Discrete Functional Analysis Tools for Some Evolution Equations

Thierry Gallouët

doi:10.1515/cmam-2017-0059

Publicly Available Published by De Gruyter February 7, 2018

Discrete Functional Analysis Tools for Some Evolution Equations

Thierry Gallouët

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0059

Abstract

We present some discrete functional analysis tools for the proof of convergence of numerical schemes, mainly for equations including diffusion terms such as the Stefan problem or the Navier–Stokes equations in the incompressible and compressible cases. Some of the results covered here have been proved in previous works, coauthored with several coworkers.

Keywords: Discrete Functional Analysis; Stefan Problem; Navier–Stokes Equations

MSC 2010: 65M08; 35Q30; 65M12

1 Introduction

The main purpose of this paper is to describe some tools, which have recently been developed, for the proof of convergence of numerical schemes for some evolution problems. We have two main examples in mind.

The first example is the case of the Navier–Stokes equations, with the three classical particular cases, namely the incompressible case, the case of variable density and the compressible case. We briefly describe the most difficult case, which is the compressible case. The main unknowns are the density ρ, the pressure p, and the velocity 𝒖. They are function of the time t which belongs to [0,T] for a given T>0, and of the space variable x which belongs to Ω, where Ω is a given bounded open connected set of ℝd, d=2 or 3. With a given source term, f, in the momentum equation, and a perfect gas equation of state (EOS for short), the equations read

(1.1)∂t⁡ρ+div⁡(ρ⁢𝒖)=0,

(1.2)∂t⁡(ρ⁢𝒖)+div⁡(ρ⁢𝒖⊗𝒖)-Δ⁢𝒖+∇⁡p=f,

(1.3)p=ργ.

The constant γ is larger than 1 for d=2 and 32 for d=3. To these equations we have to add some boundary condition, 𝒖=0 for instance, and an initial condition on ρ and 𝒖 (or on ρ⁢𝒖).

Considering the most interesting case d=3, with a convenient discretization of these equations, for instance with the so-called Marker-And-Cell scheme, MAC scheme for short, see [13] for the seminal paper, it is possible to obtain an L∞(]0,T[,Lγ(Ω))-estimate on ρn and an L2(]0,T[,L6(Ω)3)-estimate on 𝒖n, where (ρn,pn,𝒖n)n∈ℕ is a sequence of approximate solutions (with a mesh size and a time step vanishing as n→+∞). Then, up to a subsequence, it is possible to assume that, as n→+∞, ρn weakly converges to some ρ in L2(]0,T[,Lγ(Ω)) and 𝒖n weakly converges to some u in L2(]0,T[,L6(Ω)3). A first difficulty is that this weak convergence is not sufficient to prove the convergence of the product ρn⁢𝒖n to ρ⁢𝒖 (and of ρn⁢𝒖n⊗𝒖n to ρ⁢𝒖⊗𝒖, note that, since γ>32, one has ρn⁢𝒖n⊗𝒖n bounded in L1(]0,T[,Lr(Ω)3×3) for some r>1). We tackle this difficulty in this paper (at least if γ≥2 for ρn⁢𝒖n and γ≥3 for ρn⁢𝒖n⊗𝒖n). Another difficulty, not solved by the present paper, is to pass to the limit in the EOS (see, for instance, [11] for the stationary case).

The second example is the Stefan problem. The set Ω is a given bounded open set of ℝd, d≥1. The function φ is a Lipschitz continuous function from ℝ to ℝ, nondecreasing and such that φ′=0 on ]a,b[, for some real numbers a,b with a<b. We also assume that lim inf|s|→+∞⁡|φ⁢(s)||s|>0. The unknowns are the functions ρ and u, from Ω×]0,T[ to ℝ, and the equation is

(1.4)∂t⁡ρ-Δ⁢u=0,q⁢u⁢a⁢d⁢u=φ⁢(ρ).

Here also, we have to add a boundary condition, u=0, and an initial condition on ρ.

Numerous discretizations of this problem, for instance with Finite Volumes Methods (see [5]) or Gradient Discretization Methods (see [4]), lead to an L2(]0,T[,L2(Ω))-estimate on a sequence of approximate solutions (un,ρn)n∈ℕ. Then one has, up to a subsequence, that ρn→ρ weakly in L2(]0,T[,L2(Ω)) and un→u weakly in L2(]0,T[,L2(Ω)), as n→+∞.

With a mesh size and a time step vanishing as n→+∞, it is quite easy to prove that ∂t⁡ρ-Δ⁢u=0, in the distributional sense, but the difficulty is to prove that u=φ⁢(ρ). The main step for proving u=φ⁢(ρ) is to prove that

limn→+∞⁡∫0T∫Ωρn⁢un⁢𝑑x⁢𝑑t=∫0T∫Ωρ⁢u⁢𝑑x⁢𝑑t.

Then, by applying, e.g., Minty’s trick, one obtains the desired identity. This will be detailed in Sections 3.1.2 (continuous setting) and 3.2.2 (discrete setting).

The common feature of these two examples is that we have two sequences, namely (ρn)n∈ℕ and (un)n∈ℕ, weakly converging in L2(]0,T[,Lq(Ω)) for a sequence and in L2(]0,T[,Lp(Ω)) for the other sequence, with p,q>1, 1p+1q=1. We denote by ρ and u the weak limits of these two sequences. Then the objective is to present some convenient additional hypothesis in order to prove that, with Q=Ω×]0,T[,

limn→+∞⁡∫Qρn⁢(y)⁢un⁢(y)⁢𝑑y=∫Qρ⁢(y)⁢u⁢(y)⁢𝑑y,

and to show that this additional hypothesis is satisfied in the case of the two examples described above.

For the sake of clarity, we will first consider the simple case corresponding essentially to the case of stationary equations (such as the stationary compressible Stokes equations). Then we will consider the more difficult case of evolution equations, including the Stefan problem or the Navier–Stokes equations, where the time derivative plays a different role than the space derivatives.

As usual, the discrete analysis follows closely the continuous analysis and for this reason we begin by the tools in the continuous setting.

This paper uses ideas which were developed in some previous papers written in collaboration with several authors, among whom Jérôme Droniou, Robert Eymard, Raphaèle Herbin and Jean-Claude Latché. Part of the results given in this paper were presented in [7].

2 Stationary Case

2.1 Continuous Setting

Let Ω be a bounded open set of ℝd (d≥1), p,q>1, 1p+1q=1, and let (ρn)n∈ℕ and (un)n∈ℕ be sequences such that ρn→ρ weakly in Lq⁢(Ω), un→u weakly in Lp⁢(Ω) as n→+∞.

In general, one does not have, as n→+∞,

(2.1)∫Ωρn⁢un⁢𝑑x→∫Ωρ⁢u⁢𝑑x.

But, as it is well known, (2.1) is true if the sequence (un)n∈ℕ is bounded in H01⁢(Ω) and p<2⋆, with 2⋆=+∞ if d=1 or 2, and 2⋆=2⁢dd-2 if d≥3 (note that 2⋆=6 if d=3). This can be proved in two slightly different ways.

The first way is to use the compactness of the sequence (un)n∈ℕ (note that, in this paper, “compactness” always means “strong compactness”). Indeed, since H01⁢(Ω) is compactly embedded in Lp⁢(Ω), one has un→u in Lp⁢(Ω) and then (2.1) holds.

The second way is to use the compactness of the sequence (ρn)n∈ℕ in H-1⁢(Ω). Since H01⁢(Ω) is compactly embedded in Lp⁢(Ω), by duality Lq⁢(Ω) is compactly embedded in the dual space of H01⁢(Ω), namely H-1⁢(Ω). Here, one has identified, as usual, an element f of Lq⁢(Ω) with the element of Lp⁢(Ω)′ defined by g↦∫Ωf⁢g⁢𝑑x. Then one has

(2.2)ρn→ρ in ⁢H-1⁢(Ω) as ⁢n→+∞.

Since (un)n∈ℕ is bounded in H01⁢(Ω), one has also (without extraction of a subsequence)

(2.3)un→u weakly in ⁢H01⁢(Ω) as ⁢n→+∞.

Using (2.2) and (2.3), we obtain

∫Ωρn⁢un⁢𝑑x=〈ρn,un〉H-1,H01→〈ρ,u〉H-1,H01=∫Ωρ⁢u⁢𝑑x.

Remark 1.

Another way to use the compactness of the sequence (ρn)n∈ℕ in H-1⁢(Ω) is to use, for all n, the solution of the Dirichlet problem with ρn as datum, namely the function wn such that

(2.4)wn∈H01⁢(Ω),∫Ω∇⁡wn⋅∇⁡v⁢d⁢x=∫Ωρn⁢v⁢𝑑x for all ⁢v∈H01⁢(Ω).

