Abstract
This work analyzes a least-squares method in order to solve implicit time schemes associated to the 2D and 3D Navier–Stokes system, introduced in 1979 by Bristeau, Glowinksi, Periaux, Perrier and Pironneau. Implicit time schemes reduce the numerical resolution of the Navier–Stokes system to multiple resolutions of steady Navier–Stokes equations. We first construct a minimizing sequence (by a gradient type method) for the least-squares functional which converges strongly and quadratically toward a solution of a steady Navier–Stokes equation from any initial guess. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier–Stokes operator. Then, we apply iteratively the analysis on the fully implicit Euler scheme and show the convergence of the method uniformly with respect to the time discretization. Numerical experiments for 2D examples support our analysis.
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1 Introduction–motivation
Let \(\varOmega \subset {\mathbb {R}}^d\), \(d=2\) or \(d=3\) be a bounded connected open set whose boundary \(\partial \varOmega \) is Lipschitz and \(T>0\). We endow \(H_0^1(\varOmega )\) with the scalar product \(\langle v,w\rangle _{H_0^1(\varOmega )}=\int _\varOmega \nabla v\cdot \nabla w\) and the associated norm and we endow the dual \(H^{-1}(\varOmega )\) of \(H_0^1(\varOmega )\) with the dual norm of \(H_0^1(\varOmega )\). We denote by \(\varvec{{\mathcal {V}}}=\{v\in {\mathcal {D}}(\varOmega )^d, \nabla \cdot v=0\}\), \({\varvec{H}}\) the closure of \(\varvec{{\mathcal {V}}}\) in \(L^2(\varOmega )^d\) and \({\varvec{V}}\) the closure of \(\varvec{{\mathcal {V}}}\) in \(H_0^1(\varOmega )^d\) endowed with the norm of \(H_0^1(\varOmega )^d\).
The Navier–Stokes system describes a viscous incompressible fluid flow in the bounded domain \(\varOmega \) during the time interval (0, T) submitted to the external force F. It reads as follows:
where u is the velocity of the fluid, p its pressure and \(\nu \) is the viscosity constant assumed smaller than one. We refer to [19]. This work is concerned with the approximation of (1) through the time marching fully implicit Euler scheme
where \(\{t_n\}_{n=0...N}\), for a given \(N\in {\mathbb {N}}\), is a uniform discretization of the time interval (0, T). \(\delta t=T/N\) is the time discretization step. This also-called backward Euler scheme is studied for instance in [19, chapter 3, section 4]. It is proved there that the piecewise linear interpolation (in time) of \(\{y^n\}_{n\in [0,N]}\) weakly converges in \(L^2(0,T;{\varvec{V}})\) toward a solution u of (1) as \(\delta t\) goes to zero. It achieves a first order convergence with respect to \(\delta t\). We also refer to [20] for a stability analysis of the scheme in long time. We refer to [17] for Crank-Nicolson schemes achieving second order convergence.
The determination of \(y^{n+1}\) from \(y^n\) requires the resolution of a nonlinear partial differential equation. Precisely \(y^{n+1}\) together with the pressure \(\pi ^{n+1}\) solve the following problem: find \(y\in {\varvec{V}}\) and \(\pi \in L_0^2(\varOmega )\), solution of
with
Recall that for any \(f\in H^{-1}(\varOmega )^d\) and \(g\in L^2(\varOmega )^d\), there exists one solution \((y,\pi )\in {\varvec{V}}\times L_0^2(\varOmega )\) of (3), unique if \(\Vert g\Vert _{L^2(\varOmega )^d}^2 + \alpha ^{-1} \nu ^{-1}\Vert f\Vert ^2_{H^{-1}(\varOmega )^d}\) is small enough (see Proposition 1 for a precise statement). \(L_0^2(\varOmega )\) stands for the space of functions in \(L^2(\varOmega )^d\) with zero means.
A weak solution \(y\in {\varvec{V}}\) of (3) solves the formulation \(F(y,w)=0\) for all \(w\in {\varvec{V}}\) where F is defined by
If \(D_y F\) is invertible, one may approximate a weak solution through the iterative Newton method: a sequence \(\{y_k\}_{k\in {\mathbb {N}}}\in {\varvec{V}}\) is constructed as follows
If the initial guess \(y_0\) is close enough to a weak solution of (3), i.e. a solution satisfying \(F(y,w)=0\) for all w, then the sequence \(\{y_k\}_{k\in {\mathbb {N}}}\) converges. We refer to [15, Section 10.3], [4, Chapter 6]) and for some numerical aspects to [10].
Alternatively, we may also employ least-squares methods which consist in minimizing a quadratic functional, which measures how an element y is close to the solution. For instance, we may introduce the extremal problem : \(\inf _{y\in {\varvec{V}}} E(y)\) with \(E:{\varvec{V}} \rightarrow {\mathbb {R}}^+\) defined by
where the corrector v, together with the pressure, is the unique solution in \({\varvec{V}}\times L_0^2(\varOmega )\) of the linear boundary value problem:
E(y) vanishes if and only if \(y\in {\varvec{V}}\) is a weak solution of (3), equivalently a zero of \(F(y,w)=0\) for all \(w\in {\varvec{V}}\). As a matter of fact, the infimum is reached. Least-squares methods to solve nonlinear boundary value problems have been the subject of intensive developments in the last decades, as they present several advantages, notably on computational and stability viewpoints. We refer to [1, 7]. The minimization of the functional E over \({\varvec{V}}\) leads to a so-called weak least squares method. Precisely, the equality \(\sqrt{2E(y)}=\sup _{w\in {\varvec{V}}, w\ne 0} \frac{F(y,w)}{{\left| \left| \left| w \right| \right| \right| }_{{\varvec{V}}}}\)–where \({\left| \left| \left| w \right| \right| \right| }_{{\varvec{V}}}\) is defined in (9)–shows that E is equivalent to the \({\varvec{V}}^{\prime }\) norm of the Navier–Stokes equation (see Remark 1). The terminology “\(H^{-1}\)-least-squares method” is employed in [2] where the minimization of E has been introduced and numerically implemented to approximate solutions of (1) through the scheme (2). We also mention [4, Chapter 4, Section 6] which studied the use of a least-squares strategy to solve a steady Navier–Stokes equation without incompressibility constraint.
The first objective of the present work is to analyze rigorously the method introduced in [2] and show that one may construct minimizing sequences in \({\varvec{V}}\) for E that converge strongly toward a solution of (3). The second objective is to justify the use of that least-squares method to solve iteratively a weak formulation of the scheme (2), leading to an approximation of the solution of (1). This requires to show some convergence properties of a minimizing sequence for E, uniformly with respect to the parameter n related to the time discretization. As we shall see, this requires smallness assumptions on the data \(u_0\) and F.
The rest of the paper is organized as follows. Section 2 is devoted to the analyze the least-squares method (7)–(8) associated to weak solutions of (3). We show that E is differentiable over \({\varvec{V}}\) and that any critical point for E in the ball \({\mathbb {B}}:=\{y\in {\varvec{V}}, \tau _d(y)<1\}\) (see Definition 1) is also a zero of E. This is done by introducing a descent direction \(Y_1\) for E at any point \(y\in {\varvec{V}}\) for which \(E^{\prime }(y)\cdot Y_1\) is proportional to E(y). Then, assuming that there exists at least one solution of (3) in \({\mathbb {B}}\), we show that any minimizing sequence \(\{y_k\}_{(k\in {\mathbb {N}})}\) for E in \({\mathbb {B}}\) strongly converges to a solution of (14). Such limit belongs to \({\mathbb {B}}\) and is actually the unique solution. Eventually, we construct a minimizing sequence (defined in (30)) based on the element \(Y_1\) and initialized with g assumed in \({\varvec{V}}\). If \(\alpha \) is large enough, we show that this particular sequence belongs to \({\mathbb {B}}\) and converges (quadratically after a finite number of iterates related to the values of \(\nu \) and \(\alpha \)) strongly to the solution of (3) (see Theorem 1). This specific sequence coincides with the one obtained from the damped Newton method, a globally convergent generalization of (6). Then, in Sect. 3, as an application, we consider the least-squares approach to solve iteratively the backward Euler scheme (see (51)), weak formulation of (2). For each \(n>0\), in order to approximate \(y^{n+1}\), we define a minimizing sequence \(\{y_k^{n+1}\}_{k\ge 0}\) based on \(Y_1^{n+1}\) and initialized with \(y^n\). Adapting the global convergence result of Sect. 2, we then show, assuming \(\Vert u_0\Vert _{L^2(\varOmega )^d}+ \Vert F\Vert _{L^2(0,T;H^{-1}(\varOmega )^d)}\) small enough, the strong convergence of the minimizing sequences, uniformly with respect to the time discretization parameter n (see Theorem 4). The analysis is performed for \(d=2\) for both weak and regular solutions and for \(d=3\) for regular solutions. Our analysis justifies the use of Newton type methods to solve implicit time schemes for (1), as mentioned in [15, Section 10.3]. To the best of our knowledge, such analysis of convergence is original. In Sect. 4, we discuss numerical experiments based on finite element approximations in space for two 2D geometries: the celebrated example of the channel with a backward facing step and the semi-circular driven cavity introduced in [5]. We notably exhibit for small values of the viscosity constant the robustness of the damped Newton method (compared to the Newton one).
2 Analysis of a Least-squares method for a steady Navier–Stokes equation
We analyse in this section a least-squares method to solve the steady Navier–Stokes Eq. (3) assuming \(\alpha >0\): we extend [11] where the particular case \(\alpha =0\) is addressed.
