Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

Abstract

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.

Appendices

Appendix A: On the relationship between Brezzi and Babuška theories

In this short section we report some observations on the two inf-sup stability theories formulated by Babuška [1] and Brezzi [2], briefly discussing the relationship between the two theories and underlining the motivations on which we have based and developed the previous analysis. Some recent contributions, on which this section is based, can be found in the works by Xu and Zikatanov [50] and Demkowicz [8]. Considering a continuous bilinear form $\varPhi (\cdot ,\cdot ): U \times V \rightarrow \mathbb R$, the Babuška theory states that the problem⁶

$$\begin{aligned} \text{ find} \quad y \in U: \quad \varPhi (y,z) = \langle f,z \rangle \quad \forall \ z \in V \end{aligned}$$

is well posed if and only inf the following (Babuška) inf-sup condition holds:

$$\begin{aligned} \inf _{y \in U} \sup _{z \in V} \frac{\varPhi (y,z)}{\Vert y\Vert _U \Vert z\Vert _V} = \inf _{z \in V} \sup _{y \in U} \frac{\varPhi (y,z)}{\Vert y\Vert _U \Vert z\Vert _V} = \beta _{BA} > 0, \end{aligned}$$

(56)

and the unique solution of (36) satisfies

$$\begin{aligned} \Vert y\Vert _U \le \frac{\Vert f\Vert _{V^*}}{\beta _{BA}}. \end{aligned}$$

(57)

In this way, the Babuška theory can be seen as a generalization to the Petrov-Galerkin case of the Lax-Milgram result for the Galerkin-type formulation; its application to the Stokes problem is just a possible use. Our interest is to create a general framework to compute error bounds for noncoercive problems solved by reduced basis. On the other hand, the Brezzi theory applies to mixed variational problems under the form

$$\begin{aligned} \left\{ \begin{array}{ll} a(u,v) + b(p,v) = \langle f,v\rangle &{} \quad \forall \ v \in V,\\ b(q,u)= \langle g,q\rangle &{} \quad \forall \ q \in Q, \end{array} \right. \end{aligned}$$

(58)

where $a(\cdot ,\cdot ):V \times V \rightarrow \mathbb R$ and $b(\cdot ,\cdot ): Q \times V \rightarrow \mathbb R$ are continuous bilinear forms, i.e.

$$\begin{aligned} a(u,v) \le \gamma _a \Vert u\Vert _V \Vert v\Vert _V \ \ \forall \ u,v \in V, \quad b(q,v) \le \gamma _b \Vert q\Vert _Q \Vert v\Vert _V \ \ \forall \ q \in Q, \forall \ v \in V, \end{aligned}$$

and $f \in V^*,\,g \in Q^*$. Such a variational problem is well posed if and only if the following (Brezzi) inf-sup conditions hold:

$$\begin{aligned} \inf _{u \in V_0} \sup _{v \in V_0} \frac{a(u,v)}{\Vert u\Vert _V \Vert v\Vert _V} = \inf _{v \in V_0} \sup _{u \in V_0} \frac{a(u,v)}{\Vert u\Vert _V \Vert v\Vert _V} = \alpha > 0, \end{aligned}$$

(59)

where $V_0 = \{v \in {\varvec{V}}: b(q,v)= 0, \ \forall q \in Q\}$, and

$$\begin{aligned} \inf _{q \in Q} \sup _{v \in V} \frac{b(q,v)}{\Vert q\Vert _Q \Vert v\Vert _V} = \beta _{BR} > 0. \end{aligned}$$

(60)

Furthermore, under conditions (59)–(60) the unique solution $(u,p) \in V \times Q$ satisfies

$$\begin{aligned} \Vert (u,p)\Vert _{V \times Q} \le \mathcal{K }_{BR}(\alpha ^{-1}, \beta _{BR}^{-1}, \gamma _a) \Vert (f,g)\Vert _{V^* \times Q^*}. \end{aligned}$$

(61)

Moreover, it is also possible to derive the following estimates for the two variables distinctly:

$$\begin{aligned} \begin{array}{lll} \displaystyle \Vert u\Vert _X &{}\le &{} \dfrac{1}{\alpha } \left[ \Vert f\Vert _{X^*} + \dfrac{\alpha + \gamma _a}{\beta _{BR}} \Vert g\Vert _{Q^*}\right] ,\\ \displaystyle \Vert p\Vert _Q &{}\le &{} \dfrac{1}{\beta _{BR}} \left[ \left( 1 + \dfrac{\gamma _a}{\alpha } \right) \Vert f\Vert _{X^*} + \dfrac{ \gamma _a (\alpha + \gamma _a)}{\alpha + \beta _{BR}} \Vert g\Vert _{Q^*}\right] . \end{array} \end{aligned}$$

