An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes

HAL Id: hal-00693134

https://hal-upec-upem.archives-ouvertes.fr/hal-00693134

Submitted on 31 Jul 2017

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons

An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise

deterministic Markov processes

Robert Eymard, Sophie Mercier, Alain Prignet

To cite this version:

Robert Eymard, Sophie Mercier, Alain Prignet. An implicit finite volume scheme for a scalar hyper- bolic problem with measure data related to piecewise deterministic Markov processes. Journal of Com- putational and Applied Mathematics, Elsevier, 2008, 222 (2), pp.293-323. �10.1016/j.cam.2007.10.053�.

�hal-00693134�

An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov

processes

Robert Eymard, Sophie Mercier, Alain Prignet

Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 5 bd Descartes, 77454 Marne la Vallée Cedex 2, France

Keywords:Linear hyperbolic problems with measure solutions; Weak bounded variation inequalities; Chapman–Kolmogorov equations; Piecewise- deterministic Markov process; Growth–collapse Markov process

1. Introduction

Within a more and more competitive context, industrialists often have to assess as accurately as possible different quantities linked for example to economical and/or safety constraints. For instance, an industrial gas provider must be aware of the production availability of his gas plant, because of possible penalties to be paid in case of drop in its production rate. Lots of other quantities may of course be of interest to him, such as the mean functioning cost per unit time, the mean number of component failures up to some fixed horizon, and so on. All those quantities may be We are interested here in the numerical approximation of a family of probability measures, solution of the Chapman–

Kolmogorov equation associated to some non-diffusion Markov process with uncountable state space. Such an equation contains a transport term and another term, which implies redistribution of the probability mass on the whole space. An implicit finite volume scheme is proposed, which is intermediate between an upstream weighting scheme and a modified Lax–Friedrichs one.

Due to the seemingly unusual probability framework, a new weak bounded variation inequality had to be developed, in order to prove the convergence of the discretised transport term. Such an inequality may be used in other contexts, such as for the study of finite volume approximations of scalar linear or nonlinear hyperbolic equations with initial data in L¹. Also, due to the redistribution term, the tightness of the family of approximate probability measures had to be proven. Numerical examples are provided, showing the efficiency of the implicit finite volume scheme and its potentiality to be helpful in an industrial reliability context.

written as expectations of some functional of the underlying stochastic process which describes the time evolution of the studied system, or equivalently, as integrals with respect to the marginal distribution of the process. From an industrial point of view, such distributions then are essential to evaluate. Unfortunately, they often are unreachable in closed form, especially in the modern context of dynamic reliability, which is concerned with the study of so-called hybrid systems (details below). Such distributions (or their derivatives) are hence numerically evaluated, most of the time by Monte Carlo simulations which often entail very long computation times. New methods for their numerical assessment then are to be developed, from where most of the quantities with industrial interest may be derived, in the context of dynamic reliability.

The point of this paper hence is to develop some numerical scheme, to assess the marginal distribution of some process describing the time evolution of a hybrid system. Such a hybrid system is governed by two different types of dynamics: a discrete dynamic, which is related to the occurrence of events such as failures of components, some button switching, and so on, and a continuous dynamic, linked to the evolution of continuous characteristics, such as pressure, temperature, liquid level in a tank, and so on. The time evolution of such hybrid systems is modelled with so-called Piecewise-Deterministic Markov Processes (PDMPs), which are non-diffusion Markov processes with uncountable state space; see [10] or [17] for details. A PDMP hence is a Markov hybrid process(I_t,X_t)t≥0, where the discrete part I_t takes range in a finite set E, and where the continuous partX_t takes range inR^N, withN ∈ N^∗. Due to its Markovian characteristic, one may write its associated Chapman–Kolmogorov equation. A PDMP jumps at countable isolated times and such an equation comprises a transport term, which corresponds to a deterministic evolution of the process between jumps, and a redistribution term, which corresponds to jumps and entails redistribution of the probability mass on the whole space. In order to make the paper clearer, we now make the choice to specialise our exposure to so-called Markov Growth–Collapse processes (GCPs); see [5]. Such processes are PDMPs whereE is reduced to a singleton, so that the discrete part(I_t)t≥0 is constant and hence unnecessary. They typically describe the time evolution of a quantity, for instance the size of a queue, with successive phases of deterministic growth and random instantaneous collapse (or jump in the vocabulary of PDMPs). Their behaviour hence is typical of a PDMP and such a specialisation allows us to be much clearer in our exposure. Extension of our results for GCPs to PDMPs is straightforward and is exposed at the end of the paper, in Section6.

Let us now be more specific and let (X_t)t≥0 be a GCP: as already said, the process(X_t)t≥0 jumps at isolated countable times. Between jumps, the deterministic evolution of(X_t)t≥0follows an ordinary differential equation:

dX_t

dt =v(X_t) , ∀t ∈ [t₁,t₂) (1)

wherevis an application fromR^N toR^N, andt₁,t₂are two successive jump times. At jump times, transitions from X_t⁻=x∈R^NtoX_t =y∈R^Nare governed by a transition rateλ (x)and by a probability measureµ(x)(dy)which stands for the conditional distribution ofX_t given thatX_t⁻ =x.