2.1 Technical preliminary results
In the sequel \(\Vert \cdot \Vert _2\) stands for the norm \(\Vert \cdot \Vert _{L^2(\varOmega )^d}\). We shall also use the following notations
and \(\langle y,z\rangle _{{\varvec{V}}}:=\alpha \int _{\varOmega } y\cdot z +\nu \int _{\varOmega } \nabla y\cdot \nabla z\) so that \(\langle y,z\rangle _{{\varvec{V}}}\le {\left| \left| \left| y \right| \right| \right| }_{{\varvec{V}}}{\left| \left| \left| z \right| \right| \right| }_{{\varvec{V}}}\) for any \(y,z\in {\varvec{V}}\).
In the sequel, we repeatedly use the following classical estimates (see [19]).
Lemma 1
Let any \(u,v\in {\varvec{V}}\). If \(d=2\), then
If \(d=3\), then there exists a constant \(c=c(\varOmega )\) such that
Definition 1
For any \(y\in {\varvec{V}}\), \(\alpha >0\) and \(\nu >0\), we define
with \(M:=\frac{3^{3/4}}{4}c\) and c from (11).
We shall also repeatedly use the following Young type inequalities.
Lemma 2
For any \(u,v\in {\varvec{V}}\), the following inequalities hold true:
if \(d=2\) and
if \(d=3\).
Let \(f\in H^{-1}(\varOmega )^d\), \(g\in L^2(\varOmega )^d\) and \(\alpha \in {\mathbb {R}}^\star _+\). The weak formulation of (3) reads as follows: find \(y\in {\varvec{V}}\) solution of
The following result holds true (we refer to [13]).
Proposition 1
Assume \(\varOmega \subset {\mathbb {R}}^d\) is bounded and Lipschitz. There exists at least one solution y of (14) satisfying
If moreover \(\varOmega \) is \(C^2\) and \(f\in L^2(\varOmega )^d\), then any solution \(y\in {\varvec{V}}\) of (14) belongs to \(H^2(\varOmega )^d\).
Lemma 3
Assume that a solution \(y\in {\varvec{V}}\) of (14) satisfies \(\tau _d(y)<1\). Then, such solution is the unique solution of (14).
Proof
Let \(y_1\in {\varvec{V}}\) and \(y_2\in {\varvec{V}}\) be two solutions of (14). Set \(Y=y_1-y_2\). Then,
We now take \(w=Y\) and use that \(\int _\varOmega y_2\cdot \nabla Y\cdot Y=0\). If \(d=2\), we use (10) and (12) to get
leading to \((1-\tau _2(y_1)){\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}^2\le 0\). Consequently, if \(\tau _2(y_1)<1\) then \(Y=0\) and the solution of (14) is unique. In view of (15), this holds if the data satisfy \(\nu \Vert g\Vert ^2_2+{1\over \alpha }\Vert f\Vert ^2_{H^{-1}(\varOmega )^d}<2\nu ^2\).
If \(d=3\), we use (11) and (13) to obtain
leading to \(\big (1-\tau _3(y_1)\big )\Vert Y\Vert _2^2\le 0\) and to the uniqueness if \(\tau _3(y_1)<1\). In view of (15), this holds if the data satisfy \(\nu \Vert g\Vert ^2_2+{1\over \alpha }\Vert f\Vert ^2_{H^{-1}(\varOmega )^d}<M^{-2}\nu ^{7/2}\alpha ^{-1/2}\). \(\square \)
We now introduce our least-squares functional \(E: {\varvec{V}}\rightarrow {\mathbb {R}}^+\) as follows
where the corrector \(v\in {\varvec{V}}\) is the unique solution of the linear formulation
In particular, for \(d=2\), the corrector v satisfies the estimate:
Conversely, we also have
The infimum of E is equal to zero and is reached by a solution of (14). In this sense, the functional E is a so-called error functional which measures, through the corrector variable v, the deviation of the pair y from being a solution of the underlying Eq. (14).
A practical way of taking a functional to its minimum is through some (clever) use of descent directions, i.e. the use of its derivative. In doing so, the presence of local minima is something that may dramatically spoil the whole scheme. The unique structural property that discards this possibility is the strict convexity of the functional. However, for non-linear equations like (14), one cannot expect this property to hold for the functional E in (16). Nevertheless, we insist in that for a descent strategy applied to the extremal problem \(\min _{y\in {\varvec{V}}} E(y)\) numerical procedures cannot converge except to a global minimizer leading E down to zero.
Indeed, we would like to show that the critical points for E correspond to solutions of (14). In such a case, the search for an element y solution of (14) is reduced to the minimization of E.
For any \(y\in {\varvec{V}}\), we now look for an element \(Y_1\in {\varvec{V}}\) solution of the following formulation
for all \(w\in {\varvec{V}}\) where \(v\in {\varvec{V}}\) is the corrector (associated to y) solution of (17). \(Y_1\) enjoys the following property.
Proposition 2
For all \(y\in {\varvec{V}}\) satisfying \(\tau _d(y)<1\), there exists a unique solution \(Y_1\) of (20) associated to y. Moreover, this solution satisfies
Proof
We define the bilinear and continuous form \(a:{\varvec{V}}\times {\varvec{V}}\rightarrow {\mathbb {R}}\) by
so that \(a(Y,Y)={\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}^2+\int _\varOmega Y\cdot \nabla y\cdot Y\). If \(d=2\), using (12), we obtain \( a(Y,Y)\ge (1-\tau _2(y)){\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}^2\) for all \(Y\in {\varvec{V}}\). Lax-Milgram lemma leads to the existence and uniqueness of \(Y_1\) assuming that \(\tau _2(y)<1\). Then, putting \(w=Y_1\) in (20) implies
leading to (21). If \(d=3\), using (13), we obtain \(a(Y,Y)\ge (1-\tau _3(y)){\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}^2\) for all \(Y\in {\varvec{V}}\) and we conclude as before. \(\square \)
We now check the differentiability of the least-squares functional.
Proposition 3
For all \(y\in {\varvec{V}}\), the map \(Y\mapsto E(y+Y)\) is a differentiable function on the Hilbert space \({\varvec{V}}\) and for any \(Y\in {\varvec{V}}\), we have
where \(V\in {\varvec{V}}\) is the unique solution of
Proof
Let \(y\in {\varvec{V}}\) and \(Y\in {\varvec{V}}\). We have \(E(y+Y)=\frac{1}{2}{\left| \left| \left| {\overline{V}} \right| \right| \right| }^2_{{\varvec{V}}}\) where \({\overline{V}}\in {\varvec{V}}\) is the unique solution of
If \(v\in {\varvec{V}}\) is the solution of (17) associated to y, \(v'\in {\varvec{V}}\) is the unique solution of
and \(V\in {\varvec{V}}\) is the unique solution of (23), then it is straightforward to check that \({\overline{V}}-v-v'-V\in {\varvec{V}}\) is solution of
and therefore \({\overline{V}}-v-v'-V=0\). Thus
Assume \(d=2\). Then, writing (23) with \(w=V\) and using (10), we obtain
leading to \({\left| \left| \left| V \right| \right| \right| }_{{\varvec{V}}}\le {\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}(1+\frac{\sqrt{2}}{\sqrt{\alpha }\nu }{\left| \left| \left| y \right| \right| \right| }_{{\varvec{V}}})\). Similarly, using (24), we obtain \( {\left| \left| \left| v' \right| \right| \right| }_{{\varvec{V}}}\le \frac{1}{\sqrt{2\alpha }\nu }{\left| \left| \left| Y \right| \right| \right| }^2_{{\varvec{V}}}. \) It follows that \(\frac{1}{2}{\left| \left| \left| v' \right| \right| \right| }_{{\varvec{V}}}^2+\frac{1}{2}{\left| \left| \left| V \right| \right| \right| }_{{\varvec{V}}}^2+\langle V,v'\rangle _{{\varvec{V}}}+\langle v,v'\rangle _{{\varvec{V}}}=o({\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}})\) and from (25) that
Eventually, the estimate \(|\langle v,V\rangle _{{\varvec{V}}}|\le {\left| \left| \left| v \right| \right| \right| }_{{\varvec{V}}}{\left| \left| \left| V \right| \right| \right| }_{{\varvec{V}}}\le \sqrt{2}(1+\frac{\sqrt{2}}{\sqrt{\alpha }\nu }{\left| \left| \left| y \right| \right| \right| }_{{\varvec{V}}}) \sqrt{E(y)} {\left| \left| \left| Y \right| \right| \right| }_{{\varvec{V}}}\) gives the continuity of the linear map \(Y\mapsto \langle v,V\rangle _{{\varvec{V}}}\). The case \(d=3\) is similar. \(\square \)
We are now in position to prove the following result which indicates that, in the ball \({\mathbb {B}}\), any critical point for E is also a zero of E.
Proposition 4
For all \(y\in {\varvec{V}}\) satisfying \(\tau _d(y)<1\),
Proof
For any \(Y\in {\varvec{V}}\), \(E^{\prime }(y)\cdot Y=\int _\varOmega \alpha \, v\cdot V+ \nu \nabla v\cdot \nabla V\) where \(V\in {\varvec{V}}\) is the unique solution of (23). In particular, taking \(Y=Y_1\) defined by (20), we obtain an element \(V_1\in {\varvec{V}}\) solution of
for all \(w\in {\varvec{V}}\). Summing (20) and (26), we obtain that \(v-V_1\in {\varvec{V}}\) solves
This implies that v and \(V_1\) coincide and then that
It follows that
Proposition 2 allows to conclude. \(\square \)
Eventually, we prove the following coercivity type inequality for the functional E.
