# L^∞ bounds for numerical solutions of noncoercive convection-diffusion equations

In this work, we apply an iterative energy method à la de Giorgi in order to establish L^∞ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions.

## Authors

• 5 publications
• 5 publications
12/26/2021

### Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping mesh with non-periodic boundary conditions

A new local discontinuous Galerkin (LDG) method for convection-diffusion...
05/09/2019

### Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods

We introduce an efficient boundary-adapted spectral method for peridynam...
11/11/2019

### The waiting time phenomenon in spatially discretized porous medium and thin film equations

Various degenerate diffusion equations exhibit a waiting time phenomenon...
01/29/2021

### On the relation of powerflow and Telegrapher's equations: continuous and numerical Lyapunov stability

In this contribution we analyze the exponential stability of power netwo...
01/10/2020

### Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

A method of estimating all eigenvalues of a preconditioned discretized s...
11/12/2019

### Large time behavior of nonlinear finite volume schemes for convection-diffusion equations

In this contribution we analyze the large time behavior of a family of n...
08/06/2020

### Numerical Methods for a Diffusive Class Nonlocal Operators

In this paper we develop a numerical scheme based on quadratures to appr...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

The continuous problem. Let be an open bounded polygonal domain of with or . We denote by both the Lebesgue and dimensional Hausdorff measure. We assume that with and and we denote by the exterior normal to . Let be a velocity field, assumed to be nonnegative, a source term and a boundary condition.

We consider the following convection-diffusion equation with mixed boundary conditions:

 div(−∇v+Uv)+bv=f in Ω, (1a) (−∇v+Uv)⋅n=0 on ΓN, (1b) v=vD on ΓD. (1c)

This noncoercive elliptic linear problem has been widely studied by Droniou and coauthors, even with less regularity on the data, see for instance droniou_potan_2002 ; DG_M2AN_2002 ; Droniou_jnm_2003 ; DGH_sinum_2003 . Nevertheless, up to our knowledge, the derivation of explicit bounds on numerical solutions has not been done in the literature.

The numerical scheme. The mesh of the domain is denoted by and classically given by: , a set of open polygonal or polyhedral control volumes; , a set of edges or faces; a set of points. In the following, we also use the denomination “edge” for a face in dimension . As we deal with a Two-Point Flux Approximation (TPFA) of convection-diffusion equations, we assume that the mesh is admissible in the sense of Eymard2000 (Definition 9.1).

We distinguish in the interior edges, , from the exterior edges: . Among the exterior edges, we distinguish the edges included in from the edges included in : . For a given control volume , we define the set of its edges, which is also split into . For each edge , we pick one cell in the non empty set and denote it by . In the case of an interior edge , is either or .

Let denote the Euclidean distance. For all edges , we set if and if with and the transmissibility coefficient is defined by , for all . We also denote by the normal to outward . We assume that the mesh satisfies the regularity constraint:

 ∃ξ>0 such that d(xK,σ)≥ξdσ,∀K∈T,∀σ∈EK. (2)

As a consequence, we obtain that

 ∑σ∈EKm(σ)dσ≤pξm(K)∀K∈T. (3)

The size of the mesh is defined by .

Let us define

 fK=1m(K)∫Kf,bK=1m(K)∫Kb∀K∈T, UK,σ=1m(σ)∫σU⋅nK,σ,∀K∈T, ∀σ∈EK, vDσ=1m(σ)∫σvD,∀σ∈ED.

Given a Lipschitz-continuous function on which satisfies

 B(0)=1, B(s)>0 and B(s)−B(−s)=−s∀s∈R, (4)

we consider the B-scheme defined by

 ∑σ∈EKFK,σ+m(K)bKvK=m(K)fK,∀K∈T, (5)

where the numerical fluxes are defined by

 FK,σ=⎧⎨⎩0,∀K∈T,∀σ∈ENK,τσ(B(−UK,σdσ)vK−B(UK,σdσ)vK,σ),∀K∈T,∀σ∈EK∖ENK, (6)

with the convention if and if . Let us recall that the upwind scheme corresponds to the case ( is the negative part of , while is its positive part) and the Scharfetter-Gummel scheme to the case . They both satisfy (4). The centered scheme which corresponds to does not satisfy the positivity assumption. It can however be used if for all and . Thanks to the hypotheses (4), we notice that the numerical fluxes through the interior and Dirichlet boundary edges rewrite

 FK,σ=τσB(|UK,σ|dσ)(vK−vK,σ)+m(σ)(U+K,σvK−U−K,σvK,σ). (7)

Main result. The scheme (5)-(6) defines a linear system of equations whose unknown is ; It is well-known that is an M-matrix, which ensures existence and uniqueness of a solution to the scheme. Moreover, we may notice that, if and are nonnegative functions, then has nonnegative values and therefore for all . Our purpose is now to establish bounds on as stated in Theorem 1.1.