Using the weak convergence of ρn in Lq⁢(Ω), one has wn→w in H01⁢(Ω), where w is the solution of

(2.5)w∈H01⁢(Ω),∫Ω∇⁡w⋅∇⁡v⁢d⁢x=∫Ωρ⁢v⁢𝑑x for all ⁢v∈H01⁢(Ω).

Indeed, it is quite easy to prove that wn weakly converges to w in H01⁢(Ω). Then, taking v=wn in (2.4) and using the convergence of ρn to ρ in H-1⁢(Ω), one has

limn→+∞⁡∫Ω∇⁡wn⋅∇⁡wn⁢d⁢x=limn→+∞⁡∫Ωρn⁢wn⁢𝑑x=∫Ωρ⁢w⁢𝑑x=∫Ω∇⁡w⋅∇⁡w⁢d⁢x.

This proves the convergence of the H01⁢(Ω)-norm of wn to the H01⁢(Ω)-norm of w. This gives

limn→+∞⁡∫Ω(∇⁡wn-∇⁡w)⋅(∇⁡wn-∇⁡w)⁢𝑑x=limn→+∞⁡∫Ω∇⁡wn⋅∇⁡wn⁢d⁢x-2⁢limn→+∞⁡∫Ω∇⁡wn⋅∇⁡w⁢d⁢x+∫Ω∇⁡w⋅∇⁡w⁢d⁢x=0,

which proves that wn converges to w in H01⁢(Ω) (as n→+∞). In order to conclude, we now use (2.3) and obtain

∫Ωρn⁢un⁢𝑑x=∫Ω∇⁡wn⋅∇⁡un⁢d⁢x→∫Ω∇⁡w⋅∇⁡u⁢d⁢x=∫Ωρ⁢u⁢𝑑x,

since ∇⁡wn converges in L2⁢(Ω)d to ∇⁡w and ∇⁡un weakly converges in L2⁢(Ω)d to ∇⁡u.

2.2 Discrete Setting

The set Ω is always a bounded open set of ℝd (d≥1) but adapted to a space discretization. For all n∈ℕ, one has ρn∈Ln and un∈Hn, where Ln and Hn are finite-dimensional spaces included in L∞⁢(Ω). Let p,q>1 with 1p+1q=1. We assume that the sequence (un)n∈ℕ weakly converges to u in Lp⁢(Ω) and the sequence (ρn)n∈ℕ weakly converges to ρ in Lq⁢(Ω). We also assume, as in Section 2.1, that p<2⋆ and we want (as in Section 2.1) a convenient additional condition giving (2.1).

In the case of conforming discretizations (such as Finite Element Methods), namely when Hn⊂H01⁢(Ω), an easy condition is, as in Section 2.1, to assume that the sequence (un)n∈ℕ is bounded in H01⁢(Ω). We are interested here by the more involved case when Hn⊄H01⁢(Ω) but Hn is equipped with a norm, depending on n, “close” to the H01-norm. This is the case for Finite Volume Methods or Gradient Discretization Methods, see, for instance, [5], [6] and [4] for elliptic and parabolic equations and see [13] for the seminal paper on the Marker-And-Cell scheme for the Navier–Stokes equations. As in Section 2.1, we will present two different methods, using compactness of the sequence (un)n∈ℕ or compactness of the sequence (ρn)n∈ℕ.

2.2.1 Compactness of the Sequence (un)n∈ℕ

We first consider the case of classical Finite Volumes with admissible meshes, as in [5, Definition 9.1], see Figure 1. Roughly speaking for d=2 or 3 (the case d=1 is simpler), a classical Finite Volumes mesh consists in a family of disjoints “control volumes” which are open polygonal (if d=2) or polyhedral (if d=3) convex subsets of Ω whose closures cover Ω. The interface between two control volumes is contained in an hyperplane of ℝd. For each control volume K, a point xK is given (it is not necessarily the center of gravity of K). Such a mesh is “admissible” if the segment joining the points xK and xL is orthogonal to the interface σ between the control volumes K and L (see Figure 1).

Figure 1

Here is an example of admissible mesh in the sense of [5].

In this case, the space Hn is the space of functions which are constant on each control volume of the mesh ℳn. We denote by hn the maximum of the diameter of the control volumes of the mesh ℳn and we assume that limn→+∞⁡hn=0.

The space Hn is equipped with a discrete norm which mimics the H01-norm. We denote by ℰint the interfaces which are in Ω and by ℰext the interfaces which are on the boundary of Ω. If σ is the interface between K and L, we denote by dσ the distance between xK and xL. If σ is an interface of K lying on the boundary of Ω, then dσ is the distance between xK and the boundary of Ω. Finally, mσ denotes the (d-1)-Lebesgue measure of σ. With these notation, the norm on Hn, which mimics the H01-norm, reads, if uK is the value of u in the control volume K,

We assume that the sequence (∥(un)∥1,2,n)n∈ℕ is bounded. Then it is proven in [5, Theorem 9.3] that un→u in L2⁢(Ω). Now, we assume furthermore that dK,σdσ is bounded from below by a positive number (independently of K and n), where dK,σ is the distance between xK and xσ, the point belonging to σ and the segment joining xK and xL (σ=K|L). Then it is also proven in [5, Lemma 3.5] that the sequence (un)n∈ℕ is bounded in Lr⁢(Ω) for any r<+∞ if d=1 or 2, and for r=6 if d=3. It is the so-called discrete Sobolev embedding. Then un→u in Lp⁢(Ω) since p<2⋆. Finally, since ρn→ρ weakly in Lq⁢(Ω) (and q=pp-1), we obtain (2.1) as desired.

The fact that un→u in L2⁢(Ω) (and therefore in Lp⁢(Ω)) is a consequence of the Kolmogorov Compactness Theorem and of the following inequality, which is proven in [5, Lemma 9.3] with some C depending only on Ω and taking u=0 outside Ω,

(2.6)∥u(⋅+η)-u∥L2⁢(ℝd)≤C|η|∥u∥1,2,n if u∈Hn and η∈ℝd.

In order to prove (2.6) the admissibility of the mesh, namely the orthogonality between the segment joining xK and xL and K|L (the interface between K and L), is used in the proof of [5, Lemma 9.3]. Without this admissibility condition, we are not able to prove Inequality (2.6).

We consider now the case of non-admissible meshes so that we do not assume the orthogonality condition between the segment joining the points xK and xL and the interface σ between the control volumes K and L (see Fig. 1). But we assume (as before) that dK,σdσ is bounded from below by a positive number (independently of K and n). Then we can also conclude by using an inequality on the translates on u in the L1⁢(ℝd)-norm instead of the L2⁢(ℝd)-norm. This is done, for instance, in [6]. Indeed, the ∥⋅∥1,2,n-norm is the same as above (see [6, norm defined by (43) or by (74) in Lemma 5.2]) and it is proven in [6, Lemma 5.5] that

(2.7)∥u(⋅+η)-u∥L1⁢(ℝd)≤|η|d∥u∥1,2,n if u∈Hn.

Using again the Kolmogorov Compactness Theorem, we obtain the compactness of (un)n∈ℕ in L1⁢(Ω) and therefore un→u in L1⁢(Ω). But, the estimate on ∥un∥1,2,n also gives an estimate on un in Lr⁢(Ω) for any r<+∞ if d=1 or 2, and for r=6 if d=3. This is again the discrete Sobolev embedding, given for instance in [6, Lemma 5.3]. Using this estimate, together with the L1⁢(Ω)-convergence, we can deduce the convergence of un in Lp⁢(Ω) (since p<2⋆) and conclude as before that (2.1) holds.

2.2.2 Compactness of the Sequence (ρn)n∈ℕ

Although it is, of course, not necessary in this stationary case, we will try now to prove (2.1) by using compactness on ρn instead of compactness of un. The interest of this second method will appear in the evolution case where the time variable plays a different role than that of the space variables. However, we will consider here only the case p=q=2, the general case (p<2⋆, q=pp-1) needs more work).

The main difficulty is that the bound on un is on a norm which depends on n (even if this norm is “close” to the H01-norm). The trick we propose here (presented in [7]) is to use the Sobolev space Hs⁢(ℝd) with some s∈]0,1[. We first recall the definition of the space Hs⁢(ℝd).

Definition 1.

For s≥0, Hs(ℝd)={u∈L2(ℝd):(1+|⋅|2)s2u^∈L2(ℝd)}, where u^ is the Fourier transform of u. The norm for the space Hs⁢(ℝd) is given by

(2.8)∥u∥s=∥(1+|⋅|2)s2u^∥L2⁢(ℝd).

With this norm, Hs⁢(ℝd) is a Hilbert space.