Proposition 5
Assume that a solution \({\overline{y}}\in {\varvec{V}}\) of (14) satisfies \(\tau _d({\overline{y}})<1\). Then, for all \(y\in {\varvec{V}}\),
Proof
For any \(y\in {\varvec{V}}\), let v be the corresponding corrector and let \(Y=y-{\overline{y}}\). We have
for all \(w\in {\varvec{V}}\). For \(w=Y\), this equality rewrites
Repeating the arguments of the Proof of Proposition 2, the result follows. \(\square \)
Assuming the existence of a solution of (14) in the ball \({\mathbb {B}}=\{y\in {\varvec{V}}, \tau _d(y)< 1\}\), Propositions 4 and 5 imply that any minimizing sequence \(\{y_k\}_{(k\in {\mathbb {N}})}\) for E in \({\mathbb {B}}\) strongly converges to a solution of (14). Remark that, from Lemma 3, such solution is unique. In the next section, assuming the parameter \(\alpha \) large enough, we construct such sequence \(\{y_k\}_{(k\in {\mathbb {N}})}\).
Remark 1
In order to simplify notations, we have introduced the corrector variable v leading to the functional E. Instead, we may consider the functional \({\widetilde{E}}:{\varvec{V}}\rightarrow {\mathbb {R}}\) defined by
with \(B_1:{\varvec{V}}\rightarrow L^2(\varOmega )^d\) and \(B:{\varvec{V}}\times {\varvec{V}}\rightarrow L^2(\varOmega )^d\) defined by \((B_1(y),w):=(\nabla y,\nabla w)_2\) and \((B(y,z),w):=\int _{\varOmega } y\cdot \nabla z\cdot w\) respectively. E and \({\widetilde{E}}\) are equivalent. Precisely, from the definition of v (see (17)), we deduce that
Conversely,
so that \({\widetilde{E}}(y)\le (\alpha +\nu )E(y)\) for all \(y\in {\varvec{V}}\).
2.2 A strongly convergent minimizing sequence for E
We define in this section a sequence converging strongly to a solution of (14) for which E vanishes. According to Proposition 4, it suffices to define a minimizing sequence for E included in the ball \({\mathbb {B}}\). In this respect, the equality (27) shows that \(-Y_1\) (see (20)) is a descent direction for the functional E. Therefore, we can define at least formally, for any \(m\ge 1\), the minimizing sequence \(\{y_k\}_{(k\ge 0)}\):
with \(Y_{1,k}\in {\varvec{V}}\) the solution of the formulation
and \(v_k\in {\varvec{V}}\) the corrector (associated to \(y_k\)) solution of (17). The algorithm (30) can be expanded as follows:
From (19), the sequence \(\{y_k\}_{k>0}\) is bounded. However, we insist that, in order to justify the existence of the element \(Y_{1,k}\), \(y_k\) should satisfy \(\tau _d(y_k)<1\) for all k. We proceed in two steps: assuming that the sequence \(\{y_k\}_{(k>0)}\) defined by (30) satisfies \(\tau _d(y_k)\le c_1<1\) for all k, we show that \(E(y_k)\rightarrow 0\) and that \(\{y_k\}_{k\in {\mathbb {N}}}\) converges strongly in \({\varvec{V}}\) to a solution of (14). Then, we determine sufficient conditions on the initial guess \(y_0\in {\varvec{V}}\) so that \(\tau _d(y_k)<1\) for all \(k\in {\mathbb {N}}\).
We start with the following lemma which provides the main property of the sequence \(\{E(y_k)\}_{(k\ge 0)}\).
Lemma 4
Assume that the sequence \(\{y_k\}_{(k\ge 0)}\) defined by (30) satisfies
\(\tau _d(y_k)<~1\) for all \(k\ge 0\). Then, for all \(\lambda \in {\mathbb {R}}\),
if \(d=2\) and
if \(d=3\).
Proof
For any real \(\lambda \) and any \(y_k,w_k\in {\varvec{V}}\) we get the expansion
where \(v_k\), \({\overline{v}}_k\in {\varvec{V}}\) and \(\overline{{\overline{v}}}_k\in {\varvec{V}}\) solves respectively
and
Since the corrector \({\overline{v}}_k\) associated to \(Y_{1,k}\) coincides with the corrector \(v_k\) associated to \(y_k\) (see Proof of Proposition 4), expansion (35) reduces to
If \(d=2\), then (38) leads to \({\left| \left| \left| \overline{{\overline{v}}}_k \right| \right| \right| }_{{\varvec{V}}}\le \frac{{\left| \left| \left| Y_{1,k} \right| \right| \right| }^2_{{\varvec{V}}}}{\sqrt{2\alpha }\nu }\le \sqrt{2}(1-\tau _2(y_k))^{-2}\frac{E(y_k)}{\sqrt{\alpha }\nu }\) and then to (33). If \(d=3\), then
leading to (34). \(\square \)
We are now in position to prove the convergence of the sequence \(\{E(y_k)\}_{(k\ge 0)}\).
Proposition 6
Let \(\{y_k\}_{k\ge 0}\) be the sequence defined by (30). Assume that there exists a constant \(c_1\in (0,1)\) such that \(\tau _d(y_k)\le c_1\) for all k. Then \(E(y_k)\rightarrow 0\) as \(k\rightarrow \infty \). Moreover, there exists \(k_0\in {\mathbb {N}}\) such that the sequence \(\{E(y_k)\}_{(k\ge k_0)}\) decays quadratically.
Proof
Consider the case \(d=2\). The inequality \(\tau _2(y_k)\le c_1\) and (33) imply that
Let us denote the function \(p_k(\lambda )=\vert 1-\lambda \vert +\lambda ^2 c_{\alpha ,\nu }\sqrt{E(y_k)}\) for all \(\lambda \in [0,m]\). We can write
with \(p_k({\widetilde{\lambda }}_k):=\min _{\lambda \in [0,m]}p_k(\lambda )\).
Suppose first that \(c_{\alpha ,\nu }\sqrt{E(y_0)}\ge 1\) and prove that the set \(I:=\{k\in {\mathbb {N}},\ c_{\alpha ,\nu }\sqrt{E(y_k)}\ge 1\}\) is a finite subset of \({\mathbb {N}}\). For all \(k\in I\), we get
and thus, for all \(k\in I\),
Consequently, the sequence \(\{c_{\alpha ,\nu } \sqrt{E(y_k)}\}_{k\in I}\) strictly decreases and thus, there exists \(k_0\in {\mathbb {N}}\) such that for all \(k\ge k_0\), \(c_{\alpha ,\nu }\sqrt{E(y_k)}<1\). Thus I is a finite subset of \({\mathbb {N}}\). Then, for all \(k\ge k_0\), we get that
and thus, for all \(k\ge k_0\),
implying that \(c_{\alpha ,\nu }\sqrt{E(y_k)}\rightarrow 0\) as \(k\rightarrow \infty \) with a quadratic rate.
On the other hand, if \(c_{\alpha ,\nu }\sqrt{E(y_0)}< 1\) (and thus \(c_{\alpha ,\nu }\sqrt{E(y_k)}<1\) for all \(k\in {\mathbb {N}}\), since by construction the sequence \(\{E(y_k)\}_k\) decreases), then (40) holds true for all \(k\ge 0\).
In both cases, remark that \(p_k({\widetilde{\lambda }}_k)\) decreases with respect to k. The case \(d=3\) is similar with
\(\square \)
Lemma 5
Assume that the sequence \(\{y_k\}_{(k\ge 0)}\) defined by (30) satisfies \(\tau _d(y_k)\le c_1\) for all k and some \(c_1\in (0,1)\). Then \(\lambda _k\rightarrow 1\) as \(k\rightarrow \infty \).
Proof
In view of (39), we have, as long as \(E(y_k)>0\),
From the Proof of Lemma 4, \(\frac{\langle v_k, \overline{{\overline{v}}}_k \rangle _{{\varvec{V}}}}{E(y_{k})}\le C(\alpha ,\nu )(1-c_1)^{-2}\sqrt{E(y_k)}\) while \(\frac{{\left| \left| \left| \overline{{\overline{v}}}_k \right| \right| \right| }^2_{{\varvec{V}}}}{E(y_{k})}\le C(\alpha ,\nu )^2(1-c_1)^{-4}E(y_k)\). Consequently, since \(\lambda _k\in [0,m]\) and \(\frac{E(y_{k+1})}{E(y_{k})}\rightarrow 0\), we deduce that \((1-\lambda _k)^2\rightarrow 0\) as \(k\rightarrow \infty \). \(\square \)
Proposition 7
Let \(\{y_k\}_{(k\ge 0)}\) be the sequence defined by (30). Assume that there exists a constant \(c_1\in (0,1)\) such that \(\tau _d(y_k)\le c_1\) for all k. Then, \(y_k\rightarrow {\overline{y}}\) in \({\varvec{V}}\) where \({\overline{y}}\in {\varvec{V}}\) is the unique solution of (14).
Proof
Remark that we can not use Proposition 5 since we do not know yet that there exists a solution, say z, of (14) satisfying \(\tau _d(z)<1\). In view of \(y_{k+1}=y_0-\sum _{n=0}^k \lambda _n Y_{1,n}\), we write
Using that \(p_n({\widetilde{\lambda }}_n)\le p_0({\widetilde{\lambda }}_0)\) for all \(n\ge 0\), we obtain for all \(n>0\),
Recalling that \(p_{0}({\widetilde{\lambda }}_0)=min_{\lambda \in [0,1]} p_0(\lambda )<1\) since \(p_0(0)=1\) and \(p_0^{\prime }(0)=-1\), we finally obtain
from which we deduce that the series \(\sum _{k\ge 0}\lambda _k Y_{1,k}\) converges in \({\varvec{V}}\). Then, \(y_k\) converges in \({\varvec{V}}\) to \({\overline{y}}:=y_0+\sum _{k\ge 0}\lambda _k Y_{1,k}\). Eventually, the convergence of \(E(y_k)\) to 0 implies the convergence of the corrector \(v_k\) to 0 in \({\varvec{V}}\); taking the limit in the corrector Eq. (36) shows that \({\overline{y}}\) solves (14). Since \(\tau _d({\overline{y}})\le c_1<1\), Lemma 3 shows that this solution is unique. \(\square \)
As mentioned earlier, the remaining and crucial point is to show that the sequence \(\{y_k\}_{(k\ge 0)}\) satisfies the uniform property \(\tau _d(y_k)\le c_1\) for some \(c_1<1\).
