On the Discretization of the Coupled Heat and Electrical Diffusion Problems

Abstract

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with
the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear
term which lies in L
1. A finite volume scheme is proposed for the discretization of the system; we show that the approximate solution obtained
with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system.

Full text

Page 1

On the Discretization of the Coupled Heat and
Electrical Diffusion Problems
Abdallah Bradji1and Rapha` ele Herbin2
1Weierstrass Institute for Applied Analysis and Stochastics,
Mohrenstr. 39 10117 Berlin Germany
bradji@wias-berlin.de
http://www.wias-berlin.de/~bradji
2Laboratoire d’Analyse, Topologie et Probabilit´ es,
Universit´ e Aix–Marseille 1, 39 rue Joliot Curie 13453 Marseille, France
raphaele.herbin@latp.univ-mrs.fr
http://www.cmi.univ-mrs.fr/~herbin
Abstract. We consider a nonlinear system of elliptic equations, which
arises when modelling the heat diffusion problem coupled with the elec-
trical diffusion problem. The ohmic losses which appear as a source term
in the heat diffusion equation yield a nonlinear term which lies in L1. A
finite volume scheme is proposed for the discretization of the system; we
show that the approximate solution obtained with the scheme converges,
up to a subsequence, to a solution of the coupled elliptic system.
Keywords: Nonlinear elliptic system, Diffusion equation, Finite volume
scheme, L1-data, Ohmic losses.
1Introduction
It is well known that the diffusion of electricity in a resistive medium induces
some heating, known as ohmic losses. Such a situation arises for instance in the
modelling of fuel cells, see e.g. [16], [17] and references therein. Let φ denote
the electric potential, and let κ denote the electrical conductivity. Then the
ohmic losses may be written as κ∇φ · ∇φ. Since φ is the solution of a diffusion
equation, it is reasonable to seek φ in the space H1(Ω), so that ∇φ · ∇φ ∈ L1.
Hence the heat diffusion equation has a right hand–side in L1, and its analysis
falls out of the usual variational framework. Our aim in this paper is to study the
convergence of approximate solutions to the resulting coupled problem obtained
with a cell centred finite volume scheme.
The theory of elliptic and parabolic equations with irregular right–hand–side
goes back to the pioneering work of G. Stampacchia [23], where solutions to the
linear problem are defined by duality. Later on, L. Boccardo, T. Gallou¨ et and co-
authors (see [3] and references therein) introduced the tools and setting in which
one may define solutions to such problems: these so-called entropy solutions [4]
were found to be equivalent to the so-called renormalized solutions of P.-L. Lions
andF.Murat[22],aswellastothesolutionsobtainedbyapproximation,asdefined
by [10]. In the linear case, all these solutions are also equivalent to those of [23].
T. Boyanov et al. (Eds.): NMA 2006, LNCS 4310, pp. 1–15, 2007.
c ? Springer-Verlag Berlin Heidelberg 2007

