Abstract
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. Key words: finite volume schemes, uniform bounds, noncoercive elliptic equations MSC (2010): 65M08, 35B40.
Abstract
Observe that since ϕ is a non-decreasing function, one has
Abstract
Therefore, using the definition of β U we obtain that
Abstract
with
Abstract
For an interior edge, w m K and w m K,σ play a symmetric role in the preceding sum. As w m,D σ ≤ 0 for all σ ∈ E D and ϕ vanishes on R − , we can always assume that w m K ≥ w m K,σ and an edge has a contribution in the sum if at least w m K > 0. Then, under these assumptions one has
Abstract
But, w m K,σ + C m ≤ 2(1 + (w m K,σ ) + ) and applying the definition of ϕ, we get
Introduction
The continuous problem. Let Ω be an open bounded polygonal domain of R p with p = 2 or 3. We denote by m(·) both the Lebesgue and p − 1 dimensional Hausdorff measure. We assume that ∂ Ω = Γ D ∪Γ N with Γ D ∩Γ N = / 0 and m(Γ D ) > 0 and we denote by n the exterior normal to ∂ Ω . Let U ∈ C(Ω ) 2 be a velocity field, b ∈ L ∞ (Ω ) assumed to be nonnegative, f ∈ L ∞ (Ω ) a source term and v D ∈ L ∞ (Γ D ) a boundary condition.
Introduction
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 [2, 4, 3, 5] . Nevertheless, up to our knowledge, the derivation of explicit L ∞ bounds on numerical solutions has not been done in the literature.
Introduction
The numerical scheme. The mesh of the domain Ω is denoted by M = (T , E , P) and classically given by: T , a set of open polygonal or polyhedral control volumes; E , a set of edges or faces; P = (x K ) K∈T a set of points. In the following, we also use the denomination "edge" for a face in dimension 3. 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 [6] (Definition 9.1). We distinguish in E the interior edges, σ = K|L, from the exterior edges: E = E int ∪ E ext . Among the exterior edges, we distinguish the edges included in Γ D from the edges included in Γ N : E ext = E D ∪ E N . For a given control volume K ∈ T , we define E K the set of its edges, which is also split into
Introduction
For each edge σ ∈ E , we pick one cell in the non empty set {K : σ ∈ E K } and denote it by K σ . In the case of an interior edge σ = K|L, K σ is either K or L.
Introduction
Let d(·, ·) denote the Euclidean distance. For all edges σ ∈ E , we set
Introduction
and the transmissibility coefficient is defined by τ σ = m(σ )/d σ , for all σ ∈ E . We also denote by n K,σ the normal to σ ∈ E K outward K. We assume that the mesh satisfies the regularity constraint:
Introduction
As a consequence, we obtain that
Introduction
The size of the mesh is defined by h = max{diam (K) : K ∈ T }.
Introduction
Let us define
Introduction
Given a Lipschitz-continuous function on R which satisfies
Introduction
we consider the B-scheme defined by
Introduction
where the numerical fluxes are defined by
Introduction
Let us recall that the upwind scheme corresponds to the case B(s) = 1 + s − (s − is the negative part of s, while s + is its positive part) and the Scharfetter-Gummel scheme to the case B(s) = s/(e s − 1). They both satisfy (4) . The centered scheme which corresponds to B(s) = 1 − s/2 does not satisfy the positivity assumption. It can however be used if |U K,σ |d σ ≤ 2 for all K ∈ T and σ ∈ E K . Thanks to the hypotheses (4), we notice that the numerical fluxes through the interior and Dirichlet boundary edges rewrite
Introduction
Main result. The scheme (5)-(6) defines a linear system of equations Mv = S whose unknown is v = (v K ) K∈T ; It is well-known that M is an M-matrix, which ensures existence and uniqueness of a solution to the scheme. Moreover, we may notice that, if v D and f are nonnegative functions, then S has nonnegative values and therefore v K ≥ 0 for all K ∈ T . Our purpose is now to establish L ∞ bounds on v as stated in Theorem 1.
Introduction
. There exists non-negative constants M (resp. M) depending only on Ω , ξ , the function B,
Introduction
The rest of this paper is dedicated to the proof of Theorem 1. It relies on a De Giorgi iteration method (see [7] 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.
Introduction
Let us mention that from the bounds of Theorem 1, it is possible to establish global-in-time L ∞ bounds for the corresponding evolution equation by using an entropy method (see [1, Theorem 2.7] ).
Study of a particular case
In this section, we consider the particular case where the source f is non-negative and the boundary condition v D is non-negative and bounded by 1.