Lemma 6
Assume that \(y_0=g\in {\varvec{V}}\). For all \(c_1\in (0,1)\) there exists \(\alpha _0>0\), such that, for any \(\alpha \ge \alpha _0\), the unique sequence defined by (30) satisfies \(\tau _d(y_k)\le c_1\) for all \(k\ge 0\).
Proof
Let \(c_1\in (0,1)\) and assume that \(y_0\) belongs to \({\varvec{V}}\). Since \(\tau _d(y_0)\rightarrow 0\) as \(\alpha \rightarrow \infty \), there exists \(\alpha _1>0\) such that for all \(\alpha \ge \alpha _1\) \(\tau _d(y_0)\le \frac{c_1}{2}\).
Moreover, in view of the above computation and using that \(\Vert v\Vert _{H_0^1(\varOmega )^d}\le \frac{1}{\sqrt{\nu }}{\left| \left| \left| v \right| \right| \right| }_{{\varvec{V}}}\) for all \(v\in {\varvec{V}}\) and \(\alpha >0\), we obtain, for all \(k\in {\mathbb {N}}\)
where
Assume \(d=2\). From (17), we obtain for all \(y\in {\varvec{V}}\) that
In particular, taking \(y=g\) allows to remove the \(\alpha \) term and gives
If \(c_{\alpha _1,\nu }\sqrt{E(g)}\ge 1\) then for all \(\alpha \ge \alpha _1\) such that \(c_{\alpha ,\nu }\sqrt{E(g)}\ge 1\) and for all \(k\in {\mathbb {N}}\) :
If \(c_{\alpha _1,\nu }\sqrt{E(g)}< 1\) then there exists \(0<K<1\) such that for all \(\alpha \ge \alpha _1\) we have \(c_{\alpha ,\nu }\sqrt{E(g)}\le K\). We therefore have for all \(\alpha \ge \alpha _1\)
and thus for all \(k\in {\mathbb {N}}\):
On the other hand, there exists \(\alpha _0\ge \alpha _1\) such that, for all \(\alpha \ge \alpha _0\) we have
and
We then deduce from (42) and (43) that for all \(\alpha \ge \alpha _0\) and for all \(k\in {\mathbb {N}}\):
that is \(\tau _2(y_{k+1})\le c_1.\)
Assume \(d=3\). We argue as in the case \(d=2\) and deduce from (17), since \(y_0=g\), that
and thus, if \(c_{\alpha _1,\nu }\sqrt{E(g)}\ge 1\), then for all \(\alpha \ge \alpha _1\) such that \(c_{\alpha ,\nu }\sqrt{E(g)}\ge 1\) and for all \(k\in {\mathbb {N}}\):
If \(c_{\alpha _1,\nu }\sqrt{E(g)}< 1\) then there exists \(0<K<1\) such that for all \(\alpha \ge \alpha _1\) we have \(c_{\alpha ,\nu }\sqrt{E(g)}\le K\). We therefore have for all \(\alpha \ge \alpha _1\)
and thus for all \(k\in {\mathbb {N}}\) :
On the other hand, there exists \(\alpha _0\ge \alpha _1\) such that, for all \(\alpha \ge \alpha _0\) we have
and
We then deduce from (45) to (46) that for all \(\alpha \ge \alpha _0\) and for all \(k\in {\mathbb {N}}\):
that is \(\tau _3(y_{k+1})\le c_1.\) \(\square \)
Gathering the previous lemmas and propositions, we deduce the strong convergence of the sequence \(\{y_k\}_{k\ge 0}\) defined by (30), initialized by \(y_0=g\).
Theorem 1
Let \(c_1\in (0,1)\). Assume that \(y_0=g\in {\varvec{V}}\) and \(\alpha \) is large enough so that
where \(c_2(f,g)\) and \(c_3(f,g)\) are defined in (41) and (44), respectively. The sequence \(\{y_k\}_{(k\in {\mathbb {N}})}\) defined by (30) strongly converges to the unique solution y of (14). Moreover, there exists \(k_0\in {\mathbb {N}}\) such that the sequence \(\{y_k\}_{k\ge k_0}\) converges quadratically to y. Moreover, this solution satisfies \(\tau _d(y)<1\).
2.3 Additional comments
1) Estimate (15) is usually used to obtain a sufficient condition on the data f, g to ensure the uniqueness of the solution of (14) (i.e. \(\tau _d(y)<1\)): it leads to
We emphasize that such (sufficient) conditions are more restrictive than (47), as they impose smallness properties on g: precisely \(\Vert g\Vert ^2_2\le 2\nu ^2\) if \(d=2\) and \(\Vert g\Vert ^2_2\le \frac{\nu ^{5/2}}{M^2 \alpha ^{1/2}}\) if \(d=3\). This latter yields a restrictive condition for \(\alpha \) large contrary to (47).
2) Let \({\mathcal {F}}: {\varvec{V}}\rightarrow {\varvec{V}}^\prime \) the application be defined as \({\mathcal {F}}(y)=\alpha y+\nu B_1(y) + B(y,y)-f-\alpha g\). The sequence \(\{y_k\}_{(k>0)}\) associated to the Newton method to find the zero of \({\mathcal {F}}\) is formally defined as follows:
We check that this sequence coincides with the sequence obtained from (30) if \(\lambda _k\) is fixed equal to one. The algorithm (30) which consists in optimizing the parameter \(\lambda _k\in [0,m]\), \(m\ge 1\), in order to minimize \(E(y_k)\), equivalently \(\Vert {\mathcal {F}}(y_k)\Vert _{{\varvec{V}}^{\prime }}\), corresponds to the so-called damped Newton method for the application \({\mathcal {F}}\) (see [3]). As the iterates increase, the optimal parameter \(\lambda _k\) converges to one (according to Lemma 5), this globally convergent method behaves like the standard Newton method (for which \(\lambda _k\) is fixed equal to one): this explains the quadratic rate of convergence after a finite number of iterates. To the best of our knowledge, this is the first analysis of the damped Newton method for a partiel differential equation. Among the few numerical works devoted to the damped Newton method for partial differential equations, we mention [16] for computing viscoplastic fluid flows.
3) Section 6, chapter 6 of the book [4] introduces a least-squares method in order to solve an Oseen type equation (without incompressibility constraint). The convergence of any minimizing sequence toward a solution y is proved under the a priori assumption that the operator \(D{\mathcal {F}}(y):{\varvec{V}}\rightarrow {\varvec{V}}^{\prime }\)
(for some \(\alpha >0\)) is an isomorphism. y is then said to be a nonsingular point. According to Proposition 2, a sufficient condition for y to be a nonsingular point is \(\tau _d(y)<1\). Recall that \(\tau _d\) depends on \(\alpha \). As far as we know, determining a weaker condition ensuring that \(D{\mathcal {F}}(y)\) is an isomorphism is an open question. Moreover, according to Lemma 3, it turns out that this condition is also a sufficient condition for the uniqueness of (14). Theorem 1 asserts that, if \(\alpha \) is large enough, then the sequence \(\{y_k\}_{(k\in {\mathbb {N}})}\) defined in (30), initialized with \(y_0=g\), is a convergent sequence of nonsingular points.
4) We may also define a minimizing sequence for E using the derivative \(E^{\prime }\) (see (22)):
with \(g_k\in {\varvec{V}}\) such that \((g_k,w)_{{\varvec{V}}}=\langle E^{\prime }(y_k),w\rangle _{\varvec{V^{\prime }}\times {\varvec{V}}}\) for all \(w\in {\varvec{V}}\). In particular, \(\Vert g_k\Vert _{{\varvec{V}}}=\Vert E^{\prime }(y_k)\Vert _{\varvec{V^{\prime }}}\). Using the expansion (25) with \(w_k=g_k\), we can prove the linear decrease of the sequence \(\{E(y_k)\}_{k>0}\) to zero assuming however that \(E(y_0)\) is small enough, of the order of \(\nu ^2\), independently of the value of \(\alpha \) (we refer to [11, Lemma 4.1] in a similar context).
3 Application to the backward Euler scheme
We now use the results of the previous section to discuss the resolution of the backward Euler scheme (2) through a least-squares method. The weak formulation of this scheme reads as follows: given \(y^0=u_0\in {\varvec{V}}\), the sequence \(\{y^n\}_{n>0}\) in \({\varvec{V}}\) is defined by recurrence as follows:
with \(f^n\) defined by (4) in term of the external force of the Navier–Stokes model (1). We recall that a piecewise linear interpolation in time of \(\{y^n\}_{n\ge 0}\) weakly converges in \(L^2(0,T;{\varvec{V}})\) toward a solution of (1).
