Long-time behaviour of hybrid finite volume schemes for advection–diffusion equations: linear and nonlinear approaches

Abstract

We are interested in the long-time behaviour of approximate solutions to anisotropic and heterogeneous linear advection–diffusion equations in the framework of hybrid finite volume methods on general polygonal/polyhedral meshes. We consider two linear methods, as well as a new, nonlinear scheme, for which we prove the existence and the positivity of discrete solutions. We show that the discrete solutions to the three schemes converge exponentially fast in time towards the associated discrete steady-states. To illustrate our theoretical findings, we present some numerical simulations assessing long-time behaviour and positivity. We also compare the accuracy of the schemes on some numerical tests in the stationary case.

Appendices

Functional inequalities

1.1 Discrete Poincaré inequalities

We recall the following hybrid discrete Poincaré inequalities (cf. [29,  Lemmas B.25 and B.32, $p=2$]).

Proposition 7

(Discrete Poincaré inequalities) Let ${\mathcal {D}}$ be a given discretisation of $\varOmega$, with regularity parameter $\theta _{\mathcal {D}}$. There exists $C_{PW}>0$, only depending on $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \forall {\underline{v}}_{{\mathcal {D}}} \in {\underline{V}}_{{\mathcal {D}},0}^N , \qquad \Vert v_{\mathcal {M}}\Vert _{L^2(\varOmega )} \le {C_{PW}} | {\underline{v}}_{{\mathcal {D}}} |_{1,{\mathcal {D}}}. \end{aligned}$$

(A.1)

Assume that $|\varGamma ^D| > 0$. Then, there exists $C_{P,\varGamma ^D}>0$, only depending on $\varOmega$, d, $\varGamma ^D$, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \forall {\underline{v}}_{{\mathcal {D}}} \in {\underline{V}}_{{\mathcal {D}},0}^D, \qquad \Vert v_{\mathcal {M}}\Vert _{L^2(\varOmega )} \le {C_{P,\varGamma ^D}} | {\underline{v}}_{{\mathcal {D}}} |_{1,{\mathcal {D}}}. \end{aligned}$$

(A.2)

1.2 Logarithmic Sobolev inequalities

In this section, we derive logarithmic Sobolev inequalities on a bounded domain, in the continuous setting. The intermediate results of Proposition 8 below will be useful in the discrete setting. In the following, $\mu$ is a probability measure on the bounded domain $\varOmega$, and the space $L^q_\mu (\varOmega )$ denotes the space endowed with the norm $\Vert f\Vert _{L^q_\mu (\varOmega )}^q =\int _\varOmega |f|^q\mathrm {d}\mu$. We start with a preliminary lemma, which is an adaptation of part of the proof of [22,  Theorem 6.1.22] (see also [36]). We recall that $\varPhi _1(s)=s\log (s)-s+1$.

Lemma 3

For all $t\in {\mathbb {R}}$ and $\psi \in L^2_\mu (\varOmega )$ such that $\Vert \psi \Vert _{L^2_\mu (\varOmega )} = 1$, one has

$$\begin{aligned} \int _\varOmega \varPhi _1\left( (1+t\psi )^2\right) \mathrm {d}\mu\le & {} t^2\int _\varOmega \psi ^2\log (\psi ^2)\mathrm {d}\mu + (1+t^2)\log (1+t^2) \\&+(1+|\langle \psi \rangle _\mu |)t^2, \end{aligned}$$

where $\langle \psi \rangle _\mu :=\int _\varOmega \psi \,\mathrm {d}\mu$.

Proof

Let us define, for $\delta >0$,

$$\begin{aligned}f_\delta (t) = \int _\varOmega \varPhi _1\left( (1+t\psi )^2+\delta \right) \mathrm {d}\mu - t^2\int _\varOmega \psi ^2\log (\psi ^2)\mathrm {d}\mu - (1+t^2)\log (1+t^2) \,.\end{aligned}$$

Differentiating $f_\delta$ yields

$$\begin{aligned} f_\delta '(t)= & {} 2\int _\varOmega (1+t\psi )\psi \log \left( (1+t\psi )^2+\delta \right) \mathrm {d}\mu \\&-2t\int _\varOmega \psi ^2\log (\psi ^2)\mathrm {d}\mu - 2t\log (1+t^2) - 2t\,. \end{aligned}$$

In particular, $f_\delta '(0) = 2\log (1+\delta )\langle \psi \rangle _\mu$. Differentiating once more, and using that $\Vert \psi \Vert _{L^2_\mu (\varOmega )}^2 = 1$, we obtain

$$\begin{aligned} f_\delta ''(t)= & {} 2\int _\varOmega \psi ^2\log \left( \frac{(1+t\psi )^2+\delta }{(1+t^2)\psi ^2}\right) \mathrm {d}\mu \\&+ 4\int _\varOmega \psi ^2\frac{(1+t\psi )^2}{\delta +(1+t\psi )^2}\mathrm {d}\mu -\frac{4t^2}{1+t^2}-2\,. \end{aligned}$$

Therefore, using that $\log (x)\le x-1$ in the first term, that $\frac{x}{\delta +x}\le 1$ in the second, together with the fact that $\mu$ is a probability measure and that $\Vert \psi \Vert _{L^2_\mu (\varOmega )}^2 = 1$, one gets

$$\begin{aligned} f_\delta ''(t)\le \frac{2\delta }{1+t^2} + \frac{4t}{1+t^2}\langle \psi \rangle _\mu + 4-\frac{4t^2}{1+t^2}-2\le 2\delta + 2|\langle \psi \rangle _\mu |+2\,. \end{aligned}$$