We make the following assumptions on the data, which will be referred to as assumptionsH0in the following:

• the transition rateλ:R^N →R+is continuous and bounded byΛ= kλk_∞,

• the velocityv:R^N →R^N is Lipschitz continuous and bounded byV = kvk_∞>0,

• let P(R^N) be the set of probability measures on R^N; the function µ : R^N → P(R^N) is such that for all ψ ∈ Cb(R^N) (continuous and bounded from R^N toR), the function x 7→ R

R^Nψ(y)µ(x)(dy)is continuous (and bounded) fromR^NtoR,

• the measureρini(dx)is a probability measure onR^N, and stands for the initial distribution of(X_t)t≥0.

We may now write the Chapman–Kolmogorov (CK) equation associated to the Markov process(Xt)t≥0and we set ρ(t)(dx)to be the marginal distribution at timetof the process(Xt)t≥0. UnderH₀, it is then known that(ρ(t)(dx))t≥0

is the single family of probability measures solution of the CK equation, which here is written as Z

R^Nϕ(x)ρ (t) (dx)− Z

R^Nϕ(x)ρini(dx)− Z t

0

Z

R^N(v(x)· ∇ϕ(x))ρ(s)(dx)ds

= Z t

0

Z

R^Nλ(x) Z

R^Nϕ(y)µ(x)(dy)−ϕ(x)

ρ (t) (dx)ds ∀t ∈R⁺,∀ϕ∈C_c¹(R^N) (2)

whereC_c¹(R^N)stands for the set of continuously differentiable functions fromR^N toRwith a compact support;

see [10] or [17] for the CK equation associated to a PDMP, and [7] for the uniqueness result. The third term in the left side of(2)is the transport term whereas the right side is the redistribution term.

This paper is dedicated to the study of the convergence of an implicit finite volume scheme towards the solution of the CK equation(2)and we now discuss the mathematical nature of some of the arising problems. To make it clearer and only for the purpose of this introduction, we consider the case where the data measures involved in the CK equation admit density with respect to Lebesgue measure onR^N: we hence assume thatµ(x)(dy)=d_µ(x,y)dy andρini(dx)=u_ini(x) dx, whereu_ini ∈ L¹(R^N)becauseρiniis a probability measure. It may then be proved that the solutionρ(t)(dx)of the CK equation admits a density and we setρ(t)(dx)=u(x,t) dx. Writing again the CK equation, the functionuthen happens to be, at least formally, a weak solution of the equation

du

dt(x,t)+div(u(x,t)v(x))= Z

R^N

λ(y)d_µ(y,x)u(y,t)dy−λ(x)u(x,t) (3) for(x,t)∈R⁺×R^N with the initial condition

u(x,0)=u_ini(x), forx∈R^N. (4)

Problem(3)–(4) can be seen as a linear hyperbolic problem with an integral form right-hand side, and an initial condition inL¹(R^N)instead of the standard frameworku_ini ∈ L^∞(R^N). Only for the purpose of this introduction again, let us setu_h,k to be an approximate solution, whereh is a space step andk is a time step. Looking at the convergence study (see(62)–(63)in the proof ofLemma 9), one may see that the following condition is used for proving the convergence ofu_h,k:

∀R,T >0, lim

h→0h Z T

0

|u_h,k(·,t)|

B V(B(0,R))dt =0, (5)

where, for a functionv:R^N→Rand for any setΩ ⊂R^N, we denote by

|v|_{B V(Ω)}=sup

Z

R^Nv(x)divϕ(x)dx

, ϕ∈C¹(R^N)N,kϕk_∞≤1, ϕ(x)=0 for allx∈R^N\Ω

(6) the BV-semi-norm ofv.

In order to prove (5), ifu_ini were in L²_loc(R^N), one might proceed as is done in [6] or in [14] for explicit or implicit finite volumes schemes on general meshes andu_ini ∈ L^∞(R^N), and prove some weak bounded variation (BV) inequality of the following shape:

Z T 0

|u_h,k(·,t)|

B V(B(0,R))dt ≤ Cku_inik

L²(B(0,R⁰))

√

h ,

whereCandR⁰depend onRandT. Unfortunately, such an inequality is no longer valid here for the present scheme anduini ∈ L¹(R^N), which is imposed by our probabilistic context. We have hence been led to develop a new weak BV inequality:

Z T 0

|u_h,k(·,t)|

B V(B(0,R))dt ≤ C h^1/q,

for some 1 < q < 2 (seeLemma 4). This inequality, which is sufficient to get (5), is shown using the analogy between an upstream weighting scheme or a modified Lax–Friedrichs scheme, and a continuous problem including a vanishing viscosity term. We can hence use the tools developed in [11] or [16] for the convergence of finite volume approximations to the solution of elliptic problems with measure data, which themselves mimic the continuous framework provided in [4,3] for continuous parabolic problems. Nevertheless, the proofs given in such papers do not hold in the case of the classical upstream weighting scheme, since a non-vanishing viscosity term was required on the whole mesh, which has led us to introduce an intermediate scheme between an upstream weighting one and a modified Lax–Friedrichs scheme.