###### Theorem 1.1

Assume that , with a.e., and . There exists non-negative constants (resp. ) depending only on , , the function , , and (resp. and ) such that the solution to the scheme (5)-(6) verifies

 −M––– ≤ vK ≤ ¯¯¯¯¯¯M,∀K∈T.

The rest of this paper is dedicated to the proof of Theorem 1.1. It relies on a De Giorgi iteration method (see Vasseur_lectnotes and references therein). In Section 2, we start by studying a particular case where the data is normalized. Then, we give the proof of the theorem in Section 3.

Let us mention that from the bounds of Theorem 1.1, it is possible to establish global-in-time bounds for the corresponding evolution equation by using an entropy method (see (chainais_2019_large, , Theorem 2.7)).

## 2 Study of a particular case

In this section, we consider the particular case where the source is non-negative and the boundary condition is non-negative and bounded by .

Let us start with some notations. Given , we denote the -th truncation threshold by

 Cm=2(1−2−m), (8)

Then, we introduce the -th energy

 Em(v)=∑σ∈Eint∪EDτσ[log(1+(vK,σ−Cm)+)−log(1+(vK−Cm)+)]2. (9)

When there is no ambiguity we write

. The first proposition is a fundamental estimate of the energy.

###### Proposition 1

Assume that for all and for all , so that the solution to (5)-(6) satisfies for all . Then one has for all that

 Em ≤ 4pβ2U(∥U∥2L∞+∥f∥L∞)∑K∈TvK>Cmm(K). (10)

where (because of (4), ).

###### Proof

In order to shorten some expressions hereafter, let us introduce for all and for all . Let us note that we identify and the associate piecewise constant function. Therefore, we can write

 m({wm>0})=∑wmK>0m(K).

First, observe that is the discrete counterpart of

 ∫Ω|∇log(1+wm)|21{wm>0}=∫Ω∇wm⋅∇wm(1+wm)21{wm>0},  with wm=v−Cm,

where is the indicator function of . Let us define , which satisfies and let us introduce another discrete counterpart of the preceding quantity

 Fm=∑σ∈Eint∪EDτσ((wmK,σ)+−(wmK)+)(φ(wmK,σ)−φ(wmK)).

It is clear that for all , as for all we have

 (log(1+x+)−log(1+y+))2≤(x+−y+)(φ(x)−φ(y)).

Let us now multiply the scheme (5) by and sum over . Due to the non-negativity of and , we obtain, after a discrete integration by parts,

 ∑σ∈Eint∪EDFK,σ(φ(wmK)−φ(wmK,σ))≤∑K∈Tm(K)fKφ(wmK).

Using that is bounded by 1 and vanishes on , we deduce that

 ∑σ∈Eint∪EDFK,σ(φ(wmK)−φ(wmK,σ))≤∥f∥L∞m({wm>0}). (11)

We focus now on the left-hand-side of (11). Due to (7) and the definition of , we can rewrite as

 FK,σ=τσB(|UK,σ|dσ)(wmK−wmK,σ)+m(σ)(U+K,σ(wmK+Cm)−U−K,σ(wmK,σ+Cm)).

Observe that since is a non-decreasing function, one has

 (x−y)(φ(x)−φ(y))≥(x+−y+)(φ(x)−φ(y)),∀x,y∈R.

Therefore, using the definition of we obtain that

 ∑σ∈Eint∪EDFK,σ(φ(wmK)−φ(wmK,σ))≥βUFm−Gm, (12)

with

 Gm=−∑σ∈Eint∪EDm(σ)(U+K,σ(wmK+Cm)−U−K,σ(wmK,σ+Cm))(φ(wmK)−φ(wmK,σ)).

For an interior edge, and play a symmetric role in the preceding sum. As for all and vanishes on , we can always assume that and an edge has a contribution in the sum if at least . Then, under these assumptions one has

 −m(σ)(U+K,σ(wmK+Cm)−U−K,σ(wmK,σ+Cm))(φ(wmK)−φ(wmK,σ))≤∥U∥L∞m(σ)(wmK,σ+Cm)(φ(wmK)−φ(wmK,σ)).

But, and applying the definition of , we get

 (wmK,σ+Cm)(φ(wmK)−φ(wmK,σ))≤2(wmK)+−(wmK,σ)+1+(wmK)+≤2(wmK)+−(wmK,σ)+√1+(wmK)+√1+(wmK,σ)+.

Therefore,

 Gm≤2∥U∥L∞∑σ∈Eint∪EDm(σ)|(wmK)+−(wmK,σ)+|√1+(wmK)+√1+(wmK,σ)+.

We apply now Cauchy-Schwarz inequality in order to get

 Gm≤2∥U∥L∞(Fm)1/2(∑σ∈Espm(σ)dσ)1/2, (13)

where is the set of interior and Dirichlet boundary edges on which . It appears that, due to (3),

 ∑σ∈Espm(σ)dσ≤∑K∈T;wmK>0⎛⎜⎝∑σ∈EK,int∪EDKm(σ)dσ⎞⎟⎠≤pξm({wm>0}). (14)

We deduce from (11), (12), (13) and (14) that

 βUFm≤2∥U∥L∞(Fm)1/2(pξm({wm>0}))1/2+∥f∥L∞m({wm>0}),

which yields (10) using Young’s inequality and the bounds and .