One concludes by integrating this inequality twice between 0 and t, using that $f_{\delta }(0)=\varPhi _1(1+\delta )$, and letting $\delta \rightarrow 0$. $\square$

Proposition 8

For any $\xi \in L^q_\mu (\varOmega )$ with $q>2$, one has

$$\begin{aligned} \int _\varOmega \xi ^2\log \left( \frac{\xi ^2}{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu \le \frac{q}{q-2}\Vert \xi -\langle \xi \rangle _\mu \Vert _{L^q_\mu (\varOmega )}^2 + \frac{q-4}{q-2}\Vert \xi -\langle \xi \rangle _\mu \Vert _{L^2_\mu (\varOmega )}^2, \end{aligned}$$

(A.3)

where $\langle \xi \rangle _\mu :=\int _\varOmega \xi \,\mathrm {d}\mu$. Besides, one also has

$$\begin{aligned} \int _\varOmega \varPhi _1(\xi ^2)\,\mathrm {d}\mu\le & {} \frac{q}{q-2}\Vert \xi -1\Vert _{L^q_\mu (\varOmega )}^2 + \frac{2q-6}{q-2}\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2 \nonumber \\&+ \varPhi _1\left( 1+\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2\right) \,. \end{aligned}$$

(A.4)

Proof

 

• (i) Assume that $\langle \xi \rangle _\mu \ne 0$, and take t and $\psi$ such that $\xi = \langle \xi \rangle _\mu (1+t\psi )$ and $\Vert \psi \Vert _{L^2_\mu (\varOmega )} = 1$. In particular, $\langle \psi \rangle _\mu = 0$, and $1+t^2=\frac{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}{\langle \xi \rangle _\mu ^2}$. Using Lemma 3, a somewhat tedious but straightforward computation yields

  $$\begin{aligned}&\int _\varOmega \xi ^2\log \left( \frac{\xi ^2}{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu \\&\quad \le \int _{\varOmega }(\xi -\langle \xi \rangle _\mu )^2\log \left( \frac{(\xi -\langle \xi \rangle _\mu )^2}{\Vert \xi -\langle \xi \rangle _\mu \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu +2\Vert \xi -\langle \xi \rangle _\mu \Vert _{L^2_\mu (\varOmega )}^2\,. \end{aligned}$$

  Observe that the last inequality also holds if $\langle \xi \rangle _\mu = 0$. Let $\phi = \xi -\langle \xi \rangle _\mu$. Then,

  $$\begin{aligned}&\int _\varOmega \xi ^2\log \left( \frac{\xi ^2}{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu \\&\quad \le \frac{2}{q-2}\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2\int _{\varOmega }\frac{\phi ^2}{\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2}\log \left( \frac{\phi ^{q-2}}{\Vert \phi \Vert _{L^2_\mu (\varOmega )}^{q-2}}\right) \mathrm {d}\mu +2\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2\,. \end{aligned}$$

  Therefore, by Jensen’s inequality for the probability measure $\frac{\phi ^2}{\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2}\mathrm {d}\mu$ applied to the concave function $\log$, one obtains

  $$\begin{aligned} \int _\varOmega \xi ^2\log \left( \frac{\xi ^2}{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu\le & {} \frac{2}{q-2}\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2\log \left( \frac{\Vert \phi \Vert _{L^q_\mu (\varOmega )}^q}{\Vert \phi \Vert _{L^2_\mu (\varOmega )}^q}\right) +2\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2 \\= & {} \frac{q}{q-2}\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2\log \left( \frac{\Vert \phi \Vert _{L^q_\mu (\varOmega )}^2}{\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2}\right) +2\Vert \phi \Vert _{L^2_\mu (\varOmega )}^2\,, \end{aligned}$$

  and one concludes using that $\log (x)\le x-1$.

• (ii) Take t and $\psi$ such that $\xi = 1+t\psi$ and $\Vert \psi \Vert _{L^2_\mu (\varOmega )} = 1$. Remark that $t=\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}$. Using that $|\langle \psi \rangle _\mu |\le \Vert \psi \Vert _{L^2_\mu (\varOmega )}=1$, Lemma 3 yields

  $$\begin{aligned}&\int _\varOmega \varPhi _1(\xi ^2)\,\mathrm {d}\mu - \varPhi _1\left( 1+\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2\right) \\&\quad \le \int _{\varOmega }(\xi -1)^2\log \left( \frac{(\xi -1)^2}{\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu +3\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2\,. \end{aligned}$$

  Letting $\phi = \xi -1$, the proof goes on as for (i).

$\square$

From there, logarithmic Sobolev inequalities are immediate consequences of Poincaré–Sobolev inequalities, of [11,  Lemma 5.2], and of the fact that $\varPhi _1(1+s) \le s\log (1+s)$.

Corollary 1

(Logarithmic Sobolev inequalities) Assume that $\mu$ has a density (still denoted by $\mu$) with respect to the Lebesgue measure such that $0<\mu _\flat \le \mu (x) \le \mu _\sharp$ for a.e. $x\in \varOmega$. Then, for any $\xi \in H^1(\varOmega )$, one has

$$\begin{aligned} \int _\varOmega \xi ^2\log \left( \frac{\xi ^2}{\Vert \xi \Vert _{L^2_\mu (\varOmega )}^2}\right) \mathrm {d}\mu \le C(\varOmega ,d,\mu _\flat , \mu _\sharp ) \Vert \nabla \xi \Vert ^2_{L^2_\mu (\varOmega ;{\mathbb {R}}^d)}\,. \end{aligned}$$

Besides, if $|\varGamma ^D|>0$ and $\xi -1\in H^{1,D}_0(\varOmega ):=\{v\in H^1(\varOmega )\mid v=0\text { on }\varGamma ^D\}$, then

$$\begin{aligned} \int _\varOmega \varPhi _1(\xi ^2)\,\mathrm {d}\mu \le C'(\varOmega ,d,\mu _\flat , \mu _\sharp ) \left( 1+\log \left( 1+\Vert \xi -1\Vert _{L^2_\mu (\varOmega )}^2\right) \right) \Vert \nabla \xi \Vert ^2_{L^2_\mu (\varOmega ;{\mathbb {R}}^d)}\,. \end{aligned}$$

1.3 Discrete logarithmic Sobolev inequalities

Similarly to what was done in [11], one can derive discrete logarithmic Sobolev inequalities adapted to the hybrid setting.

