rewritten as:

AN(μ) +DN(μ) +CN(v_N(μ);μ) B^T_N(μ)

BN(μ) 0

v_N(μ) p_N(μ)

=

f_N(μ) g_N(μ)

(4.12) where, for 1≤i, j≤ NV and 1≤k≤ NQ,

(AN(μ))_ij =a φ^vj,φ^vi;μ

, (BN(μ))_ki=b(φ^p_k,φ^vi;μ), (CN(w;μ)_N)_ij =

NV

m=1

w^N_mc φ^vm,φ^vj,φ^vi;μ

, (4.13)

(DN(μ))ij =d φ^vj,φ^vi;μ

, (g_N(μ))k=−b φ^p_k,v_D^N;μ , (f_N(μ))_i=Fs(φ^vi;μ)−a v_D^N,φ^vi;μ

−c v_D^N,v_D^N,φ^vi;μ

. (4.14)

We solve the nonlinear saddle-point problem (4.12) by means of a fixed-point iteration since, relative to a Newton iteration, it has ahuge ball of convergence (seee.g.[9], Chapt. 7.2 and references therein). Moreover, if thesmall datacondition (4.8) is satisfied, the fixed-point method isglobally convergent. Thus, starting from an initial guess (v⁽⁰⁾_N (μ),p⁽⁰⁾_N (μ)), fork≥1 we solve

AN(μ) +DN(μ) +CN(v⁽_N^k⁻¹⁾(μ);μ) B^T_N(μ)

B_N(μ) 0

v⁽_N^k⁾(μ) p⁽_N^k⁾(μ)

=

f_N(μ) g_N(μ)

, (4.15)

to obtain (v⁽_N^k⁾(μ),p⁽_N^k⁾(μ)), untilv⁽_N^k⁾(μ)−v⁽_N^k⁻¹⁾(μ)V ≤ε^{N S}_tol, given a small tolerance ε^{N S}_tol >0. As initial guess, we take the Stokes solution of (4.12). Each Oseen system (4.15) is solved by means of a sparse LU factorization; diﬀerent approaches for solving (4.12) are based on the use ofhomotopy/continuationwith respect to the parameters, whenever interested in one solution branch, as shown in [7].

5. Reduced Basis approximation of parametrized Navier–Stokes equations

We now present the main components of the RB approximation for parametrized Navier–Stokes equations – namely, a Galerkin projection onto a low-dimensional space, built through a greedy procedure [24,27]; an efficient Offline/Online splitting; and a sharp (yet inexpensive)a posteriorierror estimation. The characterization of these ingredients in the case of Navier–Stokes equations is the goal of this section and the following one. In particular, after showing the construction of stable RB spaces, we deduce the algebraic formulation of the RB problem, as well as an efficient way to manage the storing of algebraic structures related to nonlinear terms.

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Given a positive integerN_max, let us introduce a sequence of (hierarchical³) approximation spaces: forN = 1, . . . , N_max,VN ⊂V^N andQN ⊂Q^NareN-dimensional subspaces ofV^N andQ^N, respectively, where typically N N.

The RB approximation (vN(μ), pN(μ)) of velocity and pressure ﬁelds, respectively, can be obtained by means of a Galerkin projection onto the reduced spaces VN ×QN as follows: given μ ∈ D, ﬁnd YN(μ) = (vN(μ), pN(μ))∈XN :=VN ×QN such that

A(YN(μ), WN;μ) +C(YN(μ), YN(μ), WN;μ) = ˜F(WN;μ) ∀WN ∈XN (5.1) or, equivalently,

˜a(vN(μ),wN;μ) +b(pN(μ),wN;μ) +c(vN(μ),vN(μ),wN;μ) =F(wN;μ) ∀wN ∈VN

b(qN,vN(μ);μ) =G(qN;μ) ∀qN ∈QN. (5.2) In the following subsection we extend to the NS case the construction ofstableRB spaces –i.e., spaces which fulﬁll the (Brezzi) inf-sup condition stability, by enriching the velocity space with the solutions of the supremizer problem (4.6) – and brieﬂy discuss the well-posedness of the RB approximation (5.1) (or (5.2)).

5.1. Construction of RB spaces and inf-sup stability

The reduced spaceXN =VN×QN is a global approximation space made up of well-chosen solutions (or snapshots) (v^N(μ¹), p^N(μ¹)), . . . ,(v^N(μ^N), p^N(μ^N)). To build this space, we rely on a standard greedy algorithm (seee.g.[24,27]), driven by ana posteriorierror boundΔN(μ), such thatY^N(μ)−YN(μ)X≤ΔN(μ) for all μ∈ D. Thus, starting from a given parameter valueμ¹ and the corresponding snapshot (v^N(μ¹), p^N(μ¹)), we iteratively build the setSN ={μ¹, . . . ,μ^N}by selecting, at the (n+ 1)-th iteration of the algorithm,

μⁿ⁺¹= arg max

µ∈Ξ_trainΔn(μ),

and adding to the reduced space the corresponding snapshot (v^N(μⁿ), p^N(μⁿ)). In other words, at each iteration we select, over all possible (v^N(μ), p^N(μ)),μ∈Ξ_train, the snapshot that thea posteriorierror bound Δn(μ) predicts to be worst approximated by the RB approximation associated to then−1 snapshots already retained. Here Ξ_train is a train sample which serves to select the RB space. The procedure stops at that step N =N_maxfor whichΔN(μ)≤ε^RB_tol ∀μ∈Ξ_train, beingε^RB_tol a prescribed positive tolerance.

Nevertheless, a further step is required in order to guarantee the inf-sup stability of the RB approximation.

To this aim, we deﬁne thereduced basis pressure space QN ⊂Q^N as QN = span

ζ˜n^p:=p^N(μⁿ), n= 1, . . . , N

, N = 1, . . . N_max. (5.3)

Thereduced basis velocity space VN ⊂V^N can be built as VN = span

˜ζ^vn:=v^N(μⁿ), T_p^µζ˜_n^p, n= 1, . . . , N

, N = 1, . . . N_max, (5.4) thus enriching the space of velocity snapshots with the inner (pressure) supremizers.

Thanks to these definitions, problem (5.2) fulfills an equivalent Brezzi RB inf-sup condition. In fact, by defining the RB stability factor as

βBr,N(μ) = inf

q∈Q_N sup

w∈V_N

b(q,w;μ) wVqQ

(5.5)

3In other words,V1 ⊂V2 ⊂. . .⊂VNmax and Q1 ⊂Q2 ⊂. . .⊂QNmax. This condition is important in order to ensure an eﬃcient storing of RB structures, and thus a rapid Online evaluation.