As done in [2], one may use the least-squares method (analyzed in Sect. 2) to solve iteratively (51). Precisely, in order to approximate \(y^{n+1}\) from \(y^{n}\), one may consider the following extremal problem
where the corrector \(v\in {\varvec{V}}\) solves
with \(\alpha \) and \(f^n\) given by (4). For any \(n\ge 0\), a minimizing sequence \(\{y^n_k\}_{(k\ge 0)}\) for \(E_n\) is defined as follows:
where \(Y^n_{1,k}\in {\varvec{V}}\) solves (31) for \(y_k=y_{k}^{n+1}\). For each n, algorithm (54) can be expanded as follows
for all \(w\in {\varvec{V}}\). In view of Theorem 1, the first element of the minimizing sequence is chosen equal to \(y^n\), i.e. the minimizer of \(E_{n-1}\).
The main goal of this section is to prove that for all \(n\in {\mathbb {N}}\), the minimizing sequence \(\{y^{n+1}_k\}_{k\in {\mathbb {N}}}\) converges to a solution \(y^{n+1}\) of (51). Arguing as in Lemma 6, we have to prove the existence of a constant \(c_1\in (0,1)\), such that \(\tau _d(y_k^{n})\le c_1\) for all n and k in \({\mathbb {N}}\). Remark that the initialization \(y_{0}^{n+1}\) is fixed as the minimizer of the functional \(E^{n-1}\), obtained at the previous iterate. Consequently, the uniform property \(\tau _d(y_k^{n})\le c_1\) is related to the initial guess \(y_0^0\) equal to the initial position \(u_0\), to the external force F (see (1)) and to the value of \(\alpha \). \(u_0\) and F are given a priori. On the other hand, the parameter \(\alpha \), related to the discretization parameter \(\delta t\), can be chosen as large as necessary. As we shall see, this uniform property, which is essential to set up the least-squares procedure, requires smallness properties on \(u_0\) and F.
We start with the following result analogue to Proposition 1.
Proposition 8
Let \((f^n)_{n\in {\mathbb {N}}}\) be a sequence in \(H^{-1}(\varOmega )^d\), \(\alpha >0\) and \(y^0=u_0\in {\varvec{H}}\). For any \(n\in {\mathbb {N}}\), there exists a solution \(y^{n+1}\in {\varvec{V}}\) of
for all \(w\in {\varvec{V}}\). Moreover, for all \(n\in {\mathbb {N}}\), \(y^{n+1}\) satisfies
Moreover, for all \(n\in {\mathbb {N}}^\star \):
Proof
The existence of \(y^{n+1}\) is given in Proposition 1. (58) is obtained by summing (57). \(\square \)
Remark 2
Arguing as in Lemma 3, if there exists a solution \(y^{n+1}\) in \({\varvec{V}}\) of (53) satisfying \(\tau _d(y^{n+1})<1\), then such solution is unique. In view of Proposition 8, this holds true if the quantity \({\mathcal {M}}(f,\alpha ,\nu )\) defined as follows
is small enough.
We now distinguish the case \(d=2\) from the case \(d=3\) and consider weak and regular solutions.
3.1 Two dimensional case
We have the following convergence for weak solutions of (56).
Theorem 2
Suppose \(F\in L^2(0,T;H^{-1}(\varOmega )^2)\), \(u_0\in {\varvec{V}}\). Let \(\alpha \) be large enough and \(f^n\) be given by (4) for all \(n\in \{0,\ldots ,N-1\}\). Let \(c(u_0,F)\) be defined as follows:
There exists a constant \(c>0\) such that if
then for all \(n\in \{0,\ldots ,N-1\}\) the solution \(y^{n+1}\in {\varvec{V}}\) of (56) is unique and the minimizing sequence \(\{y_k^{n+1}\}_{k\in {\mathbb {N}}}\) defined by (54) strongly converges in \({\varvec{V}}\) to \(y^{n+1}\).
Proof
According to Proposition 7, we have to prove the existence of a constant \(c_1\in (0,1)\) such that, for all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\), \(\tau _2(y_{k}^n)\le c_1\).
For \(n=0\), as in the previous section, it suffices to take \(\alpha \) large enough to ensure the conditions (47) with \(g=y_0^0=u_0\) leading to the property \(\tau _2(y_k^0)<c_1\) for all \(k\in {\mathbb {N}}\) and therefore \(\tau _2(y^1)\le c_1\).
For the next minimizing sequences, we recall (see Lemma 6) that for all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\)
where \(p_{n,0}({\widetilde{\lambda }}_{n,0})\) is defined as in the Proof of Proposition 4.
First, since for all \(n\in \{0,\ldots ,N-1\}\), \(\Vert f^n\Vert _{H^{-1}(\varOmega )^2}^2\le \alpha \Vert F\Vert ^2_{L^2(0,T; H^{-1}(\varOmega )^2)}\), we can write
Since \(y^{n}\) is solution of (56), it follows from (53) and (58), that for all n in \(\{1,\ldots ,N-1\}\):
Therefore, for all \(n\in \{0,\ldots ,N-1\}\), \({E_{n}(y^n)}\le \frac{{\alpha }}{{\nu }}c(u_0,F)\).
Let \(c_1\in (0,1)\) and suppose that \(c(u_0,F)<(1-c_1)^4\nu ^3\). Then, there exists \(K\in (0,1)\) and \(\alpha _0>0\) such that, for all \(\alpha \ge \alpha _0\), \(c_{\alpha ,\nu }\sqrt{E_n(y^n)}\le K<1\). We therefore have (see Lemma 6), for all \(\alpha \ge \alpha _0\), all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\):
From (58) we then obtain, for all \(n\in \{0,\ldots ,N-1\}\),
and since \(\frac{1}{\alpha }\sum _{k=0}^{n-1}\Vert f^k\Vert ^2_{H^{-1}(\varOmega )^2}\le \Vert F\Vert ^2_{L^2(0,T; H^{-1}(\varOmega )^2)}\), we deduce that if
then \(\Vert y^n\Vert _{H_0^1(\varOmega )^2}\le \frac{c_1}{2}\sqrt{2\alpha \nu }\). Moreover, assuming \(c(u_0,F)\le \frac{c_1^2(1-c_1)^2(1-K)^2}{4m^2}\nu ^3\), we deduce from (61), for all \(n\in \{0,\ldots ,N-1\}\) and for all \(k\in {\mathbb {N}}\) :
that is \(\tau _2(y^{n}_{k})\le c_1\). The result follows from Proposition 7. \(\square \)
We emphasize that, for each \(n\in {\mathbb {N}}\), the limit \(y^{n+1}\) of the sequence \(\{y^{n+1}_k\}_{k\in {\mathbb {N}}}\) satisfies \(\tau _2(y^{n+1})<1\) and is therefore the unique solution of (56). Moreover, for \(\alpha \) large enough, the condition (60) reads as the following smallness property on the data: \(\Vert F\Vert ^2_{L^2(0,T; H^{-1}(\varOmega )^2)}+\nu \Vert u_0\Vert _2^2\le c\nu ^3\). In contrast with the static case of Sect. 2 where the unique condition (47) on the data g is fulfilled as soon as \(\alpha \) is large, the iterated case requires a condition on the data \(u_0\) and F, whatever be the amplitude of \(\alpha \). Again, this smallness property is introduced in order to guarantees the condition \(\tau _2(y^{n})<1\) for all n. In view of (58), this condition implies notably that \(\Vert y^n\Vert _2\le c\nu ^{3/2}\) for all \(n>0\).
For regular solutions of (56) which we now consider, we may slightly improve the results, notably based on the control of two consecutive elements of the corresponding sequence \(\{y^n\}_{n\in {\mathbb {N}}}\) for the \(L^2(\varOmega )\) norm.
Proposition 9
Assume that \(\varOmega \) is \(C^2\), that \((f^n)_n\) is a sequence in \(L^2(\varOmega )^2\) and that \(u_0\in {\varvec{V}}\). Then, for all \(n\in {\mathbb {N}}\), any solution \(y^{n+1}\in {\varvec{V}}\) of (56) belongs to \(H^2(\varOmega )^2\).
If moreover, there exists \(C>0\) such that
then \(y^{n+1}\) satisfies
where P is the operator of projection from \(L^2(\varOmega )^d\) into \({\varvec{H}}\).
Proof
From Proposition 1, we know that for all \(n\in {\mathbb {N}}^*\), \(y^n\in H^2(\varOmega )^2\cap {\varvec{V}}\). Thus, integrating by part (56) we obtain, using density argument:
for all \(w\in {\varvec{H}}\). Then, taking \(w=P\varDelta y^{n+1}\) and integrating by part leads to:
Recall that
We now use (see [18, chapter 25]) that there exist three constants \(c_1, c_2\) and \(c_3\) such that
and
This implies that
with \(c=c_1c_2c_3\). Recalling (64), it follows that
But, from estimate (58), the assumption (62) implies that \(\Vert y^{n+1}\Vert _2\le \frac{\nu }{4c}\) and
Summing then implies (63) for all \(n\in {\mathbb {N}}\). \(\square \)
Remark 3
Under the hypothesis of Proposition 9, suppose that
is small (which is satisfied as soon as \(\alpha \) is large enough). Then, the solution of (56) is unique.
Indeed, let \(n\in {\mathbb {N}}\) and let \(y^{n+1}_1, y^{n+1}_2\in {\varvec{V}}\) be two solutions of (56). Then \(Y:=y^{n+1}_1-y^{n+1}_2\) satisfies
and in particular, for \(w=Y\) (since \(\int _\varOmega y^{n+1}_2\cdot \nabla Y\cdot Y=0\))
leading to
If
then \(Y=0\) and the solution is unique. But, from (58) and (63),
Therefore, there exists a constant C such that if \(B_{\alpha ,\nu }<C\), then (65) holds true.
Proposition 9 then allows to obtain the following estimate of \(\Vert y^{n+1}-y^n\Vert _2\) in term of the parameter \(\alpha \).