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2A. Bradji and R. Herbin
Other solutions obtained by approximation were defined thanks to numeri-
cal schemes. They also lead to the existence existence of a solution, but more
importantly, they yield a constructive way to compute approximate solutions of
the problem. The convergence of the finite volume scheme was proven in [20] for
the Laplace equation with right–hand–side measure; the proof was generalized in
[11] to noncoercive convection diffusion problems. The convergence of the finite
element scheme, with irregular data, on bi-dimensional polygonal domains was
proven for Delaunay triangular meshes in [18] and in [6] for three–dimensional
tetrahedral meshes under geometrical conditions. Under regularity assumptions
on the solutions, error estimates may be obtained by interpolation [8], [6]. The
convergence order of finite element solutions is also studied in [24] for ellip-
tic boundary value problems when the second member is piecewise smooth but
discontinuous along some curve.
In the present paper, we recall some of the properties which have been estab-
lished for the discretization of elliptic problems by finite volumes. We then show
how the techniques introduced in the above references may be used to prove
the convergence of a discretization scheme for the approximation of the above
mentioned heat and electricity diffusion problem, since the resulting system of
semilinear elliptic partial differential equations is such that the right-hand-side
of the second equation depends on the solution of the first one and is in L1.
The paper is organized as follows: in Section 2 we recall the main principle and
properties of finite volume schemes for elliptic problems; in Section 3, we present
the continuous problem, its weak form and the known result about existence [19].
In section 4, we describe the finite volume method for the approximation of the
system and prove the existence of a solution to the resulting discrete system for
both cases. The convergence of the finite volume cases in section 5. The proof of
convergence is based on a priori estimates, compactness result and a passage to
the limit in the scheme. Some conclusions are drawn in the last section.
2 The Cell Centered Finite Volume Method
Finite volume methods are known to be well suited for the discretization of
conservation laws; these conservationlaws may yield partial differential equations
of different nature (elliptic, parabolic or hyperbolic) and also to coupled systems
of equations of different nature.
Let Ω be a polygonal open subset of Rd, T ∈ R, and let us consider a balance
law written under the general form:
ut+ div(F(u,∇u)) + s(u) = 0 on Ω × (0,T),
where F ∈ C1(R×Rd,Rd) and s ∈ C(R,R). Let T be a finite volume mesh of Ω.
For the time being, we shall only assume that T is a collection of convex polyg-
onal control volumes K, disjoint one to another, and such that:¯Ω = ∪K∈T¯K.
The balance equation is obtained from the above conservation law by integrating
it over a control volume K and applying the Stokes formula:
(1)

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems3
?
K
utdx +
?
∂K
F(u,∇u) · nKdγ(x) +
?
K
s(u)dx = 0,
where nKstands for the unit normal vector to the boundary ∂K outward to K
and dγ denotes the integration with respect to the (d−1)–dimensional Lebesgue
measure. Let us denote by E the set of edges (faces in 3D) of the mesh, and EK
the set of edges which form the boundary ∂K of the control volume K. With
these notations, the above equation reads:
?
σ∈EK
Let k = T/M, where M ∈ N,M ≥ 1, and let us perform an explicit Euler
discretization of the above equation (an implicit or semi-implicit discretization
could also be performed, and is sometimes preferable, depending on the type of
equation). We then get:
?
σ∈EK
where u(n)denotes an approximation of u(·,t(n)), with t(n)= nk. Let us then
introduce the discrete unknowns (u(n)
time step); assuming the existence of such a set of real values, we may define a
piecewise constant function by:
K
utdx +
?
?
σ
F(u,∇u) · nKdγ(x) +
?
K
s(u)dx = 0.
K
u(n+1)− u(n)
k
dx +
?
?
σ
F(u(n),∇u(n)) · nKdγ(x) +
?
K
s(u(n))dx = 0,
K)K∈T , n∈N (one per control volume and
u(n)
T
∈ X(T ) : u(n)
T
=
?
K∈T
u(n)
K1K,
where X(T ) denotes the space of functions from Ω to R which are constant
on each control volume of the mesh T , and 1K is the characteristic function of
the control volume K, where 1K(x) = 1 if x ∈ K and 1K(x) = 0 otherwise. In
order to define the scheme, the fluxes
approximated as a function of the discrete unknowns. We denote by FK,σ(u(n)
the resulting numerical flux, the expression of which depends on the type of flux
to be approximated.
?
σF(u(n),∇u(n)) · nKdγ(x) need to be
T)
The coupled system which is the aim of our study is a system of diffusion equa-
tions. Let us now consider a linear diffusion reaction equation, that is equation
(1) with F(u,∇u) = −∇u, and s(u) = bu, b ∈ R+:
ut− Δu + bu = 0 on Ω,
the flux through a given edge then reads:
?
so that we need to discretize the term?
(2)
σ
F(u) · nK,σ=
?
σ
−∇u · nK,σ,
σ−∇u · nK,σ; this diffusion flux involves
the normal derivative to the boundary, for which a possible discretization is