Proposition 9

(Discrete logarithmic Sobolev inequality, Neumann case) Let ${\mathcal {D}}$ be a given discretisation of $\varOmega$, with regularity parameter $\theta _{\mathcal {D}}$. Let ${\underline{v}}_{\mathcal {D}}, {\underline{v}}^\infty _{\mathcal {D}}\in {\underline{V}}_{\mathcal {D}}$ be two positive vectors of unknowns such that

$$\begin{aligned} \int _{\varOmega }v_{\mathcal {M}}= \int _{\varOmega } v_{\mathcal {M}}^\infty {=}{:}{M}, \end{aligned}$$

and set $v^\infty _{{\mathcal {M}},\sharp } :=\underset{K \in {\mathcal {M}}}{\sup } v_K^\infty$. Define ${\underline{\xi }}_{\mathcal {D}}$ as the element of ${\underline{V}}_{\mathcal {D}}$ such that

$$\begin{aligned} \xi _K :=\sqrt{\frac{v_K}{v^\infty _K}}\quad \forall K\in {\mathcal {M}},\qquad \xi _\sigma :=\sqrt{\frac{v_\sigma }{v^\infty _\sigma }}\quad \forall \sigma \in {\mathcal {E}}. \end{aligned}$$

Then, there exists $C_{LS,\infty }>0$, only depending on M, $v^\infty _{{\mathcal {M}},\sharp }$, $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \int _{\varOmega } v_{\mathcal {M}}^\infty \varPhi _1 \left( \xi ^2_{\mathcal {M}}\right) \le C_{LS,\infty }^2 \,\big | {\underline{\xi }}_{\mathcal {D}}\big |_{1,{\mathcal {D}}}^2\,. \end{aligned}$$

(A.5)

Proof

By (A.3) and [11,  Lemma 5.2] applied to the probability measure $\mu (x)\,{\mathrm{d}}x=v_{\mathcal {M}}^\infty (x)\,\frac{{\mathrm{d}}x}{M}$ and to the function $\xi _{\mathcal {M}}=\sqrt{\frac{v_{\mathcal {M}}}{v_{\mathcal {M}}^\infty }}$, we first infer that, for $q>2$,

$$\begin{aligned} \int _{\varOmega }v_{\mathcal {M}}\log \left( \xi ^2_{\mathcal {M}}\right) \le C(M,v^\infty _{{\mathcal {M}},\sharp },q)\left( \Vert \xi _{\mathcal {M}}-\overline{\xi _{\mathcal {M}}}\Vert ^2_{L^q(\varOmega )}+\Vert \xi _{\mathcal {M}}-\overline{\xi _{\mathcal {M}}}\Vert ^2_{L^2(\varOmega )}\right) , \end{aligned}$$

where we let $\overline{\xi _{\mathcal {M}}}:=\frac{1}{|\varOmega |}\int _{\varOmega }\xi _{\mathcal {M}}\,{\mathrm{d}}x$. The conclusion then falls in two steps. On the one hand, since $\int _{\varOmega }v_{\mathcal {M}}= \int _{\varOmega } v_{\mathcal {M}}^\infty$, we remark that

$$\begin{aligned} \int _{\varOmega } v_{\mathcal {M}}^\infty \varPhi _1 \left( \xi ^2_{\mathcal {M}}\right) =\int _{\varOmega }v_{\mathcal {M}}\log \left( \xi ^2_{\mathcal {M}}\right) . \end{aligned}$$

On the other hand, we invoke (A.1) and the discrete Poincaré–Sobolev inequality of [29,  Lemma B.25, $p=2$] for $2<q<\frac{2d}{d-2}$:

$$\begin{aligned} \forall {\underline{w}}_{\mathcal {D}}\in {\underline{V}}_{{\mathcal {D}},0}^N, \qquad \Vert w_{\mathcal {M}}\Vert _{L^q(\varOmega )} \le C_{PS} |{\underline{w}}_{\mathcal {D}}|_{1,{\mathcal {D}}}, \end{aligned}$$

where $C_{PS}>0$ only depends on $\varOmega$, d, and $\theta _{\mathcal {D}}$, that we apply to ${\underline{w}}_{\mathcal {D}}={\underline{\xi }}_{\mathcal {D}}-\overline{\xi _{\mathcal {M}}}\,{\underline{1}}_{\mathcal {D}}\in {\underline{V}}_{{\mathcal {D}},0}^N$. This proves (A.5). $\square$

Starting from (A.4), a similar proof yields the following result. The relevant discrete Poincaré–Sobolev inequality in this case is given in [29,  Lemma B.32, $p=2$].