Also, in order to complete the convergence proof, a last technical problem is due to the redistribution term in the CK equation. Due to that term, we indeed have to control the probability mass which escapes outside compact sets for

the family of approximate distributions. In other words, we have to prove tightness of this family. With that aim, we explicitly construct a Liapounov function, which classically allows us to conclude (see [12] e.g.).

Finally, note that we had already studied the convergence of an explicit finite volume scheme towards the solution of the CK equation(2)in a previous paper [8]. The convergence proof was established there only under an inverse CFL conditionh/k→0, this condition being used in order to replace a weak bounded variation inequality which we did not have. The tools developed in the present paper would however not hold in the framework of [8], where the scheme is a purely upstream weighting one.

This paper is organised as follows. The numerical scheme is provided in Section 2 (in the case when E is a singleton). Some properties of the scheme are developed in Section 3, which are required to get some weak BV inequalities given in Section4. Such weak BV inequalities are used for proving the convergence of the scheme, which is done in Section5. We then present in Section6extensions of our results to the case of a generalE, with possible jumps between discrete states ofE. We finally conclude this paper in Section7with numerical experiments showing the efficiency and precision of the method, and its relevance in an industrial reliability context.

2. The finite volume scheme

Let us first give the definition of admissible meshes ofR^N.

Definition 1. An admissible mesh ofR^Nfor problem(2)is a partitionMofR^N such that

(1) for allK ∈M,K is bounded, the interior ofKis an open convex subset ofR^Nand theN dimensional measure ofK, denoted by m(K), is strictly positive,

(2) for all K ∈ M, denoting by∂K the boundary of K, and, for allL ∈ M, denoting by K|L =∂K ∩∂L, there existsN_K ⊂Msuch thatK 6∈N_K and∂K =S

L∈N_K K|L, and, for allL ∈N_K,K|L, called an edge ofK, is included in a hyperplane ofR^N, with a strictly positive N−1 dimensional measure equal to m(K|L); we then denote byn_{K L}the unit normal vector toK|Loriented fromK toL,

(3) the size of the mesh, defined byh_M=sup_K∈Mdiam(K), is finite, (4) there existsC1≥1 with

1 C1

h_M X

L∈N_K

m(K|L)≤m(K)≤C₁h_M X

L∈N_K

m(K|L), ∀K ∈M, (7)

1

C₁h_M≤diam(K)≤h_M, ∀K ∈M. (8)

and 1

C₁h^N_M≤m(K)≤C1h_M^N , ∀K ∈M. (9)

We then denote byCMthe infimum of allC1such that(7)–(9)hold.

Note that all classical regular meshes ofR^Nare admissible in the sense of the previous definition.

Now, letMbe a fixed admissible mesh ofR^N. For such a mesh, we set vK,L = 1

m(K|L) Z

K|L

v(x)·n_{K L}ds(x), ∀K ∈M,∀L∈N_K, (10) where ds(x)stands for theN−1 dimensional measure onK|L and

wK,L =max(|vK,L|, ε), ∀K ∈M,∀L ∈N_K (11)

for a givenε∈ [0,V]. We also set λK = 1

m(K) Z

K

λ (x)dx aK,L = 1

m(K) Z

K

λ(x) Z

L

µ(x) (dy)

dx

forK,L∈Mwith X

L∈M

aK,L =λK. (12)

For a given time stepk>0, the scheme then is written as u⁽_K⁰⁾= 1

m(K) Z

K

dρini(x), ∀K ∈M (13)

and

m(K)(u⁽_Kⁿ⁺¹⁾−u⁽_Kⁿ⁾)+k X

L∈NK

m(K|L) vK,L

u⁽_Kⁿ⁺¹⁾+u⁽_Lⁿ⁺¹⁾

2 +wK,L

2 (u⁽_Kⁿ⁺¹⁾−u⁽_Lⁿ⁺¹⁾)

!

= −km(K)λKu⁽_Kⁿ⁺¹⁾+k X

L∈M

m(L)a_L,Ku⁽_Lⁿ⁺¹⁾, ∀K ∈M,∀n ∈N.

(14)

We first prove the existence and uniqueness of a solution to this numerical scheme in the following lemma.