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4A. Bradji and R. Herbin
obtained by considering the differential quotient between the value of uT in K
and in the neighbouring control volume, let’s say L:
FK,σ(uT) = −m(σ)
dKL
where m(σ) stands for the (d −1)–dimensional Lebesgue measure of σ and dKL
is the distance between some points of K and L, which will be defined further.
We then obtain the following numerical flux:
FK,σ(uT) = −m(σ)
dKL
However, we are able to prove that this choice for the discretization of the diffu-
sion flux yields accurate results only if the mesh satisfies the so-called orthogo-
nality condition, that is, there exists a family of points (xK)K∈T, such that for
a given edge σKL, the line segment xKxLis orthogonal to this edge (see figure
1). The length dKL is then defined as the distance between xK and xL. This
geometrical feature of the mesh will be exploited to prove the consistency of the
flux. Of course, this orthogonality condition is not satisfied for any mesh. Such a
family of points exists for instance in the case of triangles, rectangles or Vorono¨ ı
meshes. We refer to [12] for more details.
(uL− uK).
(3)
(uL− uK).
xL
xK
L
K
m(σ)
σKL
dKL
dK,σ
Fig.1. Notations for a control volume
3The Continuous Coupled Problem
We wish to find some numerical approximation of solutions to the following non-
linear coupled elliptic system, which models the thermal and electrical diffusion
in a material subject to ohmic losses:
− ∇ · (κ(x,u(x))∇φ(x)) = f(x,u(x)), x ∈ Ω
φ(x) = 0, x ∈ ∂Ω,
−∇ · (λ(x,u(x))∇u(x)) = κ(x,u(x))|∇φ|2(x), x ∈ Ω,
u(x) = 0, x ∈ ∂Ω,
(4)
(5)
(6)
(7)

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems5
where Ω is a convex polygonal open subset of Rd, d = 2 or 3, with boundary ∂Ω,
φ denotes the electrical potential and u the temperature; the electrical conduc-
tivity κ, the thermal conductivity λ and the source term f are functions from
Ω × R to R satisfying the following Assumptions:
Assumption 1. The functions κ,λ and f, defined from Ω×R to R, are bounded
and continuous with respect to y ∈ R for a.e. x ∈ Ω, and measurable with respect
to x ∈ Ω for any y ∈ Ω, and such that:
∃α > 0; α ≤ κ(x,y) and α ≤ λ(x,y), ∀y ∈ R, for a.e. x ∈ Ω.
The following existence result was proven in [19]:
(8)
Theorem 1. Under Assumption 1, there exists a solution to the following weak
form of Problem (4)– (7):
⎧
⎪
where here and in the sequel, dx denotes the integration symbol with respect to
the Lebesgue measure in Rdor Rd−1.
Note that the exponents
W1,r
(9) make sense. In the case d = 2, we have u ∈ W1,p
u ?∈ H1
The proof of this theorem relies mainly on the analysis tools which were
developed for the analysis of elliptic equations with irregular right–hand–side,
see for instance [3] and references therein. We shall not need to assume this
existence result for our present analysis. Indeed, the existence of a solution to (4)–
(7) is obtained as a by–product of the convergence of the scheme. Nevertheless,
a large part of the convergence analysis of the schemes is inspired from the ideas
developed in [19] for the existence result, and we shall again use the ideas of [3]
in our proofs.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
(φ,u) ∈ H1
?
?
0(Ω) × ∩p<
d
d−1W1,p
?
?
0 (Ω),
Ω
κ(·,u)∇φ · ∇ψ dx =
λ(·,u)∇u · ∇v dx =
Ω
f(·,u)ψdx, ∀ψ ∈ H1
κ(·,u)|∇φ|2v dx, ∀v ∈ ∪r>dW1,r
0(Ω)
ΩΩ
0(Ω),
(9)
d
d−1and d are conjugate, ant that, for r > d the space
0 (Ω) is continuously imbedded in the space C(Ω,R); therefore all terms in
0
for all p < 2, but in general,
for all p <3
2.
0(Ω). Similarly, if d = 3, u ∈ W1,p
0
4The Discretization Schemes
In [17], the numerical simulation of solid oxide fuel cells relies on a mathematical
model involving a set of semilinear partial differential equations, the unknowns
of which are the temperature, the electrical potential and the concentrations of
various chemical species in the porous media of the cell. System (4)–(7) is a sub–
problem of this latter model, obtained by leaving out the chemical species diffu-
sion equations. In [17], three different discretization schemes were implemented
and compared, namely the linear finite element method, the mixed finite element