Proposition 10

(Discrete logarithmic Sobolev inequality, Dirichlet case) Assume that $|\varGamma ^D|>0$. Let ${\mathcal {D}}$ be a given discretisation of $\varOmega$, with regularity parameter $\theta _{\mathcal {D}}$. Let ${\underline{v}}_{\mathcal {D}}, {\underline{v}}^\infty _{\mathcal {D}}\in {\underline{V}}_{\mathcal {D}}$ be two positive vectors of unknowns such that

$$\begin{aligned} {\underline{v}}_{\mathcal {D}}- {\underline{v}}^\infty _{\mathcal {D}}\in {\underline{V}}_{{\mathcal {D}},0}^D, \end{aligned}$$

and set $v^\infty _{{\mathcal {M}},\sharp } :=\underset{K \in {\mathcal {M}}}{\sup } v_K^\infty$ and $M^\infty :=\int _\varOmega v_{\mathcal {M}}^\infty$. Define ${\underline{\xi }}_{\mathcal {D}}$ as the element of ${\underline{V}}_{\mathcal {D}}$ such that

$$\begin{aligned} \xi _K :=\sqrt{\frac{v_K}{v^\infty _K}}\quad \forall K\in {\mathcal {M}},\qquad \xi _\sigma :=\sqrt{\frac{v_\sigma }{v^\infty _\sigma }}\quad \forall \sigma \in {\mathcal {E}}. \end{aligned}$$

Then, letting $\mu :=\frac{v^\infty _{{\mathcal {M}}}}{M^\infty }$, there exists $C_{LS,\varGamma ^D,\infty }>0$, only depending on $M^\infty$, $v^\infty _{{\mathcal {M}},\sharp }$, $\varOmega$, d, $\varGamma ^D$, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \int _\varOmega v_{\mathcal {M}}^\infty \varPhi _1(\xi ^2_{\mathcal {M}}) \le C_{LS,\varGamma ^D,\infty }^2\left( 1+\log \left( 1 + \Vert \xi _{\mathcal {M}}-1 \Vert _{L^2_\mu (\varOmega )}^2\right) \right) | {\underline{\xi }}_{\mathcal {D}}|^2_{1,{\mathcal {D}}}. \end{aligned}$$

(A.6)

Nonlinear scheme for mixed Dirichlet-Neumann boundary conditions

In this appendix, we introduce and analyse a version of the nonlinear scheme for the evolution problem (1.1) when $|\varGamma ^D|>0$. In order to perform the asymptotic analysis, we need to assume that the data are compatible with the thermal equilibrium:

$$\begin{aligned} f= 0, \quad g^N = 0, \quad \text { and there exists } \rho ^D > 0 \text { such that } g^D = \rho ^D {{\,\mathrm{e}\,}}^{-\phi } = \rho ^D \omega . \end{aligned}$$

For such data, given $u^{in} \ge 0$, the solution u to (1.1) is positive for $t>0$, and converges towards $u^\infty = \rho ^D {{\,\mathrm{e}\,}}^{-\phi }$ when $t \rightarrow \infty$.

1.1 Scheme and well-posedness

Accordingly to this setting, we define ${\underline{u}}_{\mathcal {D}}^\infty = \rho ^D {\underline{\omega }}_{\mathcal {D}}$. One has $u^\infty _\flat {\underline{1}}_{\mathcal {D}}\le {\underline{u}}^\infty _{\mathcal {D}}\le u^\infty _\sharp {\underline{1}}_{\mathcal {D}}$, where $0< u^\infty _\flat \le u^\infty _\sharp$ only depend on $\rho ^D$, $\phi$, and $\varOmega$. Remind that, as in (3.33), given a positive ${\underline{u}}_{\mathcal {D}}\in {\underline{V}}_{\mathcal {D}}$, one defines ${\underline{w}}_{\mathcal {D}}({\underline{u}}_{\mathcal {D}}) \in {\underline{V}}_{\mathcal {D}}$ as

$$\begin{aligned} w_K:=\log \left( \frac{u_K}{u_K^\infty }\right) \quad \forall K\in {\mathcal {M}},\qquad w_\sigma :=\log \left( \frac{u_\sigma }{u_\sigma ^\infty }\right) \quad \forall \sigma \in {\mathcal {E}}. \end{aligned}$$

For mixed boundary conditions, the discrete problem reads: Find $\big ({\underline{u}}_{\mathcal {D}}^n\in {\underline{V}}_{\mathcal {D}}\big )_{n\ge 1}$ positive such that

Notice that, since for all $\sigma \in {\mathcal {E}}$, $w_\sigma ^n = \log \left( \dfrac{u^n_\sigma }{u^\infty _\sigma } \right)$, the Eq. (B.1b) only means that, for all $\sigma \in {\mathcal {E}}^D_{ext}$, $u^n_\sigma = u^\infty _\sigma$, which enforces strongly the Dirichlet boundary condition on $\varGamma ^D$. One can show the following existence result.

Theorem 5

(Existence of positive solutions and entropy dissipation) Let $u^{in} \in L^2(\varOmega )$ be a non-negative function. There exists at least one positive solution $\big ({\underline{u}}_{\mathcal {D}}^n\in {\underline{V}}_{\mathcal {D}})_{n \ge 1}$ to the nonlinear scheme (B.1). It satisfies the following entropy/dissipation relation:

$$\begin{aligned} \forall n \in {\mathbb {N}}, \qquad \frac{{\mathbb {E}}^{n+1} - {\mathbb {E}}^n }{\varDelta t} + {\mathbb {D}}^{n+1} \le 0, \end{aligned}$$

(B.2)

where ${\mathbb {E}}^n$ and ${\mathbb {D}}^n$ are, respectively, the discrete relative entropy and dissipation defined in (3.35). Moreover, there exists $\varepsilon >0$, depending on $\varLambda$, $\phi$, $u^{in}$, $\rho ^D$, $\varOmega$, d, $\varDelta t$, and ${\mathcal {D}}$ such that, for all $n\ge 1$, $u_K^n\ge \varepsilon$ for all $K\in {\mathcal {M}}$ and $u_\sigma ^n\ge \varepsilon$ for all $\sigma \in {\mathcal {E}}$.