Lemma 2. Let us assume hypothesesH₀and letMbe an admissible mesh of R^N in the sense of Definition1. Let k>0andε∈ [0,V]be given. Then there exists one and only one family of real numbers(u⁽_Kⁿ⁾)K∈M,n∈Nsuch that (13)–(14)hold andP

K∈Mm(K)|u⁽_Kⁿ⁾|<∞for all n∈N. Moreover, the following properties hold:

u⁽_Kⁿ⁾≥0, ∀K ∈M,∀n ∈N, (15)

and X

K∈M

m(K)u⁽_Kⁿ⁾=1, ∀n∈N. (16)

Proof. Letkk_L

1 be the following norm onL₁= {u :=(uK)K∈M s.t. P

K∈Mm(K)|uK|<+∞}:

kuk_L

1 = X

K∈M

m(K)|uK|. (17)

Foru∈L₁fixed, let us considerψudefined byψu(p)=rfor p∈L₁with Cm(K)r_K =Cm(K)p_K−m(K)

k (p_K−u_K)

− X

L∈N_K

m(K|L) vK,L

p_K+p_L

2 +wK,L

2 (pK−pL)

−m(K)λKpK+ X

L∈M

m(L)aL,KpL

(18)

= m(K)

k uK+m(K) C−1

k− 1

2m(K) X

L∈N_K

m(K|L) vK,L +wK,L

−λK

! pK

+1 2

X

L∈N_K

m(K|L) wK,L−vK,L

p_L + X

L∈M

m(L)a_L,Kp_L (19)

whereC>0 is a constant to be chosen such that the coefficient of p_K in(19)is non-negative.

Due to(7),vK,L+wK,L ≤2V andλK ≤Λ, we know 1

k + 1

2m(K) X

L∈NK

m(K|L) vK,L+wK,L

+λK ≤ 1

k+VCM

hM

+Λ. We then takeCsuch that

C≥ 1

k +VC_M h_M +Λ.

As the coefficients of p_L andp_Kin(19)now are non-negative (rememberwK,L−vK,L >0), we derive that kψu(p)k_L

1

≤ 1 C

X

K∈M

"

m(K)

k |u_K| +m(K) C−1 k − 1

2m(K) X

L∈NK

m(K|L) vK,L+wK,L

−λK

!

|p_K|

+1 2

X

L∈NK

m(K|L) wK,L−vK,L

|p_L| + X

L∈M

m(L)a_L,K|p_L|

#

(20)

= 1 C

"

1 kkuk_L

1+

C−1

k

kpk_L

1−1 2

X

K∈M

X

L∈NK

m(K|L) vK,L+wK,L

|pK| − X

K∈M

m(K) λK|pK|

+1 2

X

K∈M

X

L∈NK

m(K|L) wK,L−vK,L

|pL| + X

L∈M

m(L)λL|pL|

#

(21) using(12).

Noting thatP

K∈M

P

L∈NK =P

L∈M

P

K∈NL,vL,K = −vK,L, m(K|L)=m(L|K)and thatwK,L =wL,K, we easily get

X

K∈M

X

L∈NK

m(K|L) wK,L−vK,L

|p_L| = X

K∈M

X

L∈NK

m(K|L) wK,L+vK,L

|p_K| (22) and hence

kψu(p)k_L

1 ≤ 1

C 1

kkuk_L

1+

C−1

k

kpk_L

1

<+∞. (23)

As a consequence, the functionψumapsL1intoL1for anyu ∈L1. Besides, similar computations also give

ψu(p)−ψu p⁰ _L

1

≤ C−¹

k

C p−p⁰

_L

1 (24)

for all p,p⁰∈L₁. The functionψuthen is a contraction on the Banach spaceL₁and, for allu ∈L₁, the functionψu

admits a unique fixed pointp∈L₁. Noting thatψ_u(n) u⁽ⁿ⁺¹⁾

=u⁽ⁿ⁺¹⁾is equivalent to(14), the existence and uniqueness of u⁽ⁿ⁾

n∈NinL₁such that (13)–(14)are true is then clear, recursively.

Let us now set C=

u:=(u_K)K∈M∈L₁s.t.u_K ≥0 for allK ∈Mand kuk_L

1 =1 .

Takingu ∈ Cand p ∈ C, it is easy to check thatr := ψu(p)∈C, using the non-negativeness of the coefficients in (19), and noting that(20)–(21)and consequently the left inequality from(23)now are equalities.

Now, starting fromu ∈C, the sequence(p_n)n∈Nrecursively defined byp₀=uand p_n+1 =ψu(p_n)is such that p_n∈Cfor alln∈Nand converges inL1towards the single fixed point ofψu. We derive that for allu∈C, the single fixed point ofψuinL1actually is an element ofC.

Consequently, starting withu⁽⁰⁾∈C, the single sequence u⁽ⁿ⁾

n∈NinL1such that(13)–(14)are true has elements inC, which achieves the proof.

3. Properties of the scheme

For alls∈R, we denote bybscthe greatest integer lower than or equal tosand for allR ≥0, letB(0,R)⊂R^N be the open ball with centre 0 and radius R, withB(0,0)= ∅.