Before stating the main result of the section, we need a technical lemma.

###### Lemma 1

Let be a sequence of non-negative real numbers and let and . Then if for all

 un+1≤Kρnuαn,

one has

 0≤un≤(u0ρ1(α−1)2K1α−1)αnρ−n(α−1)+1(α−1)2K−1α−1

for all and the bound is optimal. In particular, if , then .

###### Proof

Just observe that the sequence satisfies for all which directly yields the result.

###### Proposition 2

Assume that for all and for all , so that for all . Then, there exists depending only on , and such that one has the implication

 E1≤ η β4U(∥U∥2L∞+∥f∥L∞)2⇒(vK≤2, ∀K∈T). (15)
###### Proof

The proof consists in establishing an induction property on which guarantees that if is small enough then . Then, as and thanks to the discrete Poincaré inequality, we deduce that

which implies for all .

For establishing the induction, first observe that as , for any we have:

 1{wm>0}≤(log(1+(wm−1)+))q(log(1+2−m+1))q1{wm−1>0}, (16)

and thus

 m({wm>0})≤1(log(1+2−m+1))q∑K∈Tm(K)(log(1+(wm−1K)+))q.

We may choose for instance and apply a discrete Poincaré-Sobolev inequality (whose constant depends only on and ), which leads to

 m({wm>0})≤1(log(1+2−m+1))3C(Ω)ξ3/2E3/2m−1. (17)

Noticing that for , , we deduce from (10) and (17) that

 Em≤4β2U(∥U∥2L∞+∥f∥L∞)~CΩ,pξ3/28m−1E3/2m−1.

Thus the sequence satisfies the hypothesis of Lemma 1 with and proportional to . We deduce the upper bound for under which .

Remark: The arguments developed in this section still hold, up to minor adaptation, for with .

## 3 Proof of Theorem 1.1

First observe that if one replaces the data and by either and , or and , in the scheme (5)-(6), then the corresponding solutions, say respectively and , are non-negative and such that is the solution to (5)-(6) in the original framework.

From there let us show that there is such that for all one has . The bound for , which is denoted by , can be obtained in the same way.

Let . First observe that satisfies the scheme (5)-(6) where the source term and boundary data have been replaced by and respectively. Moreover, one can apply Proposition 1, which yields

 E1(PM)≤4pβ2U(∥U∥2L∞+∥f+∥L∞M)m({PM>1}). (18)

Now observe that . Therefore,

 E1(PM)≤4pβ2U(∥U∥2L∞m({PVD+>M/VD+})+∥f+∥L∞Mm(Ω))≤4pβ2U(∥U∥2L∞∑K∈Tm(K)log(1+(PVD+K−1)+)2log(M/VD+)2+∥f+∥L∞Mm(Ω))≤CΩ,pξβ2U∥U∥2L∞E1(PVD+)log(M/VD+)2+4pm(Ω)β2U∥f+∥L∞M,

where we used an argument similar to (16) in the second inequality and a discrete Poincaré inequality in the third one. Then, by using (18) again we get

 E1(PVD+) ≤ 4pm(Ω)β2U(∥U∥2L∞+∥f+∥L∞VD+)

Therefore, the smallness condition of Proposition 2 is satisfied by if

 ⎡⎣∥U∥2L∞(∥U∥2L∞+∥f+∥L∞VD+)+∥f+∥L∞Mlog(MVD+)2⎤⎦(∥U∥2L∞+∥f+∥L∞M)2 ≤ CΩ,ξ,pβ4Ulog(MVD+)2. (19)

It is clear that (19) is satisfied for large enough, which permits to define . Observe that if () and , works as expected.

Acknowledgements. The authors thank the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR MOHYCON (ANR-17-CE40-0027-01) for their support. They also want to thank Alexis F. Vasseur for fruitful exchanges on the subject.

## Bibliography

• (1) Chainais-Hillairet, C., Herda, M.: Large-time behaviour of a family of finite volume schemes for boundary-driven convection-diffusion equations. IMA J. Numer. Anal. (2019).
• (2) Droniou, J.: Non-coercive linear elliptic problems. Potential Anal. 17(2), 181–203 (2002)
• (3) Droniou, J.: Error estimates for the convergence of a finite volume discretization of convection-diffusion equations. J. Numer. Math. 11(1), 1–32 (2003).
• (4) Droniou, J., Gallouët, T.: Finite volume methods for convection-diffusion equations with right-hand side in . M2AN Math. Model. Numer. Anal. 36(4), 705–724 (2002)
• (5) Droniou, J., Gallouët, T., Herbin, R.: A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal. 41(6), 1997–2031 (2003)
• (6) Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)
• (7) Vasseur, A.F.: The De Giorgi method for elliptic and parabolic equations and some applications.

In: Lectures on the analysis of nonlinear partial differential equations., vol. 4 (2016)