The proof of this theorem relies on the same arguments as the one of Theorem 1 for (homogeneous) pure Neumann boundary conditions. The major difference lies in the counterpart of Lemma 2, which is no longer based on the positivity of the mass, but on the prescribed (zero) value on the Dirichlet faces.

1.2 Long-time behaviour

In the next theorem, we state the long-time behaviour of the discrete solutions to the nonlinear scheme (B.1).

Theorem 6

(Asymptotic stability) If $\big ({\underline{u}}^n_{\mathcal {D}}\in {\underline{V}}_{\mathcal {D}}\big ) _{n \ge 1}$ is a (positive) solution to (B.1), then the discrete entropy decays exponentially fast in time: there is $\displaystyle \nu _{\mathrm {nl},\varGamma ^D}>0$, depending on $\varLambda$, $\phi$, $\varGamma ^D$, $\rho ^D$, $u^{in}$, $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \forall n \in {\mathbb {N}}, \qquad {\mathbb {E}}^{n+1} \le (1 + \nu _{\mathrm {nl},\varGamma ^D} \,\varDelta t ) ^{-1} {\mathbb {E}}^n. \end{aligned}$$

(B.3)

Consequently, the discrete solution converges exponentially fast in time towards its associated discrete steady-state.

Proof

Let $n \in {\mathbb {N}}^{\star }$. As in Sect. 4.3, one has $\displaystyle {\mathbb {D}}^{n}\ge \frac{1}{C_F}{\hat{{\mathbb {D}}}}^{n}\ge \frac{4 u_\flat ^\infty \lambda _\flat \alpha _\flat }{C_F}\big | {\underline{\xi }}_{\mathcal {D}}^{n} \big | _{1,{\mathcal {D}}}^2$, where $C_F>0$ depends on the data. Using the discrete log–Sobolev inequality (A.6) from Proposition 10, we get

$$\begin{aligned} {\mathbb {E}}^n\le & {} C^2_{LS,\varGamma ^D,\infty } \left( 1+\log \left( 1 + \Vert \xi ^n_{\mathcal {M}}-1 \Vert _{L^2_\mu (\varOmega )}^2\right) \right) | {\underline{\xi }}_{\mathcal {D}}^n|^2_{1,{\mathcal {D}}}\nonumber \\\le & {} \frac{C_{LS,\varGamma ^D,\infty }^2 C_F}{ 4 u_\flat ^\infty \lambda _\flat \alpha _\flat } \left( 1+\log \left( 1 + \Vert \xi ^n_{\mathcal {M}}-1 \Vert _{L^2_\mu (\varOmega )}^2\right) \right) {\mathbb {D}}^n. \end{aligned}$$

(B.4)

Then, there is $C> 0$ such that (recall that ${\underline{\xi }}^n_{\mathcal {D}}$ is positive)

$$\begin{aligned} \Vert \xi _{\mathcal {M}}^{n} - 1 \Vert ^2_{L^2_{\mu }(\varOmega )}\le & {} \Vert (\xi ^n_{\mathcal {M}})^2 \Vert _{{L^1_\mu (\varOmega )}}+1 \\\le & {} \left\| \varPhi _1 \left( (\xi _{\mathcal {M}}^{n})^2 \right) \right\| _{{L^1_\mu (\varOmega )}} + C = (M^\infty )^{-1}{\mathbb {E}}^n + C\,, \end{aligned}$$

where the last inequality is an application of the Fenchel–Young inequality $x\le \varPhi _1(x) + \varPhi _1^\star (1)$, where $\varPhi _1^\star$ is the convex conjugate of $\varPhi _1$ and $x = (\xi _{\mathcal {M}}^{n})^2$. But, since the entropy/dissipation relation (B.2) holds, the discrete entropy decays and ${\mathbb {E}}^n \le {\mathbb {E}}^0$. Therefore, one has

$$\begin{aligned} \Vert \xi _{\mathcal {M}}^{n} - 1 \Vert ^2_{L^2_{\mu }(\varOmega )} \le (M^\infty )^{-1}{\mathbb {E}}^0 + C\,. \end{aligned}$$

Combining this estimate with (B.4), we deduce that there exists $\nu _{\mathrm {nl},\varGamma ^D}>0$, depending on $\varLambda$, $\phi$, $\varGamma ^D$, $\rho ^D$, $u^{in}$, $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that $\displaystyle {\mathbb {E}}^n \le \nu _{\mathrm {nl},\varGamma ^D} {\mathbb {D}}^n$. Then, using the entropy/dissipation relation (B.2), we get (B.3). $\square$

Proofs of technical results

1.1 Discrete boundedness by mass and dissipation

We prove Lemma 2 from Sect. 3.3.2. To ease the reading, we first recall the result.