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6A. Bradji and R. Herbin
method, and the cell centred finite volume method. Because of interface condi-
tions involving the electrical current, a precise approximation of the electrical
flux is needed at the interfaces, the linear finite element method was found to be
less adapted than the two latter methods, so that finally the mixed finite element
method and the cell centered methods were numerically compared. The cell cen-
tred finite volume method was found to be easier to implement and comparable
to the mixed finite element method as to the ratio precision vs. computing time,
so that it was finally chosen for the simulations of different geometries of fuel
cells [16]. Here we shall give a theoretical justification of the convergence of and
the cell centred finite volume method for the discretization of system (4)–(7).
The convergence of a linear finite element scheme is proven in a forthcoming
paper [5].
To define a finite volume approximation, we introduce an admissible mesh T
in the sense of [12, Definition 9.1 page 762], which we recall here for the sake of
completeness:
Definition 1 (Admissible meshes). Let Ω be an open bounded polygonal sub-
set of Rd, d = 2 or 3. An admissible finite volume mesh of Ω, denoted by T , is
given by a family of “control volumes”, which are open polygonal convex subsets
of Ω , a family of subsets of Ω contained in hyperplanes of Rd, denoted by E
(these are the edges in two space dimensions, or faces in three space dimensions,
of the control volumes), with strictly positive (d − 1)-dimensional measure, and
a family of points of Ω satisfying the following properties:
(i) The closure of the union of all the control volumes is Ω.
(ii) For any K ∈ T , there exists a subset EK of E such that ∂K = K \ K =
∪σ∈EKσ. Furthermore, E = ∪K∈TEK.
(iii) For any (K,L) ∈ T2with K ?= L, either the (d − 1)-dimensional Lebesgue
measure of K ∩ L is 0 or K ∩ L = σ for some σ ∈ E, which will then be
denoted by σKL.
(iv) The family of points (xK)K∈T is such that xK ∈ K (for all K ∈ T ) and,
if σ = σKL, it is assumed that xK ?= xL, and that the straight line going
through xK and xLis orthogonal to σKL.
(v) For any σ ∈ E such that σ ⊂ ∂Ω, let K be the control volume such that
σ ∈ EK. We assume that if xK / ∈ σ, then the straight line going through xK
and orthogonal to σ intersects σ.
An example of two cells of such a mesh is given in Figure 1, along with some
notations. Item (iv) of the above definition was referred to in the previous section
as the “orthogonality property”:
We refer to [12] for a description of such admissible meshes, which include
triangular meshes, rectangular meshes, or Vorono¨ ı meshes. Here, for the sake of
simplicity, we assume that the points xK∈ K. The finite volume approximations
φT and uT of φ and u solution to (9) are sought in the space X(T ) of functions
from Ω to R which are constant over each control volume of the mesh, that is:
X(T ) = {u ∈ C(Ω,R);u|K∈ P0for all K ∈ T },
(10)