Study of a particular case
Let us start with some notations. Given m ≥ 1, we denote the m-th truncation threshold by
Study of a particular case
Then, we introduce the m-th energy
Study of a particular case
When there is no ambiguity we write E m = E m (v). The first proposition is a fundamental estimate of the energy.
Study of a particular case
Proposition 1. Assume that f K ≥ 0 for all K ∈ T and v D σ ∈ [0, 1] for all σ ∈ E D , so that the solution v to (5)-(6) satisfies v K ≥ 0 for all K ∈ T . Then one has for all m ≥ 1 that
Study of a particular case
Proof. In order to shorten some expressions hereafter, let us introduce w m
Study of a particular case
Let us note that we identify w m = (w m K ) K∈T and the associate piecewise constant function. Therefore, we can write
Study of a particular case
First, observe that E m is the discrete counterpart of 
Study of a particular case
It is clear that E m ≤ F m for all m ≥ 1, as for all x, y ∈ R we have
Study of a particular case
Let us now multiply the scheme (5) by ϕ(w m K ) and sum over K ∈ T . Due to the non-negativity of b and v, we obtain, after a discrete integration by parts,
Study of a particular case
Using that ϕ is bounded by 1 and vanishes on R − , we deduce that
Study of a particular case
We focus now on the left-hand-side of (11). Due to (7) and the definition of w m K , we can rewrite F K,σ as
Study of a particular case
We apply now Cauchy-Schwarz inequality in order to get
Study of a particular case
where E sp is the set of interior and Dirichlet boundary edges on which (w m K ) + − (w m K,σ ) + = 0. It appears that, due to (3),
Study of a particular case
We deduce from (11), (12), (13) and (14) that
Study of a particular case
which yields (10) using Young's inequality and the bounds E m ≤ F m and β U ≤ 1.
Study of a particular case
Before stating the main result of the section, we need a technical lemma. Proof. Just observe that the sequence v n = u n ρ n(α−1)+1
Study of a particular case
for all n ≥ 0 which directly yields the result.
Study of a particular case
Proposition 2. Assume that f K ≥ 0 for all K ∈ T and v D σ ∈ [0, 1] for all σ ∈ E D , so that v K ≥ 0 for all K ∈ T . Then, there exists η > 0 depending only on Ω , p and ξ such that one has the implication
Study of a particular case
Proof. The proof consists in establishing an induction property on E m which guarantees that if E 1 is small enough then lim E m = 0. Then, as limC m = 2 and thanks to the discrete Poincaré inequality, we deduce that
Study of a particular case
For establishing the induction, first observe that as C m = C m−1 + 2 −m+1 , for any q > 0 we have:
Study of a particular case
and thus
Study of a particular case
We may choose for instance q = 3 and apply a discrete Poincaré-Sobolev inequality (whose constant C Ω ,p depends only on Ω and p), which leads to
Study of a particular case
Noticing that for x ∈ [0, 1], (log(1 + x)) 3 ≥ (log 2) 3 x 3 , we deduce from (10) and (17) that
Study of a particular case
Thus the sequence (E m ) m≥0 satisfies the hypothesis of Lemma 1 with α = 3/2 and K proportional to
Study of a particular case
We deduce the upper bound for E 1 under which lim E m = 0.
Study of a particular case
Remark: The arguments developed in this section still hold, up to minor adaptation, for f ∈ L r (Ω ) with r > p/2.
Proof of Theorem 1
First observe that if one replaces the data f and v D by either f + and (v D ) + , or f − and (v D ) − , in the scheme (5)-(6), then the corresponding solutions, say respectively P = (P K ) K∈T and N = (N K ) K∈T , are non-negative and such that v = P − N is the solution to (5)-(6) in the original framework.
Proof of Theorem 1
From there let us show that there is M > V D + := max( (v D ) + L ∞ , 1) such that for all K ∈ T one has 0 ≤ P K ≤ M. The bound for N, which is denoted by M, can be obtained in the same way.
Proof of Theorem 1
Let M > V D + . First observe that P M := P/M satisfies the scheme (5)-(6) where the source term and boundary data have been replaced by f + /M and (v D ) + /M respectively. Moreover, one can apply Proposition 1, which yields
Proof of Theorem 1
Now observe that P = M P M = V D + P V D + . Therefore,
Proof of Theorem 1
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
Proof of Theorem 1
Therefore, the smallness condition of Proposition 2 is satisfied by
Proof of Theorem 1
It is clear that (19) is satisfied for M large enough, which permits to define M.
Proof of Theorem 1
Observe that if v D + = 0 (V D + = 1) and U = 0, M = C Ω ,ξ ,p f + L ∞ works as expected.

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