Lemma 2  Let ${\underline{w}}_{\mathcal {D}}\in {\underline{V}}_{\mathcal {D}}$, and assume that there exist $C_\sharp > 0$, and $M_\sharp \ge M_\flat >0$ such that

$$\begin{aligned} M_\flat \le \sum _{K\in {\mathcal {M}}}|K|u_K^\infty {{\,\mathrm{e}\,}}^{w_K} \le M_\sharp \qquad \text { and } \qquad {\mathbb {D}}({\underline{w}}_{\mathcal {D}}) \le C_\sharp . \end{aligned}$$

(3.39)

Then, there exists $C > 0$, depending on $\varLambda$, $u^\infty _\flat$, $u^\infty _\sharp$, $M_\flat$, $M_\sharp$, $C_\sharp$, $\varOmega$, d, and ${\mathcal {D}}$ such that

$$\begin{aligned} |w_K|\le C\quad \forall K\in {\mathcal {M}}\quad \text { and }\quad |w_\sigma |\le C \quad \forall \sigma \in {\mathcal {E}}. \end{aligned}$$

Proof

For $K\in {\mathcal {M}}$, using (2.10) and (2.11), we first infer that

$$\begin{aligned} \delta _K {\underline{w}}_K \cdot {\mathbb {A}}_K \delta _K {\underline{w}}_K = a_K^{\varLambda } ({\underline{w}}_K,{\underline{w}}_K) \ge \lambda _\flat \alpha _\flat |{\underline{w}}_K|_{1,K}^2= \lambda _\flat \alpha _\flat \sum _{\sigma \in {\mathcal {E}}_K} \frac{|\sigma |}{d_{K,\sigma }} \big (w_K-w_\sigma \big )^2 . \end{aligned}$$

By definition (2.1) of the regularity parameter $\theta _{\mathcal {D}}$, we have that $\frac{|\sigma |}{d_{K,\sigma }} \ge \frac{h_K^{d-2}}{\theta _{\mathcal {D}}}$ for all $\sigma \in {\mathcal {E}}_K$, so that

$$\begin{aligned} \delta _K {\underline{w}}_K \cdot {\mathbb {A}}_K \delta _K {\underline{w}}_K \ge \frac{\lambda _\flat \alpha _\flat }{\theta _{\mathcal {D}}} h_K^{d-2} |\delta _K {\underline{w}}_K|^2. \end{aligned}$$

(C.1)

By the expression (3.38) of ${\mathbb {D}}({\underline{w}}_{\mathcal {D}})$, and the local lower bound (C.1), we thus get

$$\begin{aligned} {\mathbb {D}}({\underline{w}}_{\mathcal {D}})&= \sum _{K \in {\mathcal {M}}} r_K\big ( {\underline{u}}_K^\infty \times \mathrm{exp}({\underline{w}}_K) \big ) \delta _K {\underline{w}}_K \cdot {\mathbb {A}}_K \delta _K {\underline{w}}_K \\&\ge \frac{\lambda _\flat \alpha _\flat }{\theta _{\mathcal {D}}}\sum _{K \in {\mathcal {M}}} h_K^{d-2}r_K\big ( {\underline{u}}_K^\infty \times \mathrm{exp}({\underline{w}}_K) \big ) |\delta _K {\underline{w}}_K |^2 \\&= \frac{\lambda _\flat \alpha _\flat }{\theta _{\mathcal {D}}}\sum _{K \in {\mathcal {M}}} \sum _{\sigma \in {\mathcal {E}}_K}h_K^{d-2} r_K\big ( {\underline{u}}_K^\infty \times \mathrm{exp}({\underline{w}}_K) \big ) (w_K - w_\sigma )^2. \end{aligned}$$

Let $K \in {\mathcal {M}}$ and $\sigma \in {\mathcal {E}}_K$ be fixed. Using, successively, the definition (3.27) of $r_K$ combined with the definition (3.29) of $f_{|{\mathcal {E}}_K|}$, the combination of (3.34) with assumptions (3.28a) and (3.28c), and the assumptions (3.28b) and (3.28d) combined with the bound (2.2) on $|{\mathcal {E}}_K|$, we infer, for $w_\sigma \ne w_K$,

$$\begin{aligned} r_K\big ( {\underline{u}}_K^\infty \times \mathrm{exp}({\underline{w}}_K) \big )(w_K - w_\sigma )^2&\ge \frac{1}{|{\mathcal {E}}_{K}|}\,m \big (u_K^\infty {{\,\mathrm{e}\,}}^{w_K}, u_\sigma ^\infty {{\,\mathrm{e}\,}}^{w_\sigma } \big ) (w_K - w_\sigma )^2 \\&\ge \frac{u^\infty _\flat }{|{\mathcal {E}}_{K}|}\, m \big ({{\,\mathrm{e}\,}}^{w_K}, {{\,\mathrm{e}\,}}^{w_\sigma } \big ) (w_K - w_\sigma )^2 \\&\ge \frac{u^\infty _\flat }{d\theta _{\mathcal {D}}^2}\left( {{\,\mathrm{e}\,}}^{w_K}- {{\,\mathrm{e}\,}}^{w_\sigma } \right) \left( w_K - w_\sigma \right) \ge 0, \end{aligned}$$

and we verify that this inequality still holds when $w_\sigma = w_K$. Since ${\mathbb {D}}({\underline{w}}_{\mathcal {D}})\le C_\sharp$ by (3.39), for all $K\in {\mathcal {M}}$, and all $\sigma \in {\mathcal {E}}_K$, we have

$$\begin{aligned} 0 \le \left( {{\,\mathrm{e}\,}}^{w_K}- {{\,\mathrm{e}\,}}^{w_\sigma } \right) \left( w_K - w_\sigma \right) \le \zeta h_K^{2-d}, \end{aligned}$$