(62)

The relationship between the Brezzi theory and the Babuška theory in the case of a Stokes problem is based on the identifications (36)–(37): in this way, we can recast the mixed variational problem (58) into the Babuška framework; the error estimation (42) derived in Sect. 6 is the Babuška estimate (57) for (41). In the same way, using the estimations (62) on (40) it is possible to derive analogous error estimates for the velocity and the pressure errors, separately. The development of separated error bounds is ongoing. Moreover, it is possible to show that the main constants derived from these two theories are related⁷ by [50]

$$\begin{aligned} \beta _{BA} \ge \frac{1}{\mathcal{K }_{BR}(\alpha ^{-1}, \beta _{BR}^{-1}, \gamma _a) } \end{aligned}$$

(63)

Thus, considering aggregate error estimates for both RB velocity and pressure of type (57) or (61), we have that the Babuška-based estimate is sharper than the Brezzi-based one since the “safety factor” $\beta _{BA}$ is in any case greater than $1/\mathcal{K }_{BR}$. Moreover, the advantage of the Babuška-based error estimator is that only a (lower bound of a) stability constant needs to be evaluated to get the error bound. Instead, the Brezzi-based error estimator would require the evaluation of the coercivity/continuity constants of $a(\cdot ,\cdot )$ and the Brezzi inf-sup constant of $b(\cdot ,\cdot )$. We remark that all the approximation stability for the Stokes RB problem is based on Brezzi theory.

Appendix B: Offline-Online procedure for error bounds construction

In order to be computed in a very rapid and efficient way, the error estimation (42) has to be based on the Offline/Online procedure already used for the RB approximation. To reach this goal, it is important to introduce the Riesz representation of $r_{\mathbf{u }}(\cdot \, ;{\varvec{\mu }})$ and $r_{p}(\cdot \, ;{\varvec{\mu }})$: $\hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }}) \in X^{\mathcal{N }}$ and $\hat{e}_{p}({\varvec{\mu }}) \in Q^{\mathcal{N }}$ satisfy

$$\begin{aligned} (\hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }}),\mathbf{w })_X \!=\! r_{\mathbf{u }}(\mathbf{w };{\varvec{\mu }}), \ \ \forall \mathbf{w }\!\in \! X^{\mathcal{N }}, \quad (\hat{e}_{p}({\varvec{\mu }}),q)_Q \!=\! r_{p}(q;{\varvec{\mu }}), \ \ \forall q \in Q^{\mathcal{N }}.\qquad \end{aligned}$$

(64)

This allows us to write (40) as

$$\begin{aligned} \begin{array}{rll} \mathcal{A }(\mathbf{e}_\mathbf{u} ({\varvec{\mu }}), \mathbf{w }; {\varvec{\mu }}) + \mathcal{B }(e_p({\varvec{\mu }}), \mathbf{w }; {\varvec{\mu }}) = &{}(\hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }}),\mathbf{w })_X &{} \quad \forall \ \mathbf{w }\in X^\mathcal{N },\\ \mathcal{B }(q,\mathbf{e}_\mathbf{u} ({\varvec{\mu }}); {\varvec{\mu }}) = &{}(\hat{e}_{p}({\varvec{\mu }}),q)_Q &{} \quad \forall \ q \in Q^\mathcal{N } \end{array} \end{aligned}$$

and it follows that the dual norm of the residual can be evaluated through the Riesz representation:

$$\begin{aligned} \Vert r_{\mathbf{u }}(\cdot \, ; {\varvec{\mu }})\Vert _{X^{\prime }} = \sup _{\mathbf{w }\in X^\mathcal{N } } \displaystyle \frac{ r_{\mathbf{u }}(\mathbf{w }; {\varvec{\mu }})}{\Vert \mathbf{w }\Vert _X} = \Vert \hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }})\Vert _X,\end{aligned}$$

(65)

$$\begin{aligned} \Vert r_p(\cdot \, ; {\varvec{\mu }})\Vert _{Q^{\prime }} = \sup _{q \in Q^\mathcal{N } } \displaystyle \frac{ r_p(q; {\varvec{\mu }})}{\Vert q\Vert _Q} = \Vert \hat{ e}_{p}({\varvec{\mu }})\Vert _Q. \end{aligned}$$

(66)