Lemma 3. Let us assumeH₀and letMbe an admissible mesh ofR^Nin the sense ofDefinition1. Let m∈(0,+∞), k >0andε ∈(0,V]be given and let(u⁽_Kⁿ⁾)K∈M,n∈Nbe the family of real numbers defined by(10),(11),(13)and

(14)such that(15)and(16)hold. Let R>0and T >0be given and letθ^(R):R+→ [0,1]be defined byθ^(R)(s)=1 for all s∈ [0,R],1+R−s for all s∈ [R,R+1]and 0 for all s≥ R+1. Let us denote

θ_K^(R)= 1 m(K)

Z

K

θ^(R)(|x|)dx, and let us define(bu⁽_Kⁿ⁾)K∈M,n∈Nby

bu⁽_Kⁿ⁾=θ_K^(R)u⁽_Kⁿ⁾, ∀K ∈M,∀n∈N. (25) Let C₁be such that C_M≤C₁, let C₂be such that h_M<C₂ and k<C₂ and let T₁be the term defined by

T1=

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L) (bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)²

(1+max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾))^m+1. (26) Then there exists C₃, only depending on N , R, T , m,v, C₁,Λ,εand C₂, such that

T₁≤C₃. (27)

Proof. Let us introduce, as in [4], the functionφm : R→Rsuch thatφm(s)=(1−1/(1+ |s|)^m)×sign(s)with φm(0)=0 andφ_m⁰ (s)=m/(1+ |s|)^m+1, for alls∈R. We define the real functionΦ_mbyΦ_m(s)=Rs

0 φm(u)du, for alls∈R. We have 0≤φm(s)≤1 and 0≤Φ_m(s)≤sfors≥0. From(14), we get

m(K)(u⁽_Kⁿ⁺¹⁾−u⁽_Kⁿ⁾)+km(K)u⁽_Kⁿ⁺¹⁾D_K+k 2

X

L∈NK

m(K|L) wK,L−vK,L

(u⁽_Kⁿ⁺¹⁾−u⁽_Lⁿ⁺¹⁾)

= −km(K)λKu⁽_Kⁿ⁺¹⁾+k X

L∈M

m(L)a_L,Ku⁽_Lⁿ⁺¹⁾, ∀K ∈M,∀n ∈N (28) denoting by

D_K = 1 m(K)

Z

K

div(v(x))dx = 1 m(K)

X

L∈N_K

m(K|L)vK,L, ∀K ∈M. (29)

Thanks toH0, we get the existence ofC4, only depending onv, such that

|DK| ≤C4. (30)

Let us multiply (28) by θ_K^(R)φm(θ_K^(R)u⁽_Kⁿ⁺¹⁾), and sum the result on K ∈ M and n = 0, . . . ,bT/kc. We get T₂+T₃+T₄+T₅=0, with

T₂=

bT/kc

X

n=0

X

K∈M

θ_K^(R)φm(θ_K^(R)u⁽_Kⁿ⁺¹⁾)m(K)(u⁽_Kⁿ⁺¹⁾−u⁽_Kⁿ⁾),

T₃=

bT/kc

X

n=0

k X

K∈M

m(K)θ_K^(R)φm(θ⁽_K^R)u⁽_Kⁿ⁺¹⁾)u⁽_Kⁿ⁺¹⁾D_K,

T₄= 1 2

bT/kc

X

n=0

k X

K∈M

θ_K^(R)φm(θ_K^(R)u⁽_Kⁿ⁺¹⁾) X

L∈NK

m(K|L) wK,L−vK,L

(u⁽_Kⁿ⁺¹⁾−u⁽_Lⁿ⁺¹⁾), and

T5=k

bT/kc

X

n=0

X

K∈M

θ_K^(R)φm(θ_K^(R)u⁽_Kⁿ⁺¹⁾) m(K)λKu⁽_Kⁿ⁺¹⁾− X

L∈M

m(L)aL,Ku⁽_Lⁿ⁺¹⁾

! . We havebu⁽_Kⁿ⁺¹⁾ ≥ 0 and P

K∈Mm(K)bu⁽_Kⁿ⁺¹⁾ ≤ 1 from Lemma 2, and bu⁽_Kⁿ⁺¹⁾ = 0 for all K ∈ M such that K 6⊂ B(0,R+1+C2). The Taylor–Lagrange expansion formula yields that, for alla,b ∈ R, there exists

Ψ_a,b∈ [min(a,b),max(a,b)]such that φm(a)(a−b)=Φ_m(a)−Φ_m(b)+1

2φm⁰ (Ψ_a,b)(a−b)²≥Φ_m(a)−Φ_m(b). (31) Applying(31)fora =

bu⁽_Kⁿ⁺¹⁾andb=

bu⁽_Kⁿ⁾, we get that T₂≥ X

K∈M

m(K)Φ_m(bu⁽_K^bT/kc+1))− X

K∈M

m(K)Φ_m(bu⁽_K⁰⁾), (32)

with

0≤ X

K∈M

m(K)Φ_m(bu⁽_K⁰⁾)≤1 due toΦ_m(s)≤sfors≥0.