Theorem 3
We assume that \(\varOmega \) is \(C^2\), that \(\{f^n\}_{n\in {\mathbb {N}}}\) is a sequence in \(L^2(\varOmega )^2\) and that \(\alpha ^{-1}\sum _{k=0}^{+\infty }\Vert f^k\Vert ^2_2<+\infty \), that \(u_0\in {\varvec{V}}\) and that for all \(n\in {\mathbb {N}}\), \(y^{n+1}\in H^2(\varOmega )^2\cap {\varvec{V}}\) is a solution of (56) satisfying \(\Vert y^{n+1}\Vert _2\le \frac{\nu }{4c}\). There exists \(C_1>0\) such that
Proof
For all \(n\in {\mathbb {N}}\), \(w=y^{n+1}-y^n\) in (56) gives:
Moreover,
Therefore,
But, (63) implies that for all \(n\in {\mathbb {N}}\)
and thus, since \(\nu <1\)
leading to \(\Vert y^{n+1}-y^n\Vert _2^2=O(\frac{1}{ \alpha \nu ^{3/2}})\) as announced. \(\square \)
This result asserts that two consecutive elements of the sequence \(\{y^n\}_{n\ge 0}\) defined by recurrence from the scheme (2) are close to each other as soon as \(\delta t\), the time step discretization, is small enough. In particular, this justifies the choice of the initial term \(y_0^{n+1}=y^n\) of the minimizing sequence in order to approximate \(y^{n+1}\).
We end this section devoted to the case \(d=2\) with the analogue of Theorem 2 for regular data.
Theorem 4
Suppose \(F\in L^2(0,T;L^2(\varOmega )^2)\), \(u_0\in {\varvec{V}}\), for all \(n\in \{0,\ldots ,N-1\}\), \(\alpha \) and \(f^n\) are given by (4) and \(y^{n+1}\in {\varvec{V}}\) solution of (56). If \(C(u_0,F):=\Vert F\Vert ^2_{L^2(0,T; L^2(\varOmega )^2)}+\nu \Vert u_0\Vert _{H_0^1(\varOmega )^2}^2\le C\nu ^{2}\) for some C and \(\alpha \) is large enough, then, for any \(n\ge 0\), the minimizing sequence \(\{y^{n+1}_k\}_{k\in {\mathbb {N}}}\) defined by (54) strongly converges to the unique of solution of (56).
Proof
As for Theorem 2, it suffices to prove that there exists \(c_1\in (0,1)\) such that, for all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\), \(\tau _2(y_{k}^n)\le c_1\). Let us recall that for all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\)
where \(p_{n,0}({\widetilde{\lambda }}_{n,0})\) is defined as in the proof of Proposition 4. From (53), since for all \(n\in \{0,\ldots ,N-1\}\), \(\Vert f^n\Vert _{2}^2\le \alpha \Vert F\Vert ^2_{L^2(0,T; L^2(\varOmega )^2)}\):
and, since \(y^{n}\) is solution of (56), then for all \(n\in \{1,\ldots ,N-1\}\):
From the Proof of Theorem 3, we deduce that for all \(n\in \{0,\ldots ,N-1\}\):
and thus, for all \(n\in \{1,\ldots ,N-1\}\)
Moreover, from (63), for all \(n\in \{0,\ldots ,N-1\}\):
Eventually, let \(c_1\in (0,1)\). Then there exists \(\alpha _0>0\) such that, for all \(\alpha \ge \alpha _0\) \(c_{\alpha ,\nu }\sqrt{E_n(y^n)}\le K<1\). We therefore have (see Theorem 2), for all \(\alpha \ge \alpha _0\), all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\):
which gives a bound of \(\Vert y^{n+1}_{k+1}\Vert _{H_0^1(\varOmega )^2}\) independent of \(\alpha \ge \alpha _0\).
Taking \(\alpha _1\ge \alpha _0\) large enough, we deduce that, for all \(\alpha \ge \alpha _1\), all \(n\in \{0,\ldots ,N-1\}\) and all \(k\in {\mathbb {N}}\), \(\Vert y^{n}_{k}\Vert _{H_0^1(\varOmega )^2}\le c_1\sqrt{2\alpha \nu }\), that is \(\tau _2(y^{n}_{k})\le c_1\). The announced convergence follows from Proposition 7. \(\square \)
3.2 Three dimensional case
We now consider regular solutions for the case \(d=3\). The following intermediate regularity result holds true.
Proposition 10
Assume that \(\varOmega \) is \(C^2\), that \((f^n)_n\) is a sequence in \(L^2(\varOmega )^3\) and that \(u_0\in {\varvec{V}}\). Then any solution \(y^{n+1}\in {\varvec{V}}\) of (56) belongs to \(H^2(\varOmega )^3\).
If moreover, there exists \(C>0\) such that
then the inequality (63) holds true.
Proof
From Proposition 1, we know that for all \(n\in {\mathbb {N}}^*\), \(y^n\in H^2(\varOmega )^3\cap {\varvec{V}}\). Let now P be the operator of projection from \(L^2(\varOmega )^3\) into \({\varvec{H}}\). Taking \(w=P\varDelta y^{n+1}\) in (56) leads to:
In view of the inequality
we use again that there exist constants \(c_1,c_2>0\) such that
so that, for \(c=c_1c_2\), we obtain
It results from (68) that
Assume that, for all \(n\in {\mathbb {N}}^*\), we have constructed by recurrence an element \(y^n\) solution of (56) such that
Then, for all \(n\in {\mathbb {N}}\)
and recursively, for all \(n\in {\mathbb {N}}^*\), we get (63).
It remains to construct a sequence \(\{y^n\}_{n\in {\mathbb {N}}^*}\) solution of (56) and satisfying for all \(n\in {\mathbb {N}}^*\) the property (70). Let us first remark that the hypothesis (67) implies that \(y^0\) satisfies (70). Let then \(n\in {\mathbb {N}}\) fixed. Assume now, that we have constructed, for \(k\in \{0,\ldots ,n\}\) a solution \(y^k\) satisfying (56) if \(k\ge 1\) and \(\frac{\nu }{ 4}-c\Vert y^{k}\Vert _3>0\) for \(c=c_1c_2\) introduced above. Let \(y_1\in {\varvec{V}}\) and let \(y_2\in H^2(\varOmega )^3\cap {\varvec{V}}\) be the unique solution of
If \(y_1\) satisfies \(\Vert y_{1}\Vert _3\le \frac{\nu }{ 4c}\), then in view of (69),
and consequently
(71) then implies
We now use that there exists a constant \(c_3>0\) such that, for all \(n\in {\mathbb {N}}\) \(\Vert y_2\Vert _3\le c_3\Vert \nabla y_2\Vert _2\) to obtain
Invoking assumption (67), we conclude that \(\Vert y_2\Vert _3\le \frac{\nu }{ 4c}\).
Eventually, we introduce the application \({\mathcal {T}}: {\mathcal {C}}\rightarrow {\mathcal {C}}\), \(y_1\mapsto y_2\) where \({\mathcal {C}}\) is the closed convex set of \({\varvec{V}}\) defined by \({\mathcal {C}}:=\{y\in {\varvec{V}},\ \frac{\nu }{ 4c}\ge \Vert y\Vert _3\}\). Let us check that \({\mathcal {T}}\) is continuous. Let \(y_1\in {\mathcal {C}}\) et \(z_1\in {\mathcal {C}}\), \(y_2={\mathcal {T}}(y_1)\) et \(z_2={\mathcal {T}}(z_1)\) so that
for all \(w\in {\varvec{V}}\) and then for \(w=z_2-y_2\)
using (72); this implies the continuity of \({\mathcal {T}}\). On the other hand, since \({\mathcal {T}}({\mathcal {C}})\) is a bounded set of \(H^2(\varOmega )^3\), \({\mathcal {T}}\) is relatively compact. The Schauder Theorem allows to affirm that \({\mathcal {T}}\) has a fixed point \(y\in {\mathcal {C}}\), that is, a solution \(y^{n+1}\in {\mathcal {C}}\) of (56). \(\square \)
Remark 4
Under the hypothesis of Proposition 10, suppose moreover that
is small enough, then, the solution of (56) is unique.
Indeed, let \(n\in {\mathbb {N}}\) and let \(y^{n+1}_1, y^{n+1}_2\in {\varvec{V}}\) be two solutions of (56). Let \(Y:=y^{n+1}_1-y^{n+1}_2\). Then,
and in particular, for \(w=Y\) (since \(\int _\varOmega y^{n+1}_2\cdot \nabla Y\cdot Y=0\))
and therefore \((\nu -\frac{c}{ \alpha }\Vert \nabla y^{n+1}_1\Vert _2\Vert P\varDelta y^{n+1}_1\Vert _2 )\Vert \nabla Y\Vert _2^2\le 0\). Moreover, from (63),
Therefore, there exists a constant \(c>0\) such that if \(C_{\alpha ,\nu }<c\), then, arguing as in the 2D case, \(\Vert \nabla Y\Vert _2^2\le 0\) and \(Y=0\).
As in the 2D case, Proposition 10 then allows, following the Proof of Theorem 3, to obtain an estimate of \(\Vert y^{n+1}-y^n\Vert _2\) in term of the parameter \(\alpha \).
Theorem 5
Assume that \(\varOmega \) is \(C^2\), that \(\{f^n\}_{n\in {\mathbb {N}}}\) is a sequence in \(L^2(\varOmega )^3\) satisfying \(\alpha ^{-1}\sum _{k=0}^{+\infty }\Vert f^k\Vert ^2_2<+\infty \). Assume moreover that \(y^0\in {\varvec{V}}\) and that for all \(n\in {\mathbb {N}}\), \(y^{n+1}\in H^2(\varOmega )^3\cap {\varvec{V}}\) is a solution of (56) satisfying (63). Then, the sequence \((y^n)_{n}\) satisfies (66).