Hence, the error bounds developed in the previous section are only useful if they allow for an efficient Offline/Online computational procedure that leads to an Online complexity independent of $\mathcal{N }$. The Offline/Online decomposition presented in the following is mainly based on the dual norm of the residual. First of all, from the affine decomposition of bilinear forms (12) we can write, equivalently,

$$\begin{aligned} \tilde{\mathcal{A }}(\mathbf{V },\mathbf W ;{\varvec{\mu }}) = \sum _{q=1}^{Q_a + 2Q_b} \tilde{\varTheta }_q({\varvec{\mu }}) \tilde{\mathcal{A }}^q(\mathbf{V },\mathbf W ), \end{aligned}$$

(67)

where

$$\begin{aligned}&\tilde{\varTheta }^q({\varvec{\mu }}) = \varTheta ^q_a, \quad q=1,\ldots ,Q_a,\\&\tilde{\varTheta }^{q+Q_a}({\varvec{\mu }}) = \tilde{\varTheta }^{q+Q_a + Q_b}({\varvec{\mu }}) = \varTheta ^q_b({\varvec{\mu }}), \quad q=1,\ldots ,Q_b, \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l@{\quad }l} \tilde{\mathcal{A }}^q(\mathbf{V },\mathbf W ) = \mathcal{A }^q(\mathbf{v },\mathbf{w }) \quad &{} q=1,\ldots , Q_a, \\ \tilde{\mathcal{A }}^q(\mathbf{V },\mathbf W ) = \mathcal{B }^q(p,\mathbf{w }) \quad &{} q=Q_a+1,\ldots ,Q_a + Q_b, \\ \tilde{\mathcal{A }}^q(\mathbf{V },\mathbf W ) = \mathcal{B }^q(q,\mathbf{v }) \quad &{} q=Q_a+Q_b+1,\ldots ,Q_a + 2Q_b. \end{array} \end{aligned}$$

In this way, denoting as $\mathbf{U }_N({\varvec{\mu }}) = (\mathbf{u }_N({\varvec{\mu }}),p_N({\varvec{\mu }})) \in \mathbb R ^{3N}$ the global vector of the RB components and recalling the expansion already used in (34), the residual can be expressed, considering the ‘global supremizer” option of Sect. 4.2, as

$$\begin{aligned} \tilde{r}(\mathbf W ;{\varvec{\mu }})&= \tilde{F}(\mathbf W ) - \tilde{\mathcal{A }}(\mathbf{U }_N^{\mathcal{N }} ({\varvec{\mu }}),\mathbf W ;{\varvec{\mu }}) \\&=\tilde{F}(\mathbf W ) - \sum _{n=1}^{3N} U_{N \, n}({\varvec{\mu }}) \sum _{q=1}^{\tilde{Q}} \tilde{\varTheta }^q ({\varvec{\mu }}) \tilde{A}^q(\varvec{\varPhi }_n,\mathbf W ), \end{aligned}$$

where $\tilde{Q} = Q_a + 2 Q_b$ and $\varvec{\varPhi }_n = (\varvec{\sigma }_n, 0)$ for $n=1,\ldots ,2N,\,\varvec{\varPhi }_n = (\mathbf{0}, \xi _n)$ for $n=2N+1,\ldots ,3N$. Together with (64) and linear superposition, this gives us

$$\begin{aligned} (\hat{\mathbf{e}}({\varvec{\mu }}), \mathbf W )_Y&= (\hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }}),\mathbf{w })_X + (\hat{e}_p({\varvec{\mu }}),q)_Q \nonumber \\&=\tilde{F}(\mathbf W ) - \sum _{n=1}^{3N} U_{N \, n}({\varvec{\mu }}) \sum _{q=1}^{\tilde{Q}} \tilde{\varTheta }^q ({\varvec{\mu }}) \tilde{A}^q(\varvec{\varPhi }_n,\mathbf W ) \end{aligned}$$

where $\hat{\mathbf{e}}({\varvec{\mu }}) = (\hat{\mathbf{e}}_{\mathbf{u }}({\varvec{\mu }}), \hat{e}_p({\varvec{\mu }})) \in Y^{\mathcal{N }}$. We thus may write $\hat{\mathbf{e}}({\varvec{\mu }}) \in Y^{\mathcal{N }}$ as

$$\begin{aligned} \hat{\mathbf{e}}({\varvec{\mu }}) = \tilde{\mathcal{F }} + \sum _{q=1}^{\tilde{Q}}\sum _{n=1}^{3N} \tilde{\varTheta }^q({\varvec{\mu }}) U_{N\, n}({\varvec{\mu }}) \tilde{\mathcal{A }}_n^q, \end{aligned}$$

where $\tilde{\mathcal{F }} \in Y^{\mathcal{N }}$ and $\tilde{\mathcal{A }}_n^q \in Y^{\mathcal{N }}$ (called FE “pseudo”-solutions) satisfy

$$\begin{aligned} (\tilde{\mathcal{F }},\mathbf W )_Y&= \tilde{F}(\mathbf W ), \quad \forall \ \mathbf W \in Y^{\mathcal{N }},\end{aligned}$$