Thanks to(30)and 0≤φm(s)≤1 for alls≥0, we have

T3≥ −(T +C2)C4. (33)

Let us writeT4=T6+T7, with T₆=1

2

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L) wK,L −vK,L

φm(bu⁽_Kⁿ⁺¹⁾)(bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾), and

T₇=1 2

bT/kc

X

n=0

k X

K∈M

φm(bu⁽_Kⁿ⁺¹⁾) X

L∈NK

m(K|L) wK,L −vK,L

u⁽_Lⁿ⁺¹⁾(θ_L^(R)−θ_K^(R)).

Changing the order of summation, we get T₇=1

2

bT/kc

X

n=0

k X

L∈M

u⁽_Lⁿ⁺¹⁾ X

K∈N_L

φm(bu⁽_Kⁿ⁺¹⁾)m(K|L) wK,L−vK,L

(θ_L^(R)−θ_K^(R)).

Using(8), we get|θ_L^(R)−θ_K^(R)| ≤2h_M. Hence, using(7), we get

0≤ X

K∈N_L

φm(bu⁽ⁿ⁺¹_K ⁾)m(K|L) wK,L −vK,L

|θ_L^(R)−θ_K^(R)| ≤m(L)4V C1. This leads to

T7≥ −2V C1 bT/kc

X

n=0

k X

L∈M

m(L)u⁽ⁿ⁺¹⁾_L ≥ −(T +C2)2V C1. (34)

Let us apply(31)toT6. We getT6=T8+T9, with T₈=1

2

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L) wK,L −vK,L

(Φ_m(bu⁽_Kⁿ⁺¹⁾)−Φ_m(bu⁽_Lⁿ⁺¹⁾)).

Thanks to the expression ofφ⁰_m, we have T₉=m

4

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L) wK,L−vK,L (bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)² (1+Ψ

bu⁽ⁿ⁺¹⁾_K ,bu⁽ⁿ⁺¹⁾_L )^m+1. We remark thatwK,L−vK,L ≥0 and

(bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)² (1+Ψ

bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾)^m+1 ≥ (bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)² (1+max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾))^m+1,

which implies that T9≥ m

4

bT/kc

X

n=0

k X

K∈M

X

L∈NK

m(K|L) wK,L−vK,L (bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)² (1+max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾))^m+1. Gathering by edges the right-hand side of the above equation leads to

T9≥ m 4

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L)wK,L

(bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾)² (1+max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾))^m+1.

Thanks to(11), we also knowwK,L ≥ε, and the termT1defined by(26)now satisfies T₉≥ m

4εT₁. (35)

We next writeT₈=T₁₀+T₁₁, with T10=

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L) vK,L

Φ_m(bu⁽ⁿ⁺¹_K ⁾)+Φ_m(bu⁽ⁿ⁺¹_L ⁾)

2 +wK,L

2 (Φ_m(bu⁽ⁿ⁺¹_K ⁾)−Φ_m(bu⁽ⁿ⁺¹_L ⁾))

! ,

and

T₁₁= −

bT/kc

X

n=0

k X

K∈M

m(K)Φ_m(bu⁽_Kⁿ⁺¹⁾)D_K. Gathering by edges, we remark that

T10=0. (36)

We also get that

T₁₁≥ −(T +C₂)C₄. (37)

T₅ = k

bT/kc

X

n=0

X

K∈M

θ_K^(R)φm(θ⁽_K^R)u⁽_Kⁿ⁺¹⁾) m(K)λKu⁽_Kⁿ⁺¹⁾− X

L∈M

m(L)a_L,Ku⁽_Lⁿ⁺¹⁾

!

≥ −k

bT/kc

X

n=0

X

K∈M

θ⁽_K^R)φm(θ_K^(R)u⁽_Kⁿ⁺¹⁾) X

L∈M

m(L)a_L,Ku⁽_Lⁿ⁺¹⁾

≥ −Λ(T+C₂).

Thanks to the relationT₂+T₃+T₇+T₉+T₁₀+T₁₁+T₅=0, and to(32),(33),(34),(36),(37), and(35), we deduce the existence ofC₃, only depending onN,R,T,m,v,C₁,ε,ΛandC₂, such that(27)holds.

4. A weak BV inequality

In this section, we apply many times H¨older’s inequality X

i∈I

aibi ≤ X

i∈I

a_i^α

!_α¹ X

i∈I

b^β_i

!_β¹

, forai,bi ≥0, α, β >1 with 1 α +1

β =1, (38)

with various choices forI,ai,bi,αandβ that we define each time.

A weak BV inequality is provided in the next lemma. Such an inequality is valid for general families (bu⁽_Kⁿ⁾)K∈M,n∈Nand does not depend on the scheme. Its proof requires some Sobolev inequalities which are given further inLemmas 6 and7. Such Sobolev inequalities were already given in [9], with alternate assumptions and proofs however (see the introduction of this paper).