(C.2)

with $\zeta :=\frac{dC_\sharp \theta _{\mathcal {D}}^3}{\lambda _\flat \alpha _\flat u^\infty _\flat }>0$ (recall that $\alpha _\flat$ depends on $\varOmega$, d, and $\theta _{\mathcal {D}}$). Besides, since $\sum _{K \in {\mathcal {M}}} |K| u_K ^\infty {{\,\mathrm{e}\,}}^{w_K} \le M_\sharp$ again by (3.39), we have $|K| u_K ^\infty {{\,\mathrm{e}\,}}^{w_K} \le M_\sharp$ for all $K \in {\mathcal {M}}$. Similarly, since $\sum _{K \in {\mathcal {M}}} |K| u_K ^\infty {{\,\mathrm{e}\,}}^{w_K} \ge M_\flat$, there exists $K_0 \in {\mathcal {M}}$ such that $|\varOmega |u_{K_0} ^\infty {{\,\mathrm{e}\,}}^{w_{K_0}} \ge M_\flat >0$. Combining these bounds, we infer that there exists $K_0\in {\mathcal {M}}$ such that

$$\begin{aligned} \log \left( \frac{M_\flat }{|\varOmega |u_\sharp ^\infty }\right) \le w_{K_0}\le \log \left( \frac{M_\sharp }{|K_0|u_\flat ^\infty }\right) . \end{aligned}$$

(C.3)

Now, let us show that we can similarly frame all the other components of ${\underline{w}}_{\mathcal {D}}$.

For $a,x \in {\mathbb {R}}$, let us define $E(a,x)=\left( {{\,\mathrm{e}\,}}^x - {{\,\mathrm{e}\,}}^a \right) (x - a)\ge 0$. Observe that $E(a,y+a){{\,\mathrm{e}\,}}^{-a} = ({{\,\mathrm{e}\,}}^y-1)y =:\xi (y)$ and that $\xi$ is continuous, strictly decreasing for $y<0$, strictly increasing for $y>0$, $\xi (0)=0$, and $\xi (y)\rightarrow +\infty$ when $y\rightarrow \pm \infty$. Let $b, a_\sharp >0$, and take $|a|\le a_\sharp$. By the properties of $\xi$, if $E(a,x)\le b$, then $|x|\le \kappa _b(a_\sharp ) := a_\sharp + \max \{|y|\ \text {s.t.}~\xi (y)=b{{\,\mathrm{e}\,}}^{a_\sharp }\}$. We can thus infer that if $(x_k)_{k=0,\dots ,m}$ is a finite sequence of real numbers such that $E(x_k, x_{k+1})\le b$ and $|x_0|\le a_\sharp$, then $|x_m| \le \kappa _b^{(m)}(a_\sharp )$ where $\kappa _b^{(m)}$ is m compositions of $\kappa _b$. In particular, the bound only depends on $a_\sharp$, m and b.

Now we can conclude the proof. Because of the connectivity of the mesh, for any cell K (respectively, face $\sigma$) there is a finite sequence of components of ${\underline{w}}_{\mathcal {D}}$, denoted $(x_k)_{k=0,\dots ,m}$, starting at $x_0 = w_{K_0}$ and finishing at $x_m = w_K$ (respectively, $x_m = w_\sigma$) such that, by (C.2), $E(x_k, x_{k+1})\le b:=\zeta h_{\mathcal {D}}^{2-d}$. The inequality (C.3) yields the initial bound on $|x_0|$, and one concludes by the above argument. $\square$

1.2 A local comparison result

We prove a local comparison result between the matrices ${\mathbb {A}}_K$ and some (local) diagonal matrices. The proof relies on arguments that are similar to those advocated in [14] to analyse the VAG scheme.

Lemma 4

For $K \in {\mathcal {M}}$, let ${\mathbb {A}}_K\in {\mathbb {R}}^{|{\mathcal {E}}_K|\times |{\mathcal {E}}_K|}$ be the matrix defined by (2.8). The matrices ${\mathbb {A}}_K$ are symmetric positive-definite, and there exists $C_A>0$, only depending on $\varLambda$, $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \forall K \in {\mathcal {M}}, \quad {\mathrm{Cond}}_2({\mathbb {A}}_K) \le C_A, \end{aligned}$$

where ${\mathrm{Cond}}_2({\mathbb {A}}_K) :=\Vert {\mathbb {A}}_K^{-1}\Vert _2\Vert {\mathbb {A}}_K\Vert _2$ is the condition number of the matrix ${\mathbb {A}}_K$. Moreover, letting for $K\in {\mathcal {M}}$, ${\mathbb {B}}_K \in {\mathbb {R}}^{|{\mathcal {E}}_K|\times |{\mathcal {E}}_K|}$ be the diagonal matrix with entries

$$\begin{aligned} B_K^{\sigma \sigma } :=\sum _{\sigma ' \in {\mathcal {E}}_K} |A_K^{\sigma \sigma '}|\qquad \text {for all }\sigma \in {\mathcal {E}}_K, \end{aligned}$$

(C.4)

there exists $C_B>0$, only depending on $\varLambda$, $\varOmega$, d, and $\theta _{\mathcal {D}}$ such that

$$\begin{aligned} \forall K \in {\mathcal {M}}, \,\forall w \in {\mathbb {R}}^{|{\mathcal {E}}_K|}, \qquad w \cdot {\mathbb {A}}_K w \le w \cdot {\mathbb {B}}_K w \le C_B \,w \cdot {\mathbb {A}}_K w. \end{aligned}$$

(C.5)

Proof

Let $K \in {\mathcal {M}}$ and $k :=|{\mathcal {E}}_K|$. As a direct consequence of its definition (2.8), the matrix ${\mathbb {A}}_K\in {\mathbb {R}}^{k\times k}$ is symmetric and positive semi-definite. Now, let $w :=(w_\sigma )_{\sigma \in {\mathcal {E}}_K} \in {\mathbb {R}}^k$, and define ${\underline{v}}_K \in {\underline{V}}_K$ such that

$$\begin{aligned} v_K = 0\qquad \text {and}\qquad v_\sigma = -w_\sigma \text { for all } \sigma \in {\mathcal {E}}_K. \end{aligned}$$

