A uniform L ∞ -estimate on s M,∆t

pⁿ⁺¹_K

K,n

m

to the scheme (2.21) thanks to formulas (2.27)–(2.28). Then

sMm,∆tm −→

m→+∞s(p) a.e inQ_tf, ξT_m,∆tm −→

m→+∞ξ(p) weakly in L²((0, t_f);H¹(Ω))and strongly inL²(Q_tf), where p is the unique solution to the continuous problem (2.1).

The proof of Theorem 2.2.4 is addressed in§2.3. The convergence of the scheme towards a weak solution is the purpose of §3.4, while the uniqueness of the weak solution is proved in appendix, cf. Proposition 2.7.4. Numerical illustrations are provided in§3.5.

2.3 Discrete properties, a priori estimates and exis-tence

In this section, we establish a priori estimates, among which the positivity of the saturation and the stability of the capillary energy. These estimates allow to prove the existence of a solution to the nonlinear system (2.21). They are also keystones in order to perform the convergence analysis later on.

2.3.1 A uniform L

-estimate on s

In what follows, (pⁿ⁺¹_K )K∈V,n≥0 denotes a solution to the scheme (2.21) (whose exis-tence will be established later). This allows to define the quantities sⁿ⁺¹_K =s(pⁿ⁺¹_K ) and ξ_Kⁿ⁺¹ =ξ(pⁿ⁺¹_K ) for all K ∈ V and alln∈ {0, . . . , N}.

2.3 Discrete properties,a priori estimates and existence

Proposition 2.3.1. For all K ∈ V, and all n ∈ {0, ..., N}, one has

0≤sⁿ_K ≤1. (2.29)

Equivalently, one has

p_? ≤pⁿ⁺¹_K , ∀K ∈ V, ∀n ∈ {0, . . . , N}. (2.30) Proof. First of all, note that there is nothing to prove if p_? = −∞. Therefore, we restrict our attention to the case of a finite p_?. The property (2.29) holds for n = 0 thanks to the discretization (2.20) of the initial data. Assume now (2.29) holds at time step n. It is equivalent to prove pⁿ⁺¹_K ≥p_? . Assume that

pⁿ⁺¹_Km = min

L∈V pⁿ⁺¹_L < p? ⇔ sⁿ⁺¹_Km <0. (2.31) In view of the definition (2.22) of ηⁿ⁺¹_K

mL, and of the fact that η(s) = 0 if s < 0, it follows from (2.21d) that

η_Kⁿ⁺¹

mL= 0 if a_KmL(uⁿ⁺¹_K

m −uⁿ⁺¹_L )≥0.

Therefore, the scheme (2.21) at vertexK_m rewrites sⁿ⁺¹_Km =sⁿ_Km− ∆t

m_Km

X

σ_KmL∈E

η_Kⁿ⁺¹_mLaKmL(uⁿ⁺¹_Km −uⁿ⁺¹_L )≥0.

This yields a contradiction with (2.31). Hence, theL^∞ estimate (2.29) holds at the time step n+ 1, thus for alln.

2.3.2 Capillary energy estimate and the control of the dissipation

The goal of this section is to get an a priori control for the capillary energy of the discrete solution and to derive some estimates coming from the dissipation of the energy. We were not able to derive the discrete counterpart of the energy/dissipation estimate (2.11). However, we can prove a discrete counterpart of (2.12) (cf. Propo-sition 2.3.2) that appears to be sufficient to establish Theorems 2.2.4 and 2.2.5. In what follows, we assume that(sⁿ_K)_K∈V is known andpⁿ⁺¹_K

K∈V denotes an arbitrary solution to the scheme (2.21).

Proposition 2.3.2. There exists C₄ depending only on θT, Λ, Ω, t_f, ksk_L¹_(p?,0), and kηk∞ such that

Chapitre 2. Nonlinear CVFE for Richards equation The proof of Proposition 2.3.2 is based on several Lemmas stated below. This section also contains technical lemmas that will be useful in the convergence proof of §3.4.

By convexity ofΓ◦s⁻¹ one deduces this estimation A ≥ ^X thanks to Jensen’s inequality and to (2.5).

2.3 Discrete properties,a priori estimates and existence

From the previous lemma, we can get an estimate on the spatial variations of the functionξ_T,∆t. In order to ease the reading, we use the shortened notation

ξ_Kⁿ⁺¹ =ξ(pⁿ⁺¹_K ), ∀K ∈ V, ∀n ∈ {0, . . . , N}, Proof. The definition (2.22) of the mobilitiesη_KLⁿ⁺¹ has been chosen so that

C₅ ≥ of uⁿ⁺¹_K and Young’s inequality leads to

C5 ≥ Using Lemma 2.2.1, we get that

Chapitre 2. Nonlinear CVFE for Richards equation The functionΓ takes non-negative values, hence so does the first term in (2.32).

But sincea_KL may become negative, we are not able to claim that the second term is non-negative (this would end the proof of Proposition 2.3.2). Nevertheless, we can prove that this term is uniformly bounded. This information, combined with Lemma 2.3.4, is sufficient to conclude the proof of Proposition 2.3.2.