Lemma4 (AWeakBVInequality).LetN ∈ N^? andletMbeanadmissiblemeshofR^N inthesenseofDefinition1.

If N ≥ 2, let q ∈ (1,^N+2_N+1)be given and let m := [(2−q) (N+1) /N] −1 >0. If N =1, let q ∈ (1,√ 2)and m∈(0,^2−q_q²)be given. Let(bu⁽_Kⁿ⁾)K∈M,n∈Nbe a family of non-negative real numbers such thatP

K∈Mm(K)bu⁽_Kⁿ⁾≤1 for all n ∈Nand such that there exists R >0withbu⁽_Kⁿ⁾ =0, for all n∈ N, for all K ∈Msuch that K 6⊂B(0,R). Let T >0and k >0be given and let the terms T₁and T₁₂be respectively defined by(26)and

T₁₂=

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L)|

bu⁽_Kⁿ⁺¹⁾− bu⁽_Lⁿ⁺¹⁾|.

Let C₁be such that C_M ≤C₁, let C₃ be such that T₁ ≤C₃ and let C₂be such that h_M <C₂and k <C₂. Then there exists C₅, only depending on N , R, T , q, m, C₁, C₂and C₃ such that

T12≤C5h⁻¹_M^/q. (39)

Remark 5. Forε∈(0,V], we know from(27)inLemma 3that the assumptions of the previous lemma are valid and hence that(39)is true for the family(bu⁽ⁿ⁾_K )K∈M,n∈Nassociated to the scheme and constructed by(25).

Proof. In the course of this proof, we denote byCi, fori >5, various positive real numbers, only depending onN, R,T,q,C₁,C₂andC₃. Let us first defineT₁₃by

T13=h^2−q_M

bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L)|

bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾|^q.

According to some ideas of [4,3], our aim is to prove thatT13 is bounded independently ofh_M (inequality(44)), which then provides the conclusion(39), applying H¨older’s inequality, as we show at the end of this proof. Letθ^(R) be associated toRas inLemma 3. We denote byM_θ(R) the set of allK ∈Msuch thatθ_K^(R) 6=0 or such that there exists L ∈ N_K withθ_L^(R) 6= 0. We then apply H¨older’s inequality(38)withI = {(n,K,L), n = 0, . . . ,bT/kc, K ∈M_θ(R),L ∈N_K},α=2/q,β=2/(2−q),

an,K,L = km(K|L) (bu⁽ⁿ⁺¹⁾_K −

bu⁽ⁿ⁺¹⁾_L )² (1+max(bu⁽ⁿ⁺¹_K ⁾,bu⁽ⁿ⁺¹_L ⁾))^m+1

!q/2

, and

b_n,K,L =(h²_Mkm(K|L))^(2−q)/2

1+max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾)(m+1)q/2

. Hence we get

T₁₃≤(T₁)^q/2(T₁₄)^1−q/2, (40)

definingT₁₄by T14=h²_M

bT/kc

X

n=0

k X

K∈M_θ(R)

X

L∈N_K

m(K|L)

1+max(bu⁽ⁿ⁺¹⁾_K ,bu⁽ⁿ⁺¹⁾_L )₍m+1)q/(2−q)

.

Note that the expression ofT14would be meaningless if we write the sum onK ∈Minstead ofK ∈M_θ(R). Thanks to q ∈(1,2)andm>0, which implyr :=(m+1)q/(2−q) >1, and thanks to the inequality(a+b)^r ≤2^r−1(a^r+b^r) for allr≥1,a,b≥0, we get the existence ofC₆, only depending onN,mandq, such that

T14≤hMC6(T15+T16), (41)

definingT15andT16by T₁₅=h_M

bT/kc

X

n=0

k X

K∈M_θ(R)

X

L∈NK

m(K|L)≤C₇, (42)

withC7 =(T +C2)C1m(B(0,R+1C2))and T16=h_M

bT/kc

X

n=0

k X

K∈M_θ(R)

X

L∈NK

m(K|L)

max(bu⁽_Kⁿ⁺¹⁾,bu⁽_Lⁿ⁺¹⁾)(m+1)q/(2−q)

.

We can then write T16≤hM

bT/kc

X

n=0

k X

K∈M_θ(R)

X

L∈N_K

m(K|L) bu⁽_Kⁿ⁺¹⁾

₍m+1)q/(2−q)

+

bu⁽_Lⁿ⁺¹⁾

₍m+1)q/(2−q) ,

which leads, thanks to(7), to T16≤2C1

bT/kc

X

n=0

k X

K∈M_θ(R)

m(K) bu⁽ⁿ⁺¹_K ⁾

₍m+1)q/(2−q)

.