Then we set, for s>0,

Hs={u∈Hs⁢(ℝd):u=0⁢ a.e. on ⁢ℝd∖Ω},

equipped with the Hs⁢(ℝd)-norm. The space Hs is a Hilbert space as a closed subspace of the Hilbert space Hs⁢(ℝd).

We begin with the case of admissible meshes. By using (2.6), it is possible to prove that the sequence (un)n∈ℕ is bounded in Hs for 0<s<12. We give the proof of this result in Lemma 4. Then, since Hs is a Hilbert space (with its natural norm), we have un→u weakly in Hs. But, since s>0, we also have compactness of Hs in L2⁢(Ω) since the Hs-norm of u allows a control on the translates of u, see Lemma 5. Then, by duality, identifying the space L2⁢(Ω) with its dual space, one has compactness of L2⁢(Ω) in (Hs)′. This gives ρn→ρ in (Hs)′ and we can conclude as in the continuous case, but with Hs instead of H01,

(2.9)∫Ωρn⁢un⁢𝑑x=〈ρn,un〉(Hs)′,Hs→〈ρ,u〉(Hs)′,Hs=∫Ωρ⁢u⁢𝑑x.

In the case of non-admissible meshes, we can also conclude but we need a little more work. We have to work with (2.7) instead of (2.6). We recall that, for u∈Hn, we also have an Lr⁢(ℝd)-estimate (in term of ∥u∥1,2,n) on u for any r<+∞ if d=1 or 2, and r=6 if d=3. Then, taking r>2, we use the inequality, for all a>0 and ε>0,

a2≤ε⁢ar+ε-1r-2⁢a.

(Actually, this inequality follows from the fact that a2>ε⁢ar implies ar-2<ε-1 which implies a<ε-1r-2. Therefore, for all ε>0 and all a>0, one has a2≤max⁡{ε⁢ar,ε-1r-2⁢a}.) Taking η∈ℝd, a=|u⁢(x+η)-u⁢(x)| and integrating on ℝd (recall that all functions are taken equal to 0 outside Ω), we obtain, for u∈Hn,

∥u(⋅+η)-u∥L2⁢(ℝd)2≤ε∥u(⋅+η)-u∥Lr⁢(ℝd)r+ε-1r-2∥u(⋅+η)-u∥L1⁢(ℝd).

It remains to choose ε=|η|r-2r-1. The Lr⁢(ℝd)-estimate on u (in terms of ∥u∥1,2,n) and (2.7) give the existence of C depending only on Ω, r and the regularity of the mesh (to be more precise, we also use an homogeneity argument) such that

(2.10)∥u(⋅+η)-u∥L2⁢(ℝd)≤C|η|r-22⁢(r-1)∥u∥1,2,n if u∈Hn and η∈ℝd.

In the case d=3, one takes r=6 and one has r-22⁢(r-1)=25. It is now possible to conclude as in the case of admissible meshes. By using (2.10), the sequence (un)n∈ℕ is bounded in Hs for 0<s<r-22⁢(r-1) (see Lemma 4). One has un→u weakly in Hs. Here also, since s>0, we have compactness of Hs in L2⁢(Ω) (see Lemma 5) and, by duality and identifying the space L2⁢(Ω) with its dual space, compactness of L2⁢(Ω) in (Hs)′. This gives ρn→ρ in (Hs)′ and we conclude with (2.9).

Remark 2.

To conclude this section, we can also remark that it is possible, similarly to the continuous case, to use the compactness of (ρn)n∈ℕ under the form ∇n⁡wn→∇⁡w in L2⁢(Ω)d (and ∇n⁡un→∇⁡u weakly in L2⁢(Ω)d) where ∇n is a discretization of ∇, wn is the solution of a discrete equivalent of (2.4) and w is the solution of (2.5). This method is used, for instance, in [4].

3 Evolution Case

3.1 Continuous Setting

Let Ω be a bounded open set of ℝd (d≥1), T>0 and let (ρn)n∈ℕ and (un)n∈ℕ be sequences such that ρn→ρ weakly in L2(]0,T[,Lq(Ω)), un→u weakly in L2(]0,T[,Lp(Ω)), with 1<p<2⋆ and q=pp-1 (recall that 2⋆=+∞ if d=1 or 2, and 2⋆=2⁢dd-2 if d≥3). As in Section 2.1, in general one does not have

(3.1)∫0T∫Ωρn⁢un⁢𝑑x⁢𝑑t→∫0T∫Ωρ⁢u⁢𝑑x⁢𝑑t as ⁢n→+∞,

even if (un)n∈ℕ is bounded in L2(]0,T[,H01(Ω)) because there is no compactness of L2(]0,T[,H01(Ω)) in L2(]0,T[,L2(Ω)) (or L2(]0,T[,L2(Ω)) in L2(]0,T[,H-1(Ω))). (We take here p=q=2 for simplicity.) Of course, (3.1) holds if (un)n∈ℕ is bounded in H1(]0,T[,H01(Ω)) since there is compactness of H1(]0,T[,H01(Ω)) in L2(]0,T[,L2(Ω)). Similarly, (3.1) holds if (ρn)n∈ℕ is bounded in H1(]0,T[,L2(Ω)) (and (un)n∈ℕ bounded in L2(]0,T[,H01(Ω))) since there is compactness of H1(]0,T[,L2(Ω)) in L2(]0,T[,H-1(Ω)). (As usual, L2⁢(Ω) is identified to its dual space.) But such hypotheses are quite strong and the objective is to obtain (3.1) using weaker hypotheses on (∂t⁡un)n∈ℕ or (∂t⁡ρn)n∈ℕ. We consider, for instance, the two examples given in Section 1.

3.1.1 Compressible Navier–Stokes Equations

Let T>0, let Ω be a bounded open connected set of ℝd, let d=2 or 3 and let γ∈ℝ such that γ>1 if d=2, γ>32 if d=3. For n∈ℕ, let (ρn,pn,𝒖n) be a (weak) solution of (1.1)–(1.3) with fn as datum instead of f and an homogeneous Dirichlet boundary condition on 𝒖n. We assume that fn→f in L2(]0,T[,L2(Ω)). Under a convenient hypothesis on the initial condition (on density and velocity), it is possible to obtain an L∞(]0,T[,Lγ(Ω))-estimate on ρn and an L2(]0,T[,H01(Ω)d)-estimate on 𝒖n. Then, up to subsequences, it is possible to assume, as n→+∞, that ρn weakly converges to some ρ in L2(]0,T[,Lγ(Ω)) and 𝒖n weakly converges to some 𝒖 in L2(]0,T[,H01(Ω)d).

In order to pass to the limit on the equation ∂t⁡ρn+div⁡(ρn⁢𝒖n)=0 for proving that ∂t⁡ρ+div⁡(ρ⁢𝒖)=0 (in the distributional sense), the main difficulty is to prove that, for any φ∈Cc∞(ℝd×]0,T[),

(3.2)limn→+∞⁡∫0T∫Ωρn⁢𝒖n⋅∇⁡φ⁢d⁢x⁢d⁢t=∫0T∫Ωρ⁢𝒖⋅∇⁡φ⁢d⁢x⁢d⁢t.

This is not easy since we only have weak convergence of ρn and 𝒖n in Lebesgue spaces.

Let un be a component of the vector-valued function 𝒖n and consider that d=3 (the case d=2 is simpler). We do not have any space-time compactness of (un)n∈ℕ since we have no condition on ∂t⁡un. But, using the fact that (ρn)n∈ℕ is bounded in L2(]0,T[,Lγ(Ω)) and assuming that (𝒖n)n∈ℕ is bounded in L2(]0,T[,L6(Ω))d, we deduce, using γ>65, that (∂t⁡ρn)n∈ℕ is bounded in L1(]0,T[,W-1,1(Ω)) and this gives a compactness result for the sequence (ρn)n∈ℕ in L2(]0,T[,H-1(Ω)) by means of an adaptation of the well-known compactness results for evolution equation due to Lions, Aubin and Simon (see, for instance, Lions [15], Aubin [2] and Simon [16]). Indeed, one uses Theorem 1 with B=H-1⁢(Ω), X=Lγ⁢(Ω) and Y=W-1,1⁢(Ω) (note that X is compactly embedded in B since γ>65, this is due, by duality, to the fact that H01⁢(Ω) is compactly embedded in Lr⁢(Ω) for r<6). Then one has ρn→ρ in L2(]0,T[,H-1(Ω)). Since 𝒖n→𝒖 weakly converges in L2(]0,T[,H01(Ω)3), we then obtain (3.1) and, similarly, since ∇⁡φ is a regular function, (3.2). More precisely, for all ψ∈Cc∞(ℝ3×]0,T[)3, one has

∫0T∫Ωρn⁢𝒖n⋅ψ⁢𝑑x⁢𝑑t=〈ρn,𝒖n⋅ψ〉L2⁢(H-1),L2⁢(H01)→〈ρ,𝒖⋅ψ〉L2⁢(H-1),L2⁢(H01)=∫0T∫Ωρ⁢𝒖⋅ψ⁢𝑑x⁢𝑑t,

which gives ∂t⁡ρ+div⁡(ρ⁢𝒖)=0 (in the distributional sense).