Lemma 2.3.5. There exists C₇ depending only on Ω, s, t_f,Λ, θT, and η such that It follows from the definition (2.22) ofη_KLⁿ⁺¹ that

η_KLⁿ⁺¹[a_KL(uⁿ⁺¹_K −uⁿ⁺¹_L )(pⁿ⁺¹_K −pⁿ⁺¹_L )]⁻ ≤η˜_KLⁿ⁺¹[a_KL(uⁿ⁺¹_K −uⁿ⁺¹_L )(pⁿ⁺¹_K −pⁿ⁺¹_L )]⁻, withη˜_KLⁿ⁺¹ defined by (2.34). Moreover, using the definition (2.21b) of uⁿ⁺¹_K together with Young’s inequality, we obtain that

˜ We deduce from Lemma 2.2.1 that

2 Then we combine (2.36), (2.37), Lemma 2.3.4, and Lemma 2.3.3 to conclude.

2.3 Discrete properties,a priori estimates and existence

Thea prioriestimate of Proposition 2.3.2 follows easily from Lemmas 2.3.3, 2.3.4, and 2.3.5. It is sufficient to prove the existence of a solution to the scheme (2.21) (see§2.3.3). Nevertheless, before going to this existence proof, we still provide some additionala priori estimates to be used later on in §3.4.

Lemma 2.3.6. There exists C₈ depending only on Ω, s, t_f,Λ, θT, and η such that Proof. The definition (2.21b) of uⁿ⁺¹_K yields

N

Thanks to Lemma 2.3.5, one has A ≤ C7. Moreover, combining once again Young inequality with Lemma 2.2.1, we get that

B ≤ 1

directly. The control onξT,∆t is provided by an argument `a laPoincar´e, cf. Appen-dix 2.7.1.

Lemma 2.3.7. There exists C₉ depending only on Ω, t_f, s,Λ, θ_T, η, s₀, and ξ_? such that

kξT,∆tk_L²_(Qtf)dxdt≤C₉, (2.40)

kξ k dxdt≤C . (2.41)

Chapitre 2. Nonlinear CVFE for Richards equation Proof. Let us first establish (2.41). Thanks to Assumption (2.4), we know that

R

Ωs₀dx<meas(Ω). The global conservativity property (2.24) allows to claim that

X XM,∆t and VT,∆t respectively such that

ξ_M,∆t⁺ (x_K, t_n+1) = ξ⁺_T,∆t(x_K, t_n+1) =ξ_K^+,n+1, ∀K ∈ V, ∀n ∈ {0, . . . , N}.

Note that ξ_T⁺_,∆t 6= (ξT,∆t)⁺ = max (0, ξT,∆t) in general, but that ξ_M,∆t⁺ = (ξM,∆t)⁺ and that ξ_M,∆t⁻ = (ξM,∆t)⁻ = max (0,−ξM,∆t). Using Assumption (A3), the 1-Lipschitz continuity ofx7→x⁺, and Lemmas 2.2.1 and 2.3.4, we obtain

ZZ Therefore, we can apply Lemma 2.7.3 stated in appendix. This provides

ZZ Combining (2.43) with (2.44) provides (2.41). In order to recover (2.40), in only remains to use (2.14) and (2.41).

2.3.3 Existence of a discrete solution

In order to prove the existence of a solution(pⁿ⁺¹_K )_K to the scheme (2.21), we need an additional mesh-depending estimate on the solution. Following [39], we introduce now the notion of transmissive path.

2.3 Discrete properties,a priori estimates and existence

Definition 2.3.8. A transmissive path w joining Ki ∈ V to Kf ∈ V consists in a list of vertices (K_q)_0≤q≤M such that K_i = K₀, K_f = K_M, with K_q 6= K<sub>`</sub> if q 6= `, and such that σ_KqKq+1 ∈ E with a_KqKq+1 > 0 for all q ∈ {0, ..., M −1}. We denote by W(K_i, K_f) the set of the transmissive path joining K_i ∈ V to K_f ∈ V.

We now state a result which is proved in [39, Lemma 3.5].

Lemma 2.3.9. For all(K_i, Kf)∈ V²there exists a transmissive pathw∈ W(Ki, Kf).

Lemma 2.3.10. There exists C? >−∞ depending only on T,∆t,Ω, s, s₀, tf,Λ, θ_T, η and z such that

pⁿ⁺¹_K ≥C?, ∀K ∈ V, ∀n∈ {0, ..., N}.

Proof. Let us prove thatpⁿ⁺¹_K ≥C_?. Assume first thatp_? >−∞, then we can choose C_? =p_? thanks to (2.30), so that we can focus on the case p_? =−∞.

In view of the global conservation property (2.24), one has that

X

K∈V

(sⁿ⁺¹_K −s₀)m_K = 0.

This ensures the existence of at least one vertex K_i such that sⁿ⁺¹_Ki ≥ s₀ > 0. In particular,

− ∞< s⁻¹(s₀)≤pⁿ⁺¹_Ki . (2.45) LetK_f ∈ V \ {K_i}, then thanks to Lemma 2.3.9, there exists a transmissive path w∈ W(K_i, K_f) = (K_q)0≤q≤M of finite length in the sense of Definition 2.3.8. Let us show that for allpⁿ⁺¹_Kq >−∞ for all q ∈ {0, ..., M}.