Let us now consider two cases:

(1) CaseN ≥2: thanks to the definition ofm, we know(m+1)q/ (2−q)=q(N+1) /N. Let us apply H¨older’s inequality(38)withI =M_θ(R), and

a_K =(m(K)bu⁽_Kⁿ⁺¹⁾)^q/N, b_K =m(K)^1−q/N(u⁽_Kⁿ⁺¹⁾)^q,

α=N/q,β=N/(N−q)(this is possible sinceN ≥2) andq^?=N q/(N−q). We get X

K∈M_θ(R)

m(K)

bu⁽_Kⁿ⁺¹⁾(m+1)q/(2−q)

≤



 X

K∈M_θ(R)

m(K)bu⁽_Kⁿ⁺¹⁾





q N 

 X

K∈M_θ(R)

m(K)

bu⁽_Kⁿ⁺¹⁾q^?





q q?

. (2) CaseN =1: let us defineq^?,αandβby 1/α+1/β =1, 1/α+q^?/β =(m+1)q/ (2−q)and ¹_β = ^q

q^?. This leads toq^? =q/(1+q−(m+1)q/(2−q))withq^? ∈(q,+∞), sinceq < (m+1)q/(2−q) <1+q due to m∈(0,^2−q_q²)withq ∈(1,√

2). Therefore, setting

a_K =(m(K)bu⁽_Kⁿ⁺¹⁾)^1/α, b_K =m(K)^1/β(u⁽_Kⁿ⁺¹⁾)^q?/β, we get

X

K∈Mθ(R)

m(K)

bu⁽_Kⁿ⁺¹⁾(m+1)q/(2−q)

≤



 X

K∈Mθ(R)

m(K)bu⁽_Kⁿ⁺¹⁾





1/α

 X

K∈Mθ(R)

m(K)

bu⁽_Kⁿ⁺¹⁾q^?





q q?

. In both cases, thanks to the hypothesisP

K∈Mm(K)bu⁽_Kⁿ⁺¹⁾≤1, we obtain T16≤2C₁

bT/kc

X

n=0

k



 X

K∈M_θ(R)

m(K) bu⁽ⁿ⁺¹_K ⁾

q^?





q/q^?

.

We now applyLemma 6or7to the values(bu⁽ⁿ⁺¹_K ⁾)K∈M: there existsC8, only depending onN,q,m,RandC1 such that



 X

K∈M_θ(R)

m(K)

bu⁽_Kⁿ⁺¹⁾q^?





q/q^?

≤C₈h^1−q_M X

K∈M

X

L∈N_K

m(K|L)|

bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾|^q, which leads to

T16≤h^1−q_M 2C1C8 bT/kc

X

n=0

k X

K∈M

X

L∈NK

m(K|L)|

bu⁽_Kⁿ⁺¹⁾−

bu⁽_Lⁿ⁺¹⁾|^q,

and therefore

h_MT₁₆ ≤2C₁C₈T₁₃. (43)

Hence we get from the hypothesisT1≤C3,(40)–(43) T₁₃≤(C₃)^q/2(C₆(h_MC₇ +2C₁C₈T₁₃))^1−q/2.

This leads, thanks tohM≤C2, to the existence ofC9, such that

T₁₃≤C₉. (44)

We now apply H¨older’s inequality(38)toT12withI = {(n,K,L), n =0, . . . ,bT/kc,K ∈M, L ∈NK},α=q, β =q/(q−1),

a_n,K,L =(h^2−q_M km(K|L))^1/q|

bu⁽_Kⁿ⁺¹⁾− bu⁽_Lⁿ⁺¹⁾|, and

b_n,K,L =h⁻¹_M^/q(h_Mkm(K|L))^(q−1)/q. Hence we get

T12≤h⁻¹_M^/qT₁₃^1/q hM bT/kc

X

n=0

k X

K∈M

X

L∈N_K

m(K|L)

!¹⁻¹/q

.

Applying(42)and(44), we get T12≤h⁻¹_M^/qC91/q(C7)^1−1/q, which implies(39).

Lemma 6 (General Discrete Sobolev Inequality, Case N =1).Let N =1and letMbe an admissible mesh in the sense of Definition1. Let(bu_K)K∈Mbe a family of real numbers such that the number of K ∈Msuch thatbu_K 6=0 is finite, let A>0such that

X

K∈M,bu_K6=0

m(K)≤ A,

and let q ∈(1,+∞)and q^? ∈(1,+∞)be given. Let C1 >CMbe given. Then there exists C10, only depending on N , q, q^?, A and C1, such that

X

K∈M

m(K)| bu_K|^q?

!1/q^?

≤C₁₀ X

K∈M

X

L∈NK

m(K|L)h_M| buK−

buL|^q h^q_M

!¹_q

. (45)

Proof. We have, for allK¯ ∈M,

|buK¯| ≤ X

K∈M

X

L∈N_K

|buK− buL|. Hence we get



 X

K∈M¯

m(K¯)| buK¯|^q?





1/q^?

≤ A^1/q? X

K∈M

X

L∈N_K

|buK − buL|.

Recalling that m(K|L)=1 for allK ∈MandL ∈N_K becauseN =1, we have X

L∈N_K

1≤ C₁ hM

m(K)