Eventually, adapting the Proof of Theorem 4, we get the following convergence result.
Theorem 6
Suppose \(F\in L^2(0,T;L^2(\varOmega )^3)\), \(y^0\in {\varvec{V}}\), for all \(n\in \{0,\ldots ,N-1\}\), \(\alpha \) and \(f^n\) are given by (4) and \(y^{n+1}\in {\varvec{V}}\) solution of (56). If \(C(y^0,F):=\Vert F\Vert ^2_{L^2(0,T; L^2(\varOmega )^3)}+\nu \Vert y^0\Vert _{H^1_0(\varOmega )^3}^2\le C\nu ^{3}\) for some \(C>0\) and \(\alpha \) is large enough, then for any \(n\ge 0\), the minimizing sequence \(\{y^{n+1}_k\}_{k\in {\mathbb {N}}}\) defined by (54) strongly converges to the unique of solution of (56).
Remark 5
We have considered regular solutions in the case \(d = 3\) in order to be able to prove the uniform property \(\tau _3(y_k^n)\le c_1<1\) for some \(c_1\) independent of k and n, i.e. \(\Vert y_k^n\Vert _{H^1_0(\varOmega )^3}\le c_1 M^{-1} (\alpha \nu ^3)^{1/4}\). Actually, for regular solutions, Proposition 10 implies that \(\Vert y_k^n\Vert _{H^1_0(\varOmega )^3}\le C\) for some C independent of \(\alpha \), which is sufficient for \(\alpha \) large enough. By considering weak solutions, we can only prove that \(\Vert y_k^n\Vert _{H^1_0(\varOmega )^3}\le C \alpha ^{1/2}\) for some \(C>0\) (see (58)) which does not imply \(\tau _3(y_k^n)\le c_1<1\).
4 Numerical illustrations
We discuss in this section numerical experiments based on finite element approximations in space for two geometries of \({\mathbb {R}}^2\): the celebrated channel with a backward facing step and the semi-circular driven cavity introduced in [5]. In both cases, the velocity of the fluid is imposed on the boundary. We first start with the case \(\alpha =0\) in (14) (discussed in [11]) allowing, first to get the solution of (1) as time becomes large and secondly, to enhance the gain of the optimization of the descent step parameter \(\lambda _k\) in (30). Then, for the semi driven cavity, we consider the cases \(\alpha =0\) and \(\alpha >0\) applied to the resolution of the backward Euler scheme (51). In a final part, we briefly compare the computational cost of this least-squares approach with standard explicit and semi-explicit scheme.
The numerical simulations are performed with the FreeFem++ package (see [8]). Regular triangular meshes are used together with the \({\mathbb {P}}_2/{\mathbb {P}}_1\) Taylor-Hood finite element, satisfying the Ladyzenskaia-Babushka-Brezzi condition of stability.
4.1 Steady case: two dimensional channel with a backward facing step
We consider in the steady situation the test problem of a two-dimensional channel with a backward facing step, described for instance in Section 45 of [6] (see also [9]). The geometry is depicted in Fig. 1. Dirichlet conditions of the Poiseuille type are imposed on the inflow and outflow sides \(\varGamma _1\) and \(\varGamma _2\) of the channel: we impose \(y=(4(H-x_2)(x_2-h)/(H-h)^2,0)\) on \(\varGamma _1\) and \(y=(4(H-h)x_2(H-x_2)/H^2,0)\) on \(\varGamma _2\), with \(h=1,H=3, l=3\) and \(L=30\). On the remaining part \(\partial \varOmega {\setminus } (\varGamma _1\cup \varGamma _2)\), the fluid flow is imposed to zero. The external force f is zero.
A two-dimensional channel with a step
We consider the extremal problem (16) to solve the steady Navier–Stokes Eq. (3) with here \(\alpha =0\). We compare the descent algorithm (30) based on the descent direction \(Y_1\) with the conjugate gradient (CG) algorithm used in [11]. In both cases, the initial guess is defined as the solution of the corresponding Stokes problem. Moreover, the scalar extremal problem with respect to \(\lambda _k\) in (30) is performed with the Newton–Rasphon method.
We start with a large value of \(\nu =1/150\). Table 1 reports the evolution of the quantity \(\Vert y_{k+1}-y_k\Vert _{H^1_0(\varOmega )^2}/\Vert y_k\Vert _{H^1_0(\varOmega )^2}\) with respect to the iterate k associated to the algorithms (30), (30) with fixed step \(\lambda _k=1\) and CG respectively. We also consider the so-called by analogy damped quasi newton method
with \({\widetilde{Y}}_{1,k}\in {\varvec{V}}\) the solution of the formulation
and \(v_k\in {\varvec{V}}\) the corrector (associated to \(y_k\)) solution of (17).
A regular mesh composed of \(14\,143\) triangles and \(7\,360\) vertices is used. Table 2 reports the evolution of the norm of the corrector \(\Vert v_k\Vert _{H^1_0(\varOmega )^2}\), an upper bound of \(\Vert y-y_k\Vert _{H^1_0(\varOmega )^2}\), according to Proposition 5. As expected in view of the results in Sect. 2.2, the descent algorithm (30) based on \(Y_{1,k}\) is much faster than the CG algorithm. Moreover, the optimal values for the optimal step \(\lambda _k\) are close to one, so that the Newton method provides a similar speed of convergence. As the norm of \(Y_{1,k}\) goes to zero with k, the term factor of \(\lambda ^2\) in (33) gets small, and the optimal \(\lambda _k\) gets close to one. Remark as well that the algorithm (73) offers an excellent speed of convergence. In term of CPU times, algorithms (30) and (73) require about 53 and 108 seconds respectively and leads to the same approximation. We have notably \(\Vert \nabla \cdot y\Vert _{L^2(\varOmega )}/ \vert \varOmega \vert \approx 1.83\times 10^{-4}\).
For smaller values of \(\nu \), the results are qualitatively differents. Table 3 reports some norms with respect to k for \(\nu =1/700\). We observe, from the last column, that the Newton method for which \(\lambda _k\) is fixed to one does not converge anymore. Actually, the Newton method, when initialized with the solution of the corresponding Stokes problem, diverges for \(\nu \le 1/250\). On the other hand, the optimization of the step \(\lambda _k\) produces a very fast convergence of the sequence \(\{y_k\}_{(k>0)}\). Observe here that the values for the optimal \(\lambda _k\) are not close to one, during the first iterations. We obtain notably \(\Vert \nabla \cdot y\Vert _{L^2(\varOmega )}/\vert \varOmega \vert \approx 5.78\times 10^{-2}\). In agreement with Theorem 1, we observe from Table 3 that the decrease of \(\sqrt{E(y_k)}\) to zero is first linear and then becomes quadratic.
The algorithm (73) is a bit more robust than the Newton one as it converges for all \(\nu \) satisfying \(\nu \ge 1/290\) approximately. Finally, as discussed in [11], the CG algorithm converges and produces similar numerical values: the convergence is however much slower since about 350 iterates are needed to achieve \(\sqrt{2E(y_k)}\) of the order \(10^{-3}\).
The algorithm (30) requires however the initial guess to be close enough to the solution. Initialized with the solution of the corresponding Stokes problem, it diverges for \(\nu \le 1/720\). A continuation method with respect to \(\nu \) is then necessary in that case. Algorithm (30) is also robust with respect to the mesh size: with a twice finer mesh composed of \(84\,707\) triangles and \(43\,069\) vertices, the convergence \(\Vert y^{k+1}-y^{k}\Vert _{H^1_0(\varOmega )^2} \le 10^{-12} \Vert y^k\Vert _{H^1_0(\varOmega )^2}\) is observed after \(k=18\) iterates (instead of 14 for the coarser mesh) leading notably to
\(\Vert \nabla \cdot y\Vert _{L^2(\varOmega )}/\vert \varOmega \vert \approx 3.91\times 10^{-2}\).
4.2 Steady case: 2D semi-circular cavity
We now consider the test discussed in [5]. The geometry is a semi-disk \(\varOmega =\{(x_1,x_2)\in {\mathbb {R}}^2, x_1^2+x_2^2< 1/4, x_2\le 0\}\) depicted on Fig. 2. The velocity is imposed to \(y=(g,0)\) on \(\varGamma _0=\{(x_1,0)\in {\mathbb {R}}^2, \vert x_1\vert <1/2\}\) with g vanishing at \(x_1=\pm 1/2\) and close to one elsewhere: we take \(g(x_1)=(1-e^{100(x_1-1/2)})(1-e^{-100(x_1+1/2)})\). On the rest \(\varGamma _1=\{(x_1,x_2)\in {\mathbb {R}}^2, x_2<0, x_1^2+x_2^2=1/4\}\) of the boundary the velocity is fixed to zero.