(68)

$$\begin{aligned} (\tilde{\mathcal{A }}_n^q,\mathbf W )_Y&= -\tilde{A}^q(\varvec{\varPhi }_n,\mathbf W ), \quad \forall \ \mathbf W \in Y^{\mathcal{N }}, \,\, 1 \le n \le 3N, \,\, 1 \le q \le \tilde{Q}. \end{aligned}$$

(69)

We note that (69) and (68) are simple parameter-independent problems and thus can be solved once in the Offline stage. It then follows that:

$$\begin{aligned} \Vert \hat{\mathbf{e}}({\varvec{\mu }})\Vert _Y^2&= \left( \tilde{\mathcal{F }} + \sum _{q=1}^{\tilde{Q}}\sum _{n=1}^{3N} \tilde{\varTheta }^q({\varvec{\mu }}) U_{Nn}({\varvec{\mu }}) \tilde{\mathcal{A }}_n^q , \, \tilde{\mathcal{F }} + \sum _{q^{\prime }=1}^{\tilde{Q}}\sum _{n^{\prime }=1}^{3N} \tilde{\varTheta }^{q^{\prime }}({\varvec{\mu }}) U_{Nn^{\prime }}({\varvec{\mu }}) \tilde{\mathcal{A }}_{n^{\prime }}^{q^{\prime }}\right) _Y \nonumber \\&= (\mathcal{F },\mathcal{F })_Y + \sum _{q=1}^{\tilde{Q}}\sum _{n=1}^{3N} \tilde{\varTheta }^q({\varvec{\mu }}) U_{Nn}({\varvec{\mu }}) \nonumber \\&\times \left\{ 2(\tilde{\mathcal{F }},\tilde{\mathcal{A }}_n^q)_Y + \sum _{q^{\prime }=1}^{Q_a}\sum _{n^{\prime }=1}^N \tilde{\varTheta }^{q^{\prime }}({\varvec{\mu }}) U_{Nn^{\prime }}({\varvec{\mu }}) (\tilde{\mathcal{A }}_n^q, \tilde{\mathcal{A }}_{n^{\prime }}^{q^{\prime }})_Y \right\} \end{aligned}$$

(70)

This expression can be related to the requested dual norm of the residual through (65)–(66). It is the sum of products of parameter-dependent known functions and parameter independent inner products, formed of more complicated but precomputable quantities. The Offline/Online decomposition is thus clear:

• (i) in the Offline stage we first solve (69), (68) for the parameter-independent FE “pseudo”-solutions $\tilde{\mathcal{F }}$ and $\tilde{\mathcal{A }}_n^q, \, 1 \le n \le 3N, \, 1 \le q \le \tilde{Q}$ and form/store the parameter-independent inner products $(\tilde{\mathcal{F }},\tilde{\mathcal{F }})_Y,\,(\tilde{\mathcal{F }},\tilde{\mathcal{A }}_n^q)_Y$ and $(\tilde{\mathcal{A }}_n^q, \tilde{\mathcal{A }}_{n^{\prime }}^{q^{\prime }})_Y, \, 1 \le n,n^{\prime } \le 3N, \, 1 \le q,q^{\prime } \le \tilde{Q}$. The Offline operation count depends on $N,\,\tilde{Q}$ and $\mathcal{N }$;

• (ii) in the Online stage—performed for each new value of ${\varvec{\mu }}$—we simply evaluate the sum (70) in terms of the $\tilde{\varTheta }^q({\varvec{\mu }}), \, 1 \le q \le \tilde{Q}$ and $U_{N\,n}({\varvec{\mu }}), \, 1 \le n \le 3N$ (already computed for the output evaluation) and the precomputed and stored (parameter-independent) $(\cdot , \cdot )_Y$ inner products. The Online operation count, and hence the marginal and asymptotic average cost, is only $\mathcal{O }(\tilde{Q}^2 9N^2)$, and thus the crucial point—the independence of $\mathcal{N }$—is achieved again.