First, we have checked in (2.45) that pⁿ⁺¹_K0 >−∞. Assume now thatpⁿ⁺¹_Kq >−∞

for someq ∈ {0, . . . , M −1}, then it follows from Lemma 2.3.5 that

N

X

n=0

∆t ^X

σKL∈E

|a_KL|ηⁿ⁺¹_KL|uⁿ⁺¹_K −uⁿ⁺¹_L ||pⁿ⁺¹_K −pⁿ⁺¹_L | ≤C₆. This ensures in particular that

∆taKqKq+1η_Kⁿ⁺¹_qKq+1(uⁿ⁺¹_Kq −uⁿ⁺¹_Kq+1)(pⁿ⁺¹_Kq −pⁿ⁺¹_Kq+1)≤C₆. Thanks to the definition (2.22) of η_Kⁿ⁺¹_qKq+1, one has

a ηⁿ⁺¹ (uⁿ⁺¹−uⁿ⁺¹ )(pⁿ⁺¹−pⁿ⁺¹ )

Chapitre 2. Nonlinear CVFE for Richards equation SinceaKqKq+1 >0, we obtain that

(uⁿ⁺¹_Kq −uⁿ⁺¹_K

q+1)(pⁿ⁺¹_Kq −pⁿ⁺¹_K

q+1)

= (pⁿ⁺¹_K

q −pⁿ⁺¹_Kq+1)²+ (pⁿ⁺¹_K

q −pⁿ⁺¹_Kq+1)(z_Kq −zKq+1) ≤ C₆

∆ta_KqKq+1η(s(pⁿ⁺¹_Kq )). Using Young inequality one has

(uⁿ⁺¹_Kq −uⁿ⁺¹_Kq+1)(pⁿ⁺¹_Kq −pⁿ⁺¹_Kq+1)≥ 1

2(pⁿ⁺¹_Kq −pⁿ⁺¹_Kq+1)²−1

2(z_Kq −z_Kq+1)², thus

pⁿ⁺¹_K

q+1 ≥pⁿ⁺¹_Kq −

v u u

t(z_Kq −z_Kq+1)²+ 2C₆

∆ta_KqKq+1η(s(pⁿ⁺¹_Kq )). This ensures thatpⁿ⁺¹_Kq+1 >−∞.

We have proved the existence of a finite quantity (C_Ki,Kf,w)_Kf∈V (depending on the data of the continuous problemΩ, s, s₀, t_f,Λ, θT, η but also on the discretization T and on∆t) such that

s(pⁿ⁺¹_Ki )≥s₀ =⇒ pⁿ⁺¹_K

f ≥ −C_Ki,Kf,w.

As a consequence, since there exists a finite number of transmissive paths between two vertices, we get the estimate

pⁿ⁺¹_K ≥ −max

Ki∈Vmax

Kf∈V min

w∈W(K_i,Kf)C_Ki,Kf,w >−∞, ∀K ∈ V, ∀n∈ {0, ..., N}.

In the previous lemma, we managed to bound the {pⁿ⁺¹_K } from below. The next lemma provides a bound from above.

Lemma 2.3.11. There exists p^? < ∞ depending only on T,∆t,Ω, t_f, s,Λ, η, s₀ and ξ_? such that

pⁿ⁺¹_K ≤p^? ∀K ∈ V, ∀n ∈ {0, ..., N}.

Proof. Sinces(p) = 1 if p≥0, one has ξ(p) = p^qη(1) if p≥0.By (2.41), one has

∆tm_Kξ(pⁿ⁺¹_K )² ≤ kξM,∆tk²_L2(Q_tf)≤(C₉)². Therefore, we getpⁿ⁺¹_K ≤ C₉

√∆tm_K 1 η(1).

Now, one can apply the same strategy as in [39, Lemma 3.11] for proving the exis-tence of a solution to the scheme (2.21).

2.4 Convergence towards a weak solution

Proposition 2.3.12. Let (sⁿ_K)_K∈V ∈[0,1]^#V be such that ^P_K∈VmKsⁿ_K = meas(Ω)s₀, there exists (at least) one solution (pⁿ⁺¹_K )_K∈V ∈ [p_?, p^?]^#V of the scheme (2.21).

Moreover, it satisfies^P_K∈Vm_Ksⁿ⁺¹_K = meas(Ω)s₀.

The proof of Proposition 2.3.12 is not detailed here since it mimics the one of [39, Lemma 3.11]. Let us just mention that it is based on a topological degree argument [61, 122].

2.4 Convergence towards a weak solution

The proof of the convergence properties stated in Theorem 2.2.5 is based on com-pactness arguments. As a first step, we show in §2.4.1 the appropriate compactness properties on the reconstructed discrete solutions. Then we identify in §3.4.2 the limit value (whose existence is ensured thanks to the compactness properties) as the unique weak solution to the problem (2.1).