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems7
where P0 denotes the set of constant functions. The finite volume scheme is
classically obtained from the balance form of Equations (4) and (6) on a control
volume K, that is:
?
−
∂K
where nKdenotes the unit normal vector to ∂K outward to K and dx denotes the
integration symbol on d dimensional domain Ω or the d−1 dimensional boundary,
with respect to the Lebesgue measure. Let EK denote the set of edges or faces
of ∂K, decomposing the boundary of K into edges or faces, ∂K = ∪σ∈EKσ, we
may rewrite (11)-(12) as:
?
−
σ∈EK
Let us write the sought approximations as φT=?
fK(uK) =
m(K)
Let E denote the set of edges (or faces in 3D) of the mesh, and Eint(resp. Eext)
the set of edges laying in Ω (resp. on ∂Ω). For σ
(resp. Fλ
(resp.?
notation uT = (uK)K∈T and φT = (φK)K∈T . With these notations, a finite
volume approximation may then be written under the form:
⎧
⎪
provided one defines the expressions Fκ,λ
discrete unknowns (φK)K∈T and (uK)K∈T.
The discrete fluxes are consistent and conservative, and are given by the clas-
sical two–points formula:
?m(σ)τκ
?m(σ)τλ
−
?
∂K
(κ(·,u)∇φ) · nKdx =
?
?
K
f(·,u)dx
(11)
(λ(·,u) · ∇u) · nKdx =
K
κ(·,u)|∇φ|2dx,
(12)
−
σ∈EK
?
?
σ
(κ(·,u)∇φ) · nK,σdx =
?
?
K
f(·,u)dx
(13)
?
σ
(λ(·,u) · ∇u) · nK,σdx =
K
κ(·,u)|∇φ|2dx.
(14)
K∈TφK1K, uT=?
f(x,uK)dx.
K∈TuK1K;
we then set
1
?
K
(15)
∈E, let Fκ
K,σ
K,σ) be an approximation of the flux?
imation of the nonlinear right-hand-side
σ(κ(x,u(x))∇φ(x)) · nK,σdγ(x)
σ(λ(x,u(x))·∇u(x))·nK,σdγ(x)), and let JK(uT,φT) denote an approx-
1
m(K)
?
Kκ(x,u(x))|∇φ|2(x)dx, with the
⎪
⎪
⎪
⎩
⎨
?
?
σ∈EK
Fκ
K,σ= m(K)fK(uK), ∀K ∈ T ,
σ∈EK
Fλ
K,σ= m(K)JK(uT,φT), ∀K ∈ T ,
(16)
K,σand JK(uT,φT) with respect to the
Fκ
K,σ=
σ(uT)(φK− φL), if σ = K|L ∈ Eint,
m(σ)τκ
σ(uT)φKif σ ∈ EK∩ Eext,
(17)
Fλ
K,σ=
σ(uT)(uK− uL) if σ = K|L ∈ Eint,
m(σ)τλ
σ(uT)(uK), if σ ∈ EK∩ Eext,
(18)

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8A. Bradji and R. Herbin
where τκ
σ(and, similarly τλ
σ) is defined through a harmonic average, that is:
τκ
σ(uT) =
⎧
⎪
⎪
⎪
⎪
⎩
⎨
κK(uK)κL(uL)
dK,σκL(uL) + dL,σκK(uK)if σ = K|L ∈ Eint,
κK(uK)
dK,σ
, if σ ∈ Eext∩ EK,
(19)
where the values κK(uK) and λK(uK) are defined by (15), replacing f by κ or λ.
The term JK(uT,φT) is defined as:
1
m(K)
σ∈EK
where, for K ∈ T and σ ∈ EK, we define the half dual cell DK,σ delimited by
xKand σ (see Figure 1) by
JK(uT,φT) =
?
m(DK,σ)Jσ(uT,φT),
(20)
DK,σ= {txK+ (1 − t)x, (x,t) ∈ σ × (0,1)},
and
Jσ(uT,φT) =τκ
σ(uT)
dσ
(Dσφ)2d
(21)
with
Dσφ =
?|φK− φL| if σ ∈ Eint,
|φK| if σ ∈ Eext
We show in Theorem 2 below the existence of (φK)K∈T and (uK)K∈T solution to
(16)–(21). This entitles us to define the functions φT and uT ∈ X(T ) with respec-
tive values φK and uK on cell K, along with the function JT(uT,φT) ∈ X(T )
with value JK(uT,φT) on cell K. To prove the existence of a finite volume solu-
tion (φT,uT) to the problem (16)-(21), we use Brouwer’s theorem. To this end
we combine Lebesgue’s theorem with the fact that X(T ) is a finite dimensional
space.
Theorem 2. Let (κ,λ,f) be three functions satisfying the Assumption 1. Let
X(T ) be the finite volume space defined in Definition 2. Then there exists at
least a solution (uT,φT) ∈ (X(T ))2to the problem (16)-(21).
The proof is based on the fixed point theorem. In fact, the existence of a solution
to (9) was proven in [19] using Schauder’s fixed point theorem; here, since the
space is finite–dimensional, we need only use Brouwer’s theorem. The proof is
an easy adaptation of that of [19] so we omit it.
5Convergence of the Finite Volume Approximation
5.1The Convergence Result
In this section, we shall prove that a solution of (16)-(21) converges, as hT =
sup{diam(K),K ∈ T } tends to 0, towards a solution of (9), as stated in the
following theorem:

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems9
Theorem 3. Under Assumption 1, let (Tn)n∈N be a sequence of admissible
meshes in the sense of Definition 1. Let (φn,un) be a solution of the system
(16)-(20) for T = Tn, and let Jn(un,φn) be defined by (20). Assume that
hn = sup{diam(K),K ∈ Tn} → 0, as n → ∞, and that there exists ζ > 0
(not depending on n), such that:
dσ≤ ζdK,σ, ∀σ ∈ En, ∀K ∈ Tn.
(22)
Then, there exists a subsequence of (Tn)n∈N, still denoted by (Tn)n∈N, such that
(φn,un) converges to a solution (φ,u) ∈ H1
in the following sense:
0(Ω)×∩q<
d
d−1W1,q
0
of (9), as n → ∞,
?φn− φ?L2(Ω)→ 0, as n → +∞,
(23)
?un− u?Lp(Ω)→ 0, as n → +∞, for all p <
d
d − 2.
(24)
Moreover,
?
Ω
Jn(un,φn)(x)dx →
?
Ω
κ(x,u(x))|∇φ|2(x)dx as n → +∞.
(25)
Proof. For the sake of clarity, we only list here the main ingredients of the proof
and refer to lemmata proven below for the details.
Let (φn)n∈N⊂ L2(Ω) and (un)n∈N⊂ L2(Ω) be such that, for any n ∈ N, the
pair (φn,un) is a solution of (16)-(20), with T = Tn (recall that this solution
exists by Theorem 2).
From Lemma 1 below, we know that the sequences (φn)n∈Nand (un)n∈Nare
bounded for respectively, the L2norm and the Lqnorm, with q <
that condition (22) is needed here since it is required when using the discrete
Sobolev inequality (see e.g. [9]) to obtain the uniform bound of (un)n∈Nin an
Lqnorm from a discrete W1,p
0
estimate. Then, following [12] Lemma 9.3 page
770 or [13], one may easily get some uniform estimates on the translates of φn
in the L2norm and of unin the Lpnorm.
We may therefore use a discrete Rellich theorem (see e.g. [11], Proposition
2.3) to obtain that the sequences (φn)n∈N and (un)n∈N are relatively compact
in, respectively, L2(Ω) and Lp(Ω), for p <
also yield the regularity of the limit (Proposition 2.4 of [11]), that is, if φ is a
limit of the sequence (φn)n∈Nin L2(Ω), then φ ∈ H1
of the sequence (un)n∈Nin Lp(Ω), then u ∈ ∩q<
Hence, for any sequence (Tn)n∈N of admissible meshes satisfying (22) and
such that size(Tn) → 0, as n → ∞, there exists a subsequence, still denoted by
(Tn)n∈N, such that:
1. unconverges to some u ∈ ∩q<
therefore in L2(Ω), as n → ∞.
2. φnconverges to some φ ∈ H1
d
d−2. Note
d
d−2. The estimates on the tranlations
0(Ω); similarly, if u is a limit
d−1W1,q
d
0 (Ω).
d
d−1W1,q
0 (Ω) in Lp(Ω), for all p <
d
d−2, and
0(Ω), in L2(Ω), as n → ∞.