Then, $\delta _K {\underline{v}}_K = (v_K-v_\sigma )_{\sigma \in {\mathcal {E}}_K}=w$. By (C.1), we immediately get that

$$\begin{aligned} w \cdot {\mathbb {A}}_K w \ge \frac{\lambda _\flat \alpha _\flat }{\theta _{\mathcal {D}}} h_K^{d-2} |w|^2, \end{aligned}$$

which implies, since $w\in {\mathbb {R}}^k$ is arbitrary, that ${\mathbb {A}}_K$ is invertible, and gives us a lower bound on its smallest eigenvalue. By the same arguments advocated to prove (C.1), noticing that $\frac{|\sigma |}{d_{K,\sigma }} \le \theta _{\mathcal {D}}h_K ^{d-2}$ for all $\sigma \in {\mathcal {E}}_K$, we infer that

$$\begin{aligned} w \cdot {\mathbb {A}}_K w \le \lambda _\sharp \alpha _\sharp \theta _{\mathcal {D}}h_K^{d-2} |w|^2. \end{aligned}$$

We eventually get, using the estimates on the eigenvalues of ${\mathbb {A}}_K$, that

$$\begin{aligned} {\mathrm{Cond}}_2({\mathbb {A}}_K)\le \frac{\lambda _\sharp \alpha _\sharp }{\lambda _\flat \alpha _\flat } \theta _{\mathcal {D}}^2{=}{:}C_A, \end{aligned}$$

(C.6)

with $C_A>0$ only depending on $\varLambda$, $\varOmega$, d, and $\theta _{\mathcal {D}}$. Now, by (C.4), since ${\mathbb {A}}_K$ is symmetric, we have

$$\begin{aligned} w \cdot {\mathbb {B}}_K w= \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} |A^{\sigma \sigma '}_K| w_\sigma ^2 = \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} |A^{\sigma \sigma '}_K| w_{\sigma '} ^2 , \end{aligned}$$

and we can use the half-sum to get

$$\begin{aligned} w \cdot {\mathbb {B}}_K w= \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} |A^{\sigma \sigma '}_K| \frac{w_\sigma ^2+ w_{\sigma '}^2}{2}. \end{aligned}$$

Using Young’s inequality, we infer

$$\begin{aligned} w \cdot {\mathbb {A}}_K w&= \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} A^{\sigma \sigma '}_K w_\sigma w_{\sigma '} \le \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} |A^{\sigma \sigma '}_K| |w_\sigma | |w_{\sigma '}| \\&\le \sum _{\sigma \in {\mathcal {E}}_K} \sum _{ \sigma ' \in {\mathcal {E}}_K} |A^{\sigma \sigma '}_K| \frac{w_\sigma ^2 +w_{\sigma '}^2}{2} = w \cdot {\mathbb {B}}_K w . \end{aligned}$$

For the second inequality, by symmetry of ${\mathbb {A}}_K$, we have

$$\begin{aligned} w \cdot {\mathbb {B}}_K w&= \sum _{\sigma \in {\mathcal {E}}_K} B_K^{\sigma \sigma } w_\sigma ^2 \le \underset{ \sigma \in {\mathcal {E}}_K}{\max } (B_K^{\sigma \sigma } ) \sum _{\sigma \in {\mathcal {E}}_K} w_\sigma ^2 = \underset{ \sigma \in {\mathcal {E}}_K}{\max } \left( \sum _ {\sigma ' \in {\mathcal {E}}_K} |A_K^{\sigma '\sigma }| \right) |w|^2 \\&= \Vert {\mathbb {A}}_K\Vert _1 |w|^2. \end{aligned}$$

The space ${\mathbb {R}}^{k\times k}$ being of finite dimension, the norms $\Vert \cdot \Vert _1$ and $\Vert \cdot \Vert _2$ are equivalent, and there exists $\gamma _k>0$ such that $\Vert \cdot \Vert _1 \le \gamma _k \Vert \cdot \Vert _2$. Moreover, since ${\mathbb {A}}_K$ is symmetric positive-definite, the following inequality holds:

$$\begin{aligned} w \cdot {\mathbb {A}}_K w \ge \frac{\Vert {\mathbb {A}}_K\Vert _2}{\text {Cond}_2({\mathbb {A}}_K) } |w|^2. \end{aligned}$$

From the previous estimates and (C.6), we deduce that

$$\begin{aligned} w \cdot {\mathbb {B}}_K w \le \gamma _{k} \,\text {Cond}_2({\mathbb {A}}_K) \,w \cdot {\mathbb {A}}_K w \le \gamma _{k} \,C_A \,w \cdot {\mathbb {A}}_K w. \end{aligned}$$

But, according to (2.2), we have $\displaystyle \underset{ K \in {\mathcal {M}}}{\max } \gamma _{k} \le \underset{(d+1) \le l \le d \theta _{\mathcal {D}}^2}{\max }\gamma _l$, therefore

$$\begin{aligned} w \cdot {\mathbb {B}}_K w \le C_B \,w \cdot {\mathbb {A}}_K w, \end{aligned}$$

where $\displaystyle C_B = C_A \underset{(d+1) \le l \le d \theta _{\mathcal {D}}^2}{\max }\gamma _{l}$ is a positive constant only depending on $\varLambda$, $\varOmega$, d, and $\theta _{\mathcal {D}}$. This completes the proof of the comparison result (C.5). $\square$