The same difficulty appears in the momentum equation in order to pass to the limit on div⁡(ρn⁢𝒖n⊗𝒖n). This will be possible if one proves that for all ψ∈Cc∞(ℝ3×]0,T[), one has

(3.3)limn→+∞⁡∫0T∫Ωρn⁢vn⁢un⁢ψ⁢𝑑x⁢𝑑t=∫0T∫Ωρ⁢v⁢u⁢ψ⁢𝑑x⁢𝑑t,

where un and vn are two components of 𝒖n (and u, v the corresponding components of 𝒖).

We consider here also the case d=3 and we use the fact that γ>32. It gives that ρn⁢un is bounded in L2(]0,T[,Lr(Ω)) with r=6⁢γ6+γ>65. Since we already know that ρn⁢un→ρ⁢u in the distributional sense, this gives that ρn⁢un→ρ⁢u weakly in L2(]0,T[,Lr(Ω)). Using now the momentum equation, we prove that (∂t⁡(ρn⁢un))n∈ℕ is bounded in L1(]0,T[,W-1,1(Ω)) and this gives a compactness result for the sequence (ρn⁢un)n∈ℕ in L2(]0,T[,H-1(Ω)) using Theorem 1 with B=H-1⁢(Ω), X=Lr⁢(Ω) and Y=W-1,1⁢(Ω) (X is compactly embedded in B since r>65). Since vn→v weakly converges in L2(]0,T[,H01(Ω)), we then obtain (3.3) and this allows us to pass to the limit in div⁡(ρn⁢𝒖n⊗𝒖n).

3.1.2 Stefan Problem

Let T>0, let Ω be a bounded open set of ℝd, d≥1. For n∈ℕ, let (ρn,un) be a (weak) solution of (1.4) with an homogeneous Dirichlet boundary condition on un and an initial condition on ρn (bounded in L2⁢(Ω)). We recall that φ is a Lipschitz continuous function from ℝ to ℝ, nondecreasing, such that φ′=0 on ]a,b[, for some real numbers a,b with a<b and lim inf|s|→+∞⁡|φ⁢(s)||s|>0.

The natural estimates for this problem are an L2(]0,T[,L2(Ω))-estimate on the functions ρn and an L2(]0,T[,H01(Ω))-estimate on the functions un. Up to a subsequence, ρn→ρ weakly in L2(]0,T[,L2(Ω)) and un→u weakly in L2(]0,T[,H01(Ω)). It is quite easy to pass to the limit in the equation ∂t⁡ρn-Δ⁢un=0 and one obtains ∂t⁡ρ-Δ⁢u=0, in the distributional sense, but it is less easy to prove that u=φ⁢(ρ).

The way to prove that u=φ⁢(ρ) consists in proving (3.1) and then to use the Minty trick given (in this simple case) in Lemma 1. In order to prove (3.1), one can use, as in Section 2.1, compactness on (un)n∈ℕ or compactness on (ρn)n∈ℕ (this was not the case in the case of the compressible Navier–Stokes equations described above).

The proof of compactness of (un)n∈ℕ in L2(]0,T[,L2(Ω)) (which leads to (3.1)) is not easy since there is no direct estimate on ∂t⁡un, but a trick due to Alt and Luckhaus [1] allows to obtain directly an estimate on the time-translates of un (without estimate on ∂t⁡un) in L2(]0,T[,L2(Ω)) and therefore gives the compactness of (un)n∈ℕ in L2(]0,T[,L2(Ω)). Then (3.1) holds and this (with the Minty trick) concludes the proof of u=φ⁢(ρ).

Instead of proving compactness of (un)n∈ℕ in L2(]0,T[,L2(Ω)), it is perhaps simpler to prove compactness of the sequence (ρn)n∈ℕ in L2(]0,T[,H-1(Ω)). Indeed, since ∂t⁡ρn-Δ⁢un=0 and since (un)n∈ℕ is bounded in L2(]0,T[,H01(Ω)), the sequence (∂t⁡ρn)n∈ℕ is bounded in L2(]0,T[,H-1(Ω)). This gives compactness of (ρn)n∈ℕ in L2(]0,T[,H-1(Ω)) using Theorem 1 with B=Y=H-1⁢(Ω) and X=L2⁢(Ω). Then one has

un→uweakly in L2(]0,T[,H01(Ω)),

ρn→ρin L2(]0,T[,H-1(Ω)),

and this gives (3.1) since

∫0T∫Ωρn⁢un⁢𝑑x⁢𝑑t=〈ρn,un〉L2⁢(H-1),L2⁢(H01)→〈ρ,u〉L2⁢(H-1),L2⁢(H01)=∫0T∫Ωρ⁢u⁢𝑑x⁢𝑑t.

Here also, it remains to use the Minty trick in order to conclude that u=φ⁢(ρ).

Remark 3.

As for the stationary case, for this Stefan problem, it is also possible to use the compactness of (ρn)n∈ℕ under the form ∇⁡wn→∇⁡w in L2(]0,T[,L2(Ω)d) (and ∇⁡un→∇⁡u weakly in L2(]0,T[,L2(Ω)d)), where wn is the solution of (2.4) and w is the solution of (2.5) (see, for instance, [4]).

Lemma 1 (Minty Trick).

Let Q be a bounded open set of RN, N≥1, and let φ be a continuous nondecreasing function from R to R. We assume that there exists C∈R+ such that

(3.4)|φ⁢(s)|≤C⁢|s|+C for all ⁢s∈ℝ.

(Note that the existence of C is true if φ is Lipschitz continuous.) Let (ρn)n∈N and (un)n∈N be sequences such that ρn→ρ weakly in L2⁢(Q), un→u weakly in L2⁢(Q). We assume that un=φ⁢(ρn) a.e., for all n, and that

limn→+∞⁡∫Qρn⁢(y)⁢un⁢(y)⁢𝑑y=∫Qρ⁢(y)⁢u⁢(y)⁢𝑑y.

Then u=φ⁢(ρ) a.e.

Proof.

Since φ is nondecreasing, one has, for all ρ¯∈L2⁢(Q),

0≤∫Q(ρn-ρ¯)⁢(φ⁢(ρn)-φ⁢(ρ¯))⁢𝑑y=∫Q(ρn-ρ¯)⁢(un-φ⁢(ρ¯))⁢𝑑y.

Inequality (3.4) gives that φ⁢(ρ¯)∈L2⁢(Q). Then

0≤∫Qρn⁢un⁢𝑑y-∫Qρn⁢φ⁢(ρ¯)⁢𝑑y-∫Qρ¯⁢un⁢𝑑y+∫Qρ¯⁢φ⁢(ρ¯)⁢𝑑y.

Passing to the limit as n→+∞ in this inequality yields

0≤∫Q(ρ-ρ¯)⁢(u-φ⁢(ρ¯))⁢𝑑y.

Let ψ∈Cc∞⁢(Q) and m>0. Taking ρ¯=ρ-1m⁢ψ in the previous inequality gives

0≤∫Q(u-φ⁢(ρ-1m⁢ψ))⁢ψ⁢𝑑y.

The function (u-φ⁢(ρ-1m⁢ψ))⁢ψ converges a.e. to (u-φ(ρ)ψ, as m→+∞, and is dominated by the function (|u|+C⁢|ρ|+C⁢|ψ|+C)⁢ψ which belongs to L1⁢(Q). Then, using the Dominated Convergence Theorem, we obtain, as m→+∞,

0≤∫Q(u-φ⁢(ρ))⁢ψ⁢𝑑y.

This inequality with ψ and -ψ gives 0=∫Q(u-φ⁢(ρ))⁢ψ⁢𝑑y and then, since ψ is arbitrary, u=φ⁢(ρ) a.e. in Q. ∎

In the previous examples, we use a compactness result for evolution equations which is essentially due to [15, 2, 16], a proof is given in [10]. We now give this theorem.

Theorem 1.

Let X, B, Y be three Banach spaces, X⊂B, X⊂Y, such that the following hold:

X is compactly embedded in B.
For any bounded sequence (wn)n∈ℕ of X, if ∥wn-w∥B→0 and ∥wn∥Y→0, then w=0.

Let T>0, 1≤p<+∞ and let (un)n∈N be a sequence such that

(un)n∈ℕ is bounded in Lp(]0,T[,X),
(∂t⁡un)n∈ℕ is bounded in L1(]0,T[,Y).

