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:
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:
As a consequence, we obtain that
The size of the mesh is defined by .
Let us define
Given a Lipschitz-continuous function on which satisfies
we consider the B-scheme defined by
where the numerical fluxes are defined by
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
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.
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.
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
Then, we introduce the -th energy
When there is no ambiguity we write
. The first proposition is a fundamental estimate of the energy.
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
First, observe that is the discrete counterpart of
where is the indicator function of . Let us define , which satisfies and let us introduce another discrete counterpart of the preceding quantity
It is clear that for all , as for all we have
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,
Using that is bounded by 1 and vanishes on , we deduce that
Observe that since is a non-decreasing function, one has
Therefore, using the definition of we obtain that
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
But, and applying the definition of , we get
We apply now Cauchy-Schwarz inequality in order to get
where is the set of interior and Dirichlet boundary edges on which . It appears that, due to (3),
which yields (10) using Young’s inequality and the bounds and .
Before stating the main result of the section, we need a technical lemma.
Let be a sequence of non-negative real numbers and let and . Then if for all
for all and the bound is optimal. In particular, if , then .
Just observe that the sequence satisfies for all which directly yields the result.
Assume that for all and for all , so that for all . Then, there exists depending only on , and such that one has the implication
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:
We may choose for instance and apply a discrete Poincaré-Sobolev inequality (whose constant depends only on and ), which leads to
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.
Now observe that . Therefore,
Therefore, the smallness condition of Proposition 2 is satisfied by if
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.
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