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10A. Bradji and R. Herbin
Thanks to the Lebesgue dominated theorem, we also get that κ(·,un) → κ(·,u)
and f(·,un) → f(·,u) as n → ∞, in the Lpnorm for any p <
One then obtains by a straightforward adaptation of the proof of Theorem 2
in [13], that the function φ ∈ H1
u) weak solution of the first equation of (9), that is:
?
A straightforward adaptation of the proof of the convergence of the discrete H1
norm in [12] (Theorem 9.1, proof page 776) yields that the discrete ohmic losses
converge to the continuous ones, that is:
?
d
d−2.
0(Ω) is the (unique, for the considered function
Ω
κ(x,u(x))∇φ(x) · ∇ψ(x)dx =
?
Ω
f(x,u(x))ψ(x)dx, ∀ψ ∈ H1
0(Ω).
0
?
σ∈E
m(σ)τκ
K|L(un
K)(φn
L− φn
K)2→
Ω
κ(x,u(x))|∇φ|2(x)dx as n → +∞,
which proves (25).
In order to prove (23) and (24), there now only remains to show that u satisfies
the second equation of (9). In order to do so, we proceed in a now classical way,
that is, we multiply the second equation of the scheme (16) by ψ(xK) where ψ is
a smooth function with compact support in Ω, we sum over K ∈ Tn, and obtain:
?
K∈T
?
σ∈E
m(σ)τλ
σ(un
K)(un
K− un
L)ψ(xK) =
?
K∈T
m(K)Jn
K(un,φn)ψ(xK)(26)
Let us now pass to the limit as n → +∞. A straightforward adaptation of the
proof of e.g. Theorem 2 in [13] yields that the left hand side of (26) tends to
−?
by density of C∞(Ω) we get that u satisfies
?
for all ψ ∈ ∪q>dW1,q
In the following sections, we shall derive the estimates and the intermediate
convergence results which were used in the above proof.
Ωu∇· (λ(·,u)∇ψ)(x)dx, as n → +∞. Moreover, we show in Lemma 2 below
that the right hand side of (26) tends to?
Ωκ(x,u(x))|∇φ|2(x)ψ(x)dx, so that,
−
Ω
λ(x,u(x))∇u(x) · ∇ψ(x)dx =
?
Ω
κ(x,u(x))|∇φ|2(x)ψ(x)dx
0 (Ω). This concludes the proof of the theorem.
5.2 Estimate on the Approximate Solutions and Compactness
Recall that the approximate finite volume solutions are piecewise constant; hence
they are not in the spaces W1,p, and we need therefore to define a discrete W1,p
norm (see also [9,12]) in order to obtain some compactness results.
Definition 2 (Discrete W1,pnorm). Let Ω be an open bounded subset of Rd,
d = 1,2 or 3, and let T be an admissible finite volume mesh in the sense of

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems11
Definition 1. For uT ∈ X(T ), uT =?
K∈TuK1K, and p ∈ [1,+∞),
?uT?1,p,T =
??
?|uK− uL| if σ ∈ Eint,
σ∈E
m(σ)dσ(Dσu
dσ
)p
?1
p
,
with the notation
Dσu =
|uK| if σ ∈ Eext
To prove the convergence of (φT,uT), we prove at first some estimates on φT
and uT.
Lemma 1. Under Assumption 1, let T be an admissible mesh in the sense of
Definition 1, and let ζT > 0 be such that:
dσ≤ ζT dK,σ, ∀σ ∈ E, and for any K ∈ T .
Let (φT,uT) be a solution of (16)-(20). Then there exists C1∈ R?
ing on Ω, d, ?f?L∞(Ω×R,R), ?κ?L∞(Ω×R,R), and α such that
?φT?1,2,X(T )≤ C1
(27)
+, only depend-
(28)
?φT?L2(Ω)≤ C1,
(29)
and
?JT(uT,φT)?L1(Ω)≤ C1.
d
d−1), there exists C2 ∈ R?
(30)
Moreover, for all q ∈ [1,
?f?L∞(Ω×R,R),?κ?L∞(Ω×R,R), ?λ?L∞(Ω×R,R), ζT, q, α such that
?uT?1,q,X(T )≤ C2,
and
+only depending on Ω, d,
(31)
?uT?Lp∗ ≤ C2.
(32)
Proof. The proof of (28) follows [12], Lemma 9.2 page 768 and the estimate
(29) is then obtained by the discrete Poincar´ e inequality [12] Lemma 9.1 page
765. Let us then prove the L1estimate (30) on the right hand side JT(uT,φT).
Indeed, by definition of JT(uT,φT),
?JK(uT,φT)?L1(Ω)=
K∈T
where Dσ denotes the “diamond cell” around σ, that is Dσ = DK,σ∪ DL,σ
if σ = K|L ∈ Eint, and Dσ = DK,σ if σ ∈ Eext∩ EK. From the definition of
Jσ(uT,φT), noting that m(Dσ) =1
obtains that:
?JK(uT,φT)?L1(Ω)≤?κ?
which proves (30). Since JT(uT,φT) ∈ L1(Ω), one obtains (31) by a straight-
forward adaptation of [20], Lemma 1 (see also [11], Theorem 2.2). The estimate
(32) follows from a discrete Sobolev inequality [