Then there exists u∈Lp(]0,T[,B) such that, up to a subsequence, un→u in Lp(]0,T[,B).

Hypothesis (ii) of Theorem 1 on the spaces X, B, Y is perhaps a little bit curious. Indeed, we can distinguish two cases. A simple situation appears when ∥⋅∥B≤C∥⋅∥Y for some C>0 (this is the case, in particular, when Y=X and hypothesis (i) holds). Another situation is discussed in [16], when B is continuously embedded in Y, that is, ∥⋅∥Y≤C∥⋅∥B (for some C>0). Hypothesis (ii) in Theorem 1 covers a more general framework.

An example of Banach spaces X, B, Y satisfying the two hypotheses (i)–(ii) is X=W01,1⁢(Ω), B=L1⁢(Ω), Y=W⋆-1,1⁢(Ω)=(W01,∞⁢(Ω))′, where, as usual, we identify an element of L1⁢(Ω) with the corresponding linear form on W01,∞⁢(Ω).

A main tool, crucial for the case where ∥⋅∥B≰C∥⋅∥Y is the following lemma essentially due to Lions [15], the proof of which is quite easy by contradiction.

Lemma 2.

Let X, B, Y are three Banach spaces, X⊂B, X⊂Y, satisfying hypotheses (i)–(ii) of Theorem 1. Then, for any ε>0, there exists Cε such that, for w∈X,

∥w∥B≤ε⁢∥w∥X+Cε⁢∥w∥Y.

A particular case for which the conclusion of Lemma 2 is true, simpler than Lemma 2, is when B is a Hilbert space, X is a Banach space continuously embedded in B and Y=X but the norm in Y, denoted by ∥⋅∥Y, is the dual norm of ∥⋅∥X with respect to the scalar product of B, namely

∥u∥Y=sup{(u|v)B:v∈X,∥v∥X≤1}.

Then, for any ε>0 and w∈X, one has

∥w∥B≤ε⁢∥w∥X+1ε⁢∥w∥Y.

The proof is simple since

∥u∥B=(u|u)B12≤(∥u∥Y∥u∥X)12≤ε∥w∥X+1ε∥w∥Y.

In this simple case, the compactness of X in B is not needed for the conclusion of Lemma 2, but nevertheless this compactness is needed for Theorem 1 even in this case. We can also remark that in this simple case, the space Y is the space X with the norm ∥⋅∥Y and it is not necessarily a Banach space. Indeed, the fact that Y is a Banach space is never needed in Theorem 1, we only need Y to be a normed space since we can always complete this normed space in order to have a Banach space.

We recall the three spaces X, B, Y which were used in this section for the examples describes above (Navier–Stokes equations and Stefan problem).

For the compressible Navier–Stokes equations, in order to prove the compactness of the sequence (ρn)n∈ℕ in L2(]0,T[,H-1(Ω)) we choose B=H-1⁢(Ω), X=Lγ⁢(Ω) and Y=W-1,1⁢(Ω). In order to prove the compactness of (ρn⁢un)n∈ℕ in L2(]0,T[,H-1(Ω)), we choose B=H-1⁢(Ω), X=Lr⁢(Ω) (with r=6⁢γ6+γ) and Y=W-1,1⁢(Ω).

For the Stefan problem, we choose X=L2⁢(Ω), B=Y=H-1⁢(Ω).

Remark 4.

Another interesting example is the case of the incompressible Navier–Stokes equations for d=2 or 3 (see [3]), where we have to pass to the limit on div⁡(𝒖n⊗𝒖n). We set H={𝒖∈H01⁢(Ω)d:div⁡𝒖=0} (which is the natural space for the velocity) and we can use, for this case, Theorem 1 with p=2, B=L2⁢(Ω), X=H, Y=H′ (with the identification of L2⁢(Ω) with its dual space). It gives the compactness of the sequence of approximate velocities (𝒖n)n∈ℕ in L2(]0,T[,L2(Ω)d) which allows to pass to the limit in div⁡(𝒖n⊗𝒖n). In this example one has X continuously embedded in B (and, even, compactly embedded in B) but X is not dense in B so that B′ (which is identified with B) is not included in Y. However, one has H⊂H′ (using the identification of B′ with B), which is the hypothesis needed in Theorem 1. This is a situation quite general. If E is a Banach space, continuously embedded in the Hilbert F, and if F′ is identified with F, then one has E⊂E′ but one does not have F⊂E′ except if E is dense in F (see, for instance, [10, Chapter 4]).

3.2 Discrete Setting

The objective of this subsection is to adapt the methods of Section 3.1 (in particular Theorem 1) to a discrete setting, in order to prove the convergence of numerical schemes. The set Ω is now a bounded open set of ℝd adapted to a space discretization. The time interval is [0,T], T>0.

Let n∈ℕ; one has a time step kn such that T=kn⁢Nn with some positive integer Nn and one has a space discretization which gives two finite-dimensional spaces Ln and Hn. As in the stationary case, Section 2.2, we consider the case where Ln and Hn are spaces of functions constant on control volumes defined by some meshes (which can be different for Ln and Hn as in interesting case of the MAC-scheme)

Assume that ρn and un are functions constant in time on each interval ](l-1)kn,lkn[, l=1,…,Nn (but we could also assume that these functions are continuous in time and affine on each interval ](l-1)kn,lkn[, this will not change the results given hereafter). For all t∈](l-1)kn,lkn[, l=1,…,Nn, one has

ρn⁢(⋅,t)=ρn(l)∈Ln and un⁢(⋅,t)=un(l)∈Hn.

Since we consider functions which are constant in time on each interval ](l-1)kn,lkn[, we have also to define discrete derivatives, namely for l∈{2,…,Nn},

∂t,n⁡un⁢(⋅,t)=∂t,kn(l)⁡u=1kn⁢(un(l)-un(l-1))for t∈](l-1)kn,lkn[,

∂t,n⁡ρn⁢(⋅,t)=∂t,kn(l)⁡ρn=1kn⁢(ρn(l)-ρn(l-1))for t∈](l-1)kn,lkn[,

and, for l=1,

∂t,nun(⋅,t)=∂t,nρn(⋅,t)=0 for t∈]0,kn[.

We assume that limn→+∞⁡kn=0 and limn→+∞⁡hn=0, where hn is the maximum of the diameter of the control volumes of the meshes defining Ln and Hn. The sequences (ρn)n∈ℕ and (un)n∈ℕ weakly converge to ρ and u in L2(]0,T[,L2(Ω)) (we do not consider in this subsection the more general case of weak convergence in L2(]0,T[,Lq(Ω)) and L2(]0,T[,Lp(Ω)), with 1p+1q=1). We want, as in Section 3.1, a convenient additional condition giving (3.1). We will mimic the method of Section 3.1.

We begin with a discrete version of Lemma 2.

Lemma 3.

Let B be a Banach space and let (Bn)n∈N be a sequence of finite-dimensional subspaces of B. Let ∥⋅∥Xn and ∥⋅∥Yn be two norms on Bn such that, if (∥wn∥Xn)n∈N is bounded, the following hold:

Up to a subsequence, there exists w∈B such that wn→w in B.
If ∥wn-w∥B→0 and ∥wn∥Yn→0, then w=0.

Then, for any ε>0, there exists C¯ε such that, for all n∈N and all w∈Bn,

(3.5)∥w∥B≤ε⁢∥w∥Xn+C¯ε⁢∥w∥Yn.

Proof.

We first remark that Bn is a finite-dimensional subspace of B. Then, for all n and for all ε>0, there exists C¯ε satisfying (3.5) for all u∈Bn. The problem is to find C¯ε independently of n.

We argue by contradiction. Assume that there exists ε>0 such that C¯ε does not exists. Then, for a subsequence of (Bn)n∈ℕ, still denoted (Bn)n∈ℕ, there exists (wn)n∈ℕ such that wn∈Bn and

∥wn∥B>ε⁢∥wn∥Xn+Cn⁢∥wn∥Yn,

with limn→+∞⁡Cn=+∞.

By homogeneity, it is possible to assume that ∥wn∥B=1. Then (∥wn∥Xn)n∈ℕ is bounded and, by (i), up to a subsequence, wn→w in B (so that ∥w∥B=1). But ∥wn∥Yn→0, so that w=0 (by ii), in contradiction with ∥w∥B=1. ∎

We now give a discrete version of Theorem 1, the proof of which uses Lemma 3.

Theorem 2.

Let B be a Banach space, let 1≤q<+∞ and let (Bn)n∈N a family of finite-dimensional subspaces of B. Let ∥⋅∥Xn and ∥⋅∥Yn be two norms on Bn such that, if (∥wn∥Xn)n∈N is bounded, the following hold:

Up to a subsequence, there exists w∈B such that wn→w in B.
If ∥wn-w∥B→0 and ∥wn∥Yn→0, then w=0.