Semi-disk geometry
For a regular triangular mesh, composed of \(79\,628\) triangles and \(40\,205\) vertices, leading to a mesh size \(h\approx 6.23\times 10^{-3}\), the Newton method (\(\lambda _k=1\)) initialized with the corresponding Stokes solution, converges up to \(\nu ^{-1} \approx 500\). On the other hand, the algorithm (30) still converges up to \(\nu ^{-1}\approx 910\). Figures 3 depicts the streamlines of the steady state solution corresponding to \(\nu ^{-1}=500\) and to \(\nu ^{-1}=i\times 10^{3}\) for \(i=1,\ldots , 8\). The values used to plot the stream function are given in Table 6. The figures are in very good agreements with those depicted in [5]. The solution corresponding to \(\nu ^{-1}=500\) is obtained from the sequence given (30) initialized with the Stokes solution. Eight iterates are necessary to achieve \(\sqrt{2E(y)}\approx 3.4\times 10^{-17}\). The stopping criterion is \(\Vert y_{k+1}-y_k\Vert _{H^1_0(\varOmega )^2}\le 10^{-12} \Vert y_k\Vert _{H^1_0(\varOmega )^2}\). Tables 4 and 5 collect some values for \(\nu =1/500\) and \(\nu =1/700\). Then, the other solutions are obtained by a continuation method with respect to \(\nu \) taking \(\delta \nu ^{-1}=500\). For instance, the solution corresponding to \(\nu ^{-1}=5000\) is obtained from the algorithm (30) initialized with the steady solution corresponding to \(\nu ^{-1}=4500\). Table 9 reports the history of the continuation method and highlights the efficiency of the algorithm (30): up to \(\nu ^{-1}=9500\), few iterations achieve the convergence of the minimizing sequence \(\{y_k\}_{k\in {\mathbb {N}}}\). From \(\nu ^{-1}=10^4\), with a finer mesh (for which the mesh size is \(h\approx 4.37\times 10^{-3}\)), \(\delta \nu \) is reduced to \(\delta \nu ^{-1}=100\) and leads to convergence beyond \(\nu ^{-1}=15,000\). Table also reports the minimal values of the streamline function \(\psi \) which compare very well with those of [5].
Streamlines of the steady state solution for \(\nu ^{-1}=500, 1000,2000,3000,4000,5000,6000,7000\) and \(\nu ^{-1}=8000\)
The case \(\alpha >0\) leads to similar results, in full agreement with the theoretical Sect. 2. For \(\nu =1/1000\), Table 7 reports results of the algorithm (30) for \(\alpha \in \{10^{-1},1.,10.,100.\}\). As expected, the gain of coercivity of the functional E involves a notable robustness and speed up of the algorithm. Recall that for \(\alpha =0.\) and \(\nu =1/1000\), algorithm (30) does not converge. For \(\alpha =10^{-1}\), we observe the convergence after 8 iterates. For a fixed value of \(\nu \), this number of iterates decreases as \(\alpha \) gets larger. Actually, we observe that when \(\alpha \sqrt{\nu }={\mathcal {O}}(1)\), algorithm (30) converges after few iterates with \(\lambda _k\) close to 1 for all k. Moreover, when \(\alpha \nu ={\mathcal {O}}(1)\), the convergence is achieved after one iterate only. This behavior suggests that one may recover the solution of the steady Navier–Stokes system (corresponding to \(\alpha =0\)) by using a continuation procedure with respect to the parameter \(\alpha \) decreasing to zero. We easily check, for any \(\alpha \ge 0\), the estimate \(\Vert \nabla (y_{\alpha }-y_{\alpha =0})\Vert _{2}\le c_p \sqrt{\frac{\alpha }{\nu }}\Vert y_{\alpha =0}\Vert _2\) where \(y_{\alpha }\) solves (5) with \(g=0\), and \(c_p\) the Poincaré constant. For \(\nu =1/5000\), Table 8 reports the history of the continuation approach starting from \(\alpha =1.\) to \(\alpha =0\) with intermediate steps \(\alpha \in \{10^{-1},10^{-2},10^{-3}, 10^{-4}\}\). Figure 4 depicts the evolution of the sequences \(\{\sqrt{2E(y_k)}\}_k\) and \(\{\lambda _k\}_k\) obtained from the algorithm (30) with \(\alpha =10^{-3}\). The algorithm is initialized with the solution corresponding to \(\alpha =10^{-2}\). The behavior of these sequences fully illustrates Theorem 1 and the robustness of the method: as \(\lambda _k\) increases, the decay of \(\sqrt{2E(y_k)}\), initially low, gets larger and becomes very fast when \(\lambda _k\) is close to one. Figure 5 reports the streamlines of the solution for various values of \(\alpha \).
Semi-disk geometry; \(\nu =1/5000\); evolution of \(\sqrt{2E(y_k)}\) and \(\lambda _k\) w.r.t. k for \(\alpha \) from \(10^{-2}\) to \(10^{-3}\)
Streamlines of the \(\alpha \)-steady state solution for \(\alpha =1.,10^{-1}, 10^{-2}\) (Top) and \(\alpha =10^{-3}, 10^{-4}\) and 0. (Bottom)
4.3 Unsteady case: 2D semi-circular cavity
We now use the least-squares method in order to solve iteratively the implicit Euler scheme (2). The parameter \(\alpha =1/\delta t\) is strictly positive. We remind that for \(\nu \) approximatively larger than 1/6600, the unsteady solution converges as time evolves to the steady solution (corresponding to \(\alpha =0\)) obtained in the previous section by a continuation technique. Actually, the iterative process due to the time discretization can also be seen as a continuation approach. We consider the value \(\nu =1/1000\). Following [5], we take as initial condition \(u_0\) the steady-state solution corresponding to \(\nu =1/500\). For the value \(\alpha =200\) corresponding to the time discretization parameter \(\delta t=5\times 10^{-3}\), we observe the convergence of the sequence \(\{y_k^{n+1}\}_{k>0}\) after at most three iterations, for each n (except for \(n=0\) requiring 6 iterations). For the value \(\alpha =2000\) corresponding to \(\delta t=5\times 10^{-4}\) (used in [5]), we observe the convergence of the sequence after one iterate. At time \(T=10\), the unsteady state solution is close to the solution of the steady Navier–Stokes equation: we compute that the sequence \(\{y^{n}\}_{n=0,\ldots ,2000}\) satisfies \(\Vert y^{2000}-y^{1999}\Vert _{L^2(\varOmega )}/\Vert y^{2000}\Vert _{L^2(\varOmega )}\approx 1.19\times 10^{-5}\). \(n=2000\) corresponds to \(T=2000\times \delta t=10\). Figures 6 display the streamlines of the unsteady state solution corresponding to \(\nu =1/1000\) at time 0, 1, 2, 3, 4, 5, 6 and 7 s to be compared with the streamlines of the steady solution depicted in Fig. 5. These figures are in full agreement with [5].
Streamlines of the unsteady state solution for \(\nu ^{-1}=1000\) at time \(t=i\), \(i=0,\ldots ,8\)s
Before comparing with standard time marching schemes, let us make a comment on the algorithm (55). For \(\alpha \) large, the optimal step \(\lambda _k\) in (55) equals one and the convergence with respect to k is achieved after one iterate. The convergent approximation \(y^{n+1}:=y_1^{n+1}\) then simply solves, for each n the following semi-implicit scheme (mentioned in [15, section 13.4])
For \(\delta t\) of the order \(10^{-3}\), this first-order in time scheme displays, for \(\nu =1/1000\) and the semi-disk geometry of Fig. 2, very similar results than the conditionally stable partially explicit scheme (we refer to [19, Section 5.1])
for all \(w\in {\varvec{V}}\) and than the unconditionally stable scheme
for all \(w\in {\varvec{V}}\). In term of computational cost, the scheme (75) is as expected faster than the scheme (74): the ratio of the computational time to perform 10000 iterates (leading to \(T=10\)) between (74) and (75) is approximatively equal to 1.65. The regular triangulation used corresponds to a mesh size h of the order of \(6.23\times 10^{-3}\) making (75) stable. On the other hand, the computational times of (74) and (76) are equivalent: we observe a ratio equal to 1.05. For \(\delta t = 10^{-2}\), scheme (75) is unstable. The convergence with respect to k of (55) (not anymore equivalent to (74)) is observed after two iterates. The ratio of computational time between (55) and (76) raises to 1.89. We observe that the approximation for (55) is much less sensitive to the variation of \(\delta t\): we observe similar results than with \(\delta t=10^{-3}\) and than [5] where \(\delta t=5\times 10^{-4}\) is used.
5 Conclusions and perspectives
We have rigorously analyzed a weak least-squares method introduced forty years ago in [2] allowing to solve a steady nonlinear Navier–Stokes equation, in the incompressible regime. This equation with a zero order term appears after any fully implicit time discretization of the unsteady Navier–Stokes equation. We have constructed a sequence converging strongly to the solution of the steady equation. Using a particular descent direction very appropriate for the analysis, this convergent sequence turns out to coincide with the sequence obtained using the damped Newton method to solve the underlying variational weak formulation. This globally convergent approach enjoys a quadratic rate of convergence after a finite number of iterates and is in particular much faster than the conjugate gradient method used in [2, 11]. Then, we have shown the convergence of the method, uniformly with respect to the time discretization, to solve the fully implicitly Euler scheme associated to the unsteady Navier–Stokes equation. When the time discretization is fine enough, each step of the damped Newton method simply reduced to the Newton one. In such a case, we obtained a proof of convergence of the Newton scheme to solve the unsteady Navier–Stokes. As far as we know, this proof is original. Numerical experiments have highlighted the robustness of the method, including for values of the viscosity coefficient of order \(10^{-4}\). We also emphasize that the least-squares approaches, employed here to treat the Navier–Stokes nonlinearity, can be used to solve other nonlinear equations, as formally done in [14] for a sublinear heat equation. Eventually, we may solve the unsteady Navier–Stokes system by a fully \(L^2(0,T;H^{-1}(\varOmega ))\) least-squares approach. The underlying corrector solves an unsteady Stokes type equation; we refer to [12].
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Lemoine, J., Münch, A. Resolution of the implicit Euler scheme for the Navier–Stokes equation through a least-squares method. Numer. Math. 147, 349–391 (2021). https://doi.org/10.1007/s00211-021-01171-1
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DOI: https://doi.org/10.1007/s00211-021-01171-1