The space Xn is the space Bn with norm ∥⋅∥Xn, the space Yn is Bn with norm ∥⋅∥Yn. Let T>0, kn>0, T=kn⁢Nn, and let (un)n∈N be a sequence such that

for all n, un⁢(⋅,t)=un(l)∈Bn for t∈((l-1)⁢kn,l⁢kn), l=1,…,Nn,
the sequence (un)n∈ℕ is bounded in Lq(]0,T[,Xn), that is to say that there exists C1>0 such that for all n, ∑l=1Nnkn⁢∥un(l)∥Xnq≤C1,
the sequence (∂t,kn⁡un)n∈ℕ is bounded in L1(]0,T[,Yn), that is to say that there exists C2>0 such that for all n, ∑l=2Nnkn⁢∥∂t,kn⁡un(l)∥Yn≤C2.

Then there exists u∈Lq(]0,T[,B) such that, up to a subsequence, un→u in Lq(]0,T[,B).

See, for instance, [10, Theorem 4.51] for a proof of Theorem 2. Similar theorems are also in [3, 12].

Of course, the main example for the present paper is Bn=Ln or Bn=Hn. But it remains to choose B, ∥⋅∥Xn and ∥⋅∥Yn. We present these choices for the discretization of the two examples of Section 3.1, namely the compressible Navier–Stokes equations and the Stefan problem.

3.2.1 Navier–Stokes Equations

We begin with the case of the compressible Navier–Stokes equations as in Section 3.1.1 with a discretization using the MAC scheme, see [13, 3]. In this example, un is one component of the vector-valued function 𝒖n, which is the discrete velocity field, and each component of 𝒖n is a constant function on each control volume of its own mesh and for all time interval ](l-1)kn,lkn[ (it is the so-called staggered discretization). The discrete density ρn is a constant function on the control volumes of another grid, generally called the primal grid, denoted Ln, and for all time interval ](l-1)kn,lkn[.

We denote by H¯n the spatial space for the discrete velocity field. This space is equipped with a norm, denoted ∥⋅∥1,2,n, which mimics the (H01)d-norm. Indeed, this norm contains, for each component of 𝒖n the norm ∥⋅∥1,2,n defined in Section 2.2 in the case of admissible meshes.

The estimates that can be obtained on the approximate solutions mimics the ones of the continuous setting. One obtain an estimate in L∞(]0,T[,Lγ(Ω)) for ρn and an estimate on L2(]0,T[,H¯n) for 𝒖n, where H¯n is equipped with the norm ∥⋅∥1,2,n (the way to obtain this estimate on 𝒖n is roughly explained below). In particular, this gives that the sequence (𝒖n)n∈ℕ is bounded in L2(]0,T[,L2(Ω)d). We also assume that γ≥2 so that the sequence (ρn)n∈ℕ is bounded in L2(]0,T[,L2(Ω)). Then we can assume, up to a subsequence, that ρn→ρ weakly in L2(]0,T[,L2(Ω)) and 𝒖n→𝒖 weakly in L2(]0,T[,L2(Ω)d). We recall that we want to obtain (3.1) or, more generally, to prove the convergence of ρn⁢un to ρ⁢u in a convenient Lebesgue space. In order to obtain (3.1), we will use, as in Section 3.1.1, a compactness result on ρn.

Taking, for instance, an implicit discretization of the mass balance (namely ∂t⁡ρ+div⁡(ρ⁢𝒖)=0), we have with some convenient upwind discretization of div⁡(ρ⁢𝒖), for all l∈{2,…,Nn},

(3.6)∂t,kn(l)⁡ρn+divn⁡(ρn(l)⁢𝒖n(l))=0.

A crucial idea in the discretization of Navier–Stokes Equations with staggered grids is to deduce from (3.6) a discrete mass balance on the mesh, or on the meshes (in the case of the MAC scheme), associated to the velocity field. In the case of the MAC scheme, this discrete mass balance reads for each i=1,…,d and for all l∈{2,…,Nn},

∂t,kn(l)⁡ρn,i+divn,i⁡(ρn(l)⁢𝒖n(l))=0,

where the function ρn,i is a reconstruction of ρn on the mesh associated to the corresponding component of 𝒖𝒏 and divn,i a discretization of div deduced from divn. We refer to [8, Section 3.3] for the first application of this idea, but not with the MAC scheme, and, for instance, [14] for the application of this idea with the MAC scheme. A main interest of this idea is that it gives, together with the discretization of the momentum equation, a kinetic energy balance and therefore an estimate on 𝒖n in L2(]0,T[,H¯n) where H¯n is equipped with the norm ∥⋅∥1,2,n.

In order to apply Theorem 2, one takes Bn=Ln and ∥⋅∥Xn=∥⋅∥L2⁢(Ω) so that the sequence (ρn)n∈ℕ is bounded in L2(]0,T[,Xn). For the choice of ∥⋅∥Yn, we remark that a discrete integration by part leads to, for all l∈{2,…,Nn},

∫Ωv⁢∂t,kn(l)⁡ρn⁢d⁢x=-∫Ωv⁢divn⁡(ρn(l)⁢𝒖n(l))⁢𝑑x=∫Ω(ρ⁢𝒖)n(l)⋅∇n⁡v⁢d⁢x for all ⁢v∈Ln,

where ∇n is a convenient discretization of ∇. Then a natural choice of ∥⋅∥Yn is, for all w∈Ln,

∥w∥Yn=max⁡{∫Ωw⁢φ⁢𝑑x:φ∈Ln,∥∇n⁡φ∥L∞⁢(Ω)d+∥φ∥L∞⁢(Ω)=1}.

With this norm, one has ∥∂t,kn(l)⁡ρn∥Yn≤∥(ρ⁢𝒖)n(l)∥L1⁢(Ω)d. Since (ρn)n is bounded in L2(]0,T[,L2(Ω)) and (𝒖n)n is bounded in L2(]0,T[,L2(Ω)d), one has a bound for (ρ⁢𝒖)n in L1(]0,T[,L1(Ω)d) which gives a bound for ∂t,kn⁡ρn in L1(]0,T[,Yn).

We choose B=(Hs)′ for s such that 0<s<12 (see Section 2.2 for the definition of Hs) and we now prove that hypotheses (i)–(ii) of Theorem 2 are satisfied. We already know that L2⁢(Ω) is compactly embedded in (Hs)′ (see Section 2.2 and Lemma 5 in Section 4). This gives hypothesis (i). In order to prove hypothesis (ii), let wn∈Ln such that (wn)n∈ℕ is bounded in L2⁢(Ω), wn→w in (Hs)′ and ∥wn∥Yn→0. We want to prove that w=0. Let φ∈Cc∞⁢(Ω). We define φn in Ln taking, for instance, the values of φ at the centers of the control volumes defining the space Ln. It is quite easy to prove that the L∞-norm of ∇n⁡φn is bounded by the L∞-norm of ∇⁡φ. Then one has

|∫Ωwnφndx|≤∥wn∥Yn(∥∇nφn∥L∞⁢(Ω)d+∥φn∥L∞⁢(Ω))≤∥wn∥Yn(∥∇φ∥L∞⁢(Ω)d+∥φ∥L∞⁢(Ω)).

This gives

limn→+∞⁡∫Ωwn⁢φn⁢𝑑x=0.

But, since (wn)n∈ℕ is bounded in L2⁢(Ω) and wn→w in (Hs)′, one has also wn→w weakly in L2⁢(Ω) (we use here the uniqueness of the limit in (Hs)′). Then, since φn→φ uniformly in Ω, one has

limn→+∞⁡∫Ωwn⁢φn⁢𝑑x=∫Ωw⁢φ⁢𝑑x.

We then conclude that ∫Ωw⁢φ⁢𝑑x=0 for all φ∈Cc∞⁢(Ω) and therefore that w=0 a.e. on Ω.

We can now apply Theorem 2, it gives ρn→ρ in L2(]0,T[,(Hs)′). Indeed, the previous proof shows that it is also possible to prove that the sequence (∂t,kn⁡ρn)n∈ℕ is bounded in L1(]0,T[,W⋆-1,1(Ω)) and then to apply Theorem 2 with ∥⋅∥Yn=∥⋅∥W⋆-1,1⁢(Ω), where W⋆-1,1⁢(Ω)=W01,∞⁢(Ω)′.

We now conclude. Taking 0<s<12, we recall, see Section 2.2 and Lemma 4, that an estimate on ∥v∥1,2,n gives an estimate on v in Hs. Then the sequence (𝒖𝒏)n∈ℕ is bounded in L2(]0,T[,(Hs)d) and therefore weakly convergent in this space (up to a subsequence). By the uniqueness of the weak limit in L2(]0,T[,L2(Ω)d), its limit is necessarily 𝒖 (and the convergence holds without extracting a subsequence) so that we finally obtain (3.1) for any component un of 𝒖n. Furthermore, it is possible to prove that for all φ∈C∞(ℝd×]0,T[,ℝ) one has

∫0T∫Ωρn⁢𝒖n⋅∇n⁡φ⁢d⁢x⁢d⁢t=〈ρn,𝒖n⋅∇⁡φ〉L2⁢((Hs)′),L2⁢(Hs)+Rn→〈ρ,𝒖⋅∇⁡φ〉L2⁢((Hs)′),L2⁢(Hs)=∫0T∫Ωρ⁢𝒖⋅∇⁡φ⁢d⁢x⁢d⁢t.

This gives, in particular, ∂t⁡ρ+div⁡(ρ⁢𝒖)=0 in the distributional sense.

The previous proof gives also, for all ψ∈C∞(ℝd×]0,T[,ℝd),

∫0Tρn⁢𝒖n⁢ψ⁢𝑑x⁢𝑑t→∫0Tρ⁢𝒖⁢ψ⁢𝑑x⁢𝑑t.

In the case d=3 (the most interesting case), one has an estimate on 𝒖n in L2(]0,T[,L6(Ω)d) (this is due to the discrete Sobolev inequality, see Section 2.1, proven in [5, Lemma 3.5]) and then an estimate on ρn⁢𝒖n in L2(]0,T[,Lr(Ω)d). with r=6⁢γ6+γ. This allows us to conclude that ρn⁢𝒖n→ρ⁢𝒖 weakly in L2(]0,T[,Lr(Ω)). If γ≥3, one has r≥2. Then it is also possible, if γ≥3, to pass to the limit (in the momentum equation) in the term div⁡(ρn⁢𝒖n⊗𝒖n) as we did in Section 3.1.1. We use Theorem 2 with each component of 𝒖n and the corresponding reconstruction of ρn on the associated mesh (and this gives the space Bn). We choose B=(Hs)′ (with 0<s<12), the L2-norm for Xn and, for instance, the W⋆-1,1⁢(Ω)-norm for Yn. This gives the compactness in L2(]0,T[,(Hs)′) for each component of the sequence (ρn⁢𝒖n)n∈ℕ and we conclude using the boundedness in L2(]0,T[,Hs) of each component of the sequence (𝒖n)n∈ℕ.

3.2.2 Stefan Problem

For the case of the Stefan problem, the spatial discretization is the same for ρn and for un (with the notation of the beginning of Section 3.2, one has Ln=Hn). We recall that we are interested by the case where the space Hn is not included in H01⁢(Ω) but Hn is equipped with a norm, depending on n, “close” to the H01-norm and denoted ∥⋅∥1,2,n. We refer, for instance, to [5, 6, 4].

The discretization of the Stefan problem described in Section 1 gives that the couple (ρn,un) satisfy

∂t,kn⁡ρn-Δn⁢un=0,un=φ⁢(ρn).

The discrete operator Δn from Hn to Hn is a convenient discretization of Δ as it is done, for instance, in [5, 6, 4]. We recall that φ is a Lipschitz continuous function from ℝ to ℝ, nondecreasing such that φ′=0 on ]a,b[, for some real numbers a,b with a<b and lim inf|s|→+∞⁡|φ⁢(s)||s|>0.

A natural estimate for this problem gives that the sequence (un)n∈ℕ is bounded in L2(]0,T[,Zn), where Zn is the space Hn with the norm ∥⋅∥1,2,n. Then, using a discrete Poincaré estimate (see [5] for the case of admissible meshes or [6] for a more general case of meshes), one obtains that the sequence (un)n∈ℕ is bounded in L2(]0,T[,L2(Ω)). From this estimate on un, one deduces, with the hypothesis lim inf|s|→+∞⁡|φ⁢(s)||s|>0, that (ρn)n∈ℕ is also bounded in L2(]0,T[,L2(Ω)). A consequence of these estimates is that we can assume, up to a subsequence, that ρn→ρ weakly in L2(]0,T[,L2(Ω)) and un→u weakly in L2(]0,T[,L2(Ω)).

The weak convergence of ρn and un and some consistency property of the discretization of Δ lead to

∂t⁡ρ-Δ⁢u=0.

By using the estimate of un is L2(]0,T[,Zn), it is also classical to prove that u∈L2(]0,T[,H01(Ω)) (see [5] for the case of admissible meshes or [6] for a more general case of meshes).

Our purpose here is to give two ways to prove that u=φ⁢(ρ). As in the continuous case, the first step is to prove (3.1) and then to conclude with the Minty trick (Lemma 1).

For proving (3.1), as in the continuous case (Section 3.1.2), we can use compactness of (un)n∈ℕ or compactness of (ρn)n∈ℕ.

The compactness of (un)n∈ℕ in not given by an application of Theorem 2 because we do not have any estimate on ∂t,kn⁡un. However, it is sometimes possible to adapt the method of Alt–Luckhaus [1] to this discrete setting in order to obtain some estimates on the time-translates of un and then compactness of the sequence (un)n∈ℕ in L2(]0,T[,L2(Ω)). This gives (3.1) and then, using the Minty trick, u=φ⁢(ρ). This way is, for instance, used in [5].

The second way is to prove some compactness on the sequence (ρn)n∈ℕ. We recall that the norm ∥⋅∥1,2,n control the Hs-norm for some convenient s>0 (see Section 2.2 and Lemma 4 in Section 4). In the case of admissible meshes as in [5], we can take any s<12. in the case of more general meshes as in [6], we can take, for d=3, any s<25. Then one has un→u weakly in L2(]0,T[,Hs) (since L2(]0,T[,Hs) is a Hilbert space). In order to prove (3.1), it suffices to prove that (ρn)n∈ℕ converges in L2(]0,T[,(Hs)′) (as usual, L2⁢(Ω) is identified with its dual space). We will prove this compactness on (ρn)n∈ℕ with Theorem 2 applied with

B=(Hs)′,Bn=Hn,∥⋅∥Xn=∥⋅∥L2⁢(Ω),∥⋅∥Yn=∥⋅∥-1,2,n,

where ∥⋅∥-1,2,n is the dual norm of the norm ∥⋅∥1,2,n, that is, for v∈Hn,

∥v∥-1,2,n=max⁡{∫Ωv⁢w⁢𝑑x:w∈Hn,∥w∥1,2,n=1}.

By multiplying, for w∈Hn, the equation ∂t,kn⁡ρn-Δn⁢un=0 by w and using a discrete integration by part, the estimate on un in L2(]0,T[,Zn) gives an estimate on ∂t,kn⁡ρn in L2(]0,T[,Yn). In order to apply Theorem 2, it remains to verify hypotheses (i)–(ii) of Theorem 2. Hypothesis (i) is due to compactness of L2⁢(Ω) in (Hs)′ (which is a consequence, by duality, of the compact embedding of Hs in L2⁢(Ω), Lemma 5 in Section 4). For proving hypothesis (ii), let (wn)n∈ℕ be a bounded sequence of L2⁢(Ω) converging to w in (Hs)′ and to 0 for the Yn-norm. We first remark that wn→w weakly in L2⁢(Ω). Let ψ∈Cc∞⁢(Ω). We can define ψn in Ln taking, for instance, the mean values of ψ on the control volumes defining the space Ln. It is possible to prove that ∥ψn∥1,2,n≤C⁢∥ψ∥H1⁢(Ω), where C depends only the regularity of the mesh (see, for instance, [5, Lemma 9.4] for the case of admissible meshes). Then, under a regularity hypothesis on the sequence of meshes, one has

|∫Ωwnψndx|≤∥wn∥Yn∥ψn∥1,2,n≤C∥wn∥Yn∥ψ∥H1⁢(Ω).

This gives

limn→+∞⁡∫Ωwn⁢ψn⁢𝑑x=0.

But, since wn→w weakly in L2⁢(Ω) and ψn→ψ uniformly in Ω, one has

limn→+∞⁡∫Ωwn⁢ψn⁢𝑑x=∫Ωw⁢ψ⁢𝑑x.

We then conclude that ∫Ωw⁢ψ⁢𝑑x=0 for all ψ∈Cc∞⁢(Ω) and therefore that w=0 a.e. on Ω.

All the hypotheses of Theorem 2 are satisfied and we obtain the convergence of the sequence (ρn)n∈ℕ in L2(]0,T[,(Hs)′). It gives (3.1) and we conclude with the Minty trick (Lemma 1) that u=φ⁢(ρ).

Remark 5.

Here also, as in Section 2.2, it is possible to use the compactness of (ρn)n∈ℕ under the form ∇n⁡wn→∇⁡w in L2(]0,T[,L2(Ω)d) (and ∇n⁡un→∇⁡u weakly in L2(]0,T[,L2(Ω)d)), where ∇n is a discretization of ∇, wn is the solution of a discrete equivalent of (2.4) and w is the solution of (2.5). This method is used, for instance, in [4].

4 Appendix

We prove here that the space Nα,2⁢(ℝd) (which is a Nikolsky-space) is continuously embedded in the space Hs⁢(ℝd) for 0≤s<α≤1 (Lemma 4) and that the space Hs (with s>0 and Ω bounded) is compactly embedded in L2⁢(ℝd) (Lemma 5).

Definition 2.

Let 0<α≤1 and d≥1. The space Nα,2⁢(ℝd) is the set of elements u in L2⁢(Ω) such that there exists C satisfying

∥u(⋅+η)-u∥L2⁢(ℝd)≤C|η|α for all η∈ℝd.

The norm in the space Nα,2⁢(ℝd) is defined by

∥u∥Nα,2=∥u∥L2⁢(ℝd)+maxη∈ℝd,η≠0⁡∥u(⋅+η)-u∥L2⁢(ℝd)|η|α.

With this norm, Nα,2 is a Banach space.

Lemma 4.

Let d≥1, 0≤s<α≤1. Then the space Nα,2⁢(Rd) is continuously embedded in the space Hs⁢(Rd) (see Definition 2.8).

Proof.

Let u∈Nα,2⁢(ℝd) and C=∥u∥Nα,2. For all η∈ℝd, one has

∥u(⋅+η)^-u^∥L2⁢(ℝd)=∥u(⋅+η)-u∥L2⁢(ℝd)≤C|η|α.

We recall that for all η∈ℝd, u(⋅+η)^⁢(ξ)=ei⁢η⋅ξ⁢u^⁢(ξ) a.e. Then

(4.1)∫ℝd|ei⁢η⋅ξ-1|2⁢|u^⁢(ξ)|2⁢𝑑ξ≤C2⁢|η|2⁢α.

Let e1,…,ed be the canonical basis of ℝd. Let j∈{1,…,d} and t>0. With η=t⁢ej in (4.1) one obtains

∫ℝd|ei⁢t⁢ξj-1|2t2⁢α⁢|u^⁢(ξ)|2⁢𝑑ξ≤C2,

and then, for ε>0 and for all t>0,

∫ℝd|ei⁢t⁢ξj-1|2t2⁢α⁢|u^⁢(ξ)|2⁢𝑑ξ⁢1t1-ε≤C2⁢1t1-ε.

Integrating this inequality between 0 and 1 and using the Fubini–Tonelli Theorem lead to

∫ℝd|u^⁢(ξ)|2⁢(∫01|ei⁢t⁢ξj-1|2t2⁢α+1-ε⁢𝑑t)⁢𝑑ξ≤C2ε.

We use now the change of variable t⁢|ξj|=τ so that (since |ei⁢τ-1|=|e-i⁢τ-1|)

(4.2)∫ℝd|u^⁢(ξ)|2⁢|ξj|2⁢α-ε⁢(∫0|ξj||ei⁢τ-1|2τ2⁢α+1-ε⁢𝑑τ)⁢𝑑ξ≤C2ε.

We set

aε=∫01|ei⁢τ-1|2τ2⁢α+1-ε⁢𝑑τ.

Note that 0<aε<+∞ since |ei⁢τ-1|2=(cos⁡(τ)-1)2+(sin⁡(τ))2≤2⁢τ2 and α≤1. With this definition of aε, one has

∫ℝd|u^⁢(ξ)|2⁢|ξj|2⁢α-ε⁢𝑑ξ≤∫|ξj|<1|u^⁢(ξ)|2⁢𝑑ξ+1aε⁢∫|ξj|≥1|u^⁢(ξ)|2⁢|ξj|2⁢α-ε⁢(∫0|ξj||ei⁢τ-1|2τ2⁢α+1-ε⁢𝑑τ)⁢𝑑ξ,

and then we deduce from (4.2) that

∫ℝd|u^⁢(ξ)|2⁢|ξj|2⁢α-ε⁢𝑑ξ≤∥u∥L2⁢(ℝd)2+C2ε⁢aε≤bε⁢∥u∥Nα,22,

with bε=1+1ε⁢aε. This gives

∥u∥α-ε22=∫ℝd(1+|ξ|2)α-ε2⁢|u^⁢(ξ)|2⁢𝑑ξ≤d⁢bε⁢∥u∥Nα,22

and concludes the proof. ∎

Lemma 5.

Let Ω be a bounded open set of Rd, d≥1, let s>0 and let

Hs={u∈Hs⁢(ℝd):u=0⁢ a.e. on ⁢ℝd∖Ω}

be equipped with the norm ∥⋅∥s given in (2.8). Then Hs is compactly embedded in L2⁢(Rd).

Proof.

Since Hs¯ is continuously embedded in Hs for 0≤s<s¯, we have only to prove the lemma for 0<s≤1.

Let 0<s≤1, we are going to prove that for all u∈Hs and η∈ℝd,

(4.3)∥u(⋅+η)-u∥L2⁢(ℝd)≤2∥u∥s|η|s.

Inequality (4.3) gives that Hs is compactly embedded in L2⁢(ℝd). It is a consequence of the Kolmogorov Theorem in Lp-Spaces (see, for instance, [9, Theorem 8.16]). To this end, let η∈ℝd, η≠0. One has, as in Lemma 4,

∥u(⋅+η)-u∥L2⁢(ℝd)2=∥u(⋅+η)^-u^∥L2⁢(ℝd)2=∫ℝd|ei⁢η⋅ξ-1|2|u^(ξ)|2dξ.

We now remark that for all η, ξ∈ℝd, one has

(4.4)|ei⁢η⋅ξ-1|2≤4⁢(1+|ξ|2)s⁢|η|2⁢s.

Indeed, since (for all x∈ℝ) |ei⁢x-1|≤2, estimate (4.4) is true if |η|≥1 and if |η|<1 but |ξ|⁢|η|≥1. If |η|<1 and |ξ|⁢|η|<1, estimate (4.4) is also true since |ei⁢x-1|2≤2⁢x2 and then |ei⁢η⋅ξ-1|2≤2⁢|ξ|2⁢|η|2≤2⁢|ξ|2⁢s⁢|η|2⁢s (since s≤1), which gives (4.4). Using (4.4), we then obtain that

∥u(⋅+η)-u∥L2⁢(ℝd)2≤∫ℝd4(1+|ξ|2)s|η|2⁢s|u^(ξ)|2dξ

and, finally, ∥u(⋅+η)-u∥L2⁢(ℝd)≤2∥u∥s|η|s. This concludes the proof. ∎

5 Conclusion

This paper presents some tools useful for proving the convergence of numerical schemes. The main result is Theorem 2. Two applications are presented. The first one appears for the discretization of the Navier–Stokes equations and the second one for the Stefan problem. However, one may find other problems where the tools discussed here can be applied for proving the convergence of the numerical schemes. The ideas that are developed in this paper can probably also been extended to the analysis of high-order nonconforming methods (e.g., for the convergence analysis of discretization schemes for the Navier–Stokes equations, Leray–Lions problems, nonlinear elasticity models, etc.).

In the case of the discrete setting of the compressible Navier–Stokes equations when the equation of state does not allow an L2(]0,T[,L2(Ω))-estimate on ρ, it will be interesting to work with a sequence (ρn)n∈ℕ weakly convergent in L2(]0,T[,Lq(Ω)) with some 1<q<+∞ and a sequence (un)n∈ℕ weakly convergent in L2(]0,T[,Lp(Ω)) with 1p+1q=1 (we only consider in this paper the case p=q=2).

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Received: 2017-05-16

Revised: 2017-11-10

Accepted: 2017-11-21

Published Online: 2018-02-07

Published in Print: 2018-07-01

Discrete Functional Analysis Tools for Some Evolution Equations

Abstract

1 Introduction

2 Stationary Case

2.1 Continuous Setting

Remark 1.

2.2 Discrete Setting

2.2.1 Compactness of the Sequence (un)n∈ℕ

2.2.2 Compactness of the Sequence (ρn)n∈ℕ

Definition 1.

Remark 2.

3 Evolution Case

3.1 Continuous Setting

3.1.1 Compressible Navier–Stokes Equations

3.1.2 Stefan Problem

Remark 3.

Lemma 1 (Minty Trick).

Proof.

Theorem 1.

Lemma 2.

Remark 4.

3.2 Discrete Setting

Lemma 3.

Proof.

Theorem 2.

3.2.1 Navier–Stokes Equations

3.2.2 Stefan Problem

Remark 5.

4 Appendix

Definition 2.

Lemma 4.

Proof.

Lemma 5.

Proof.

5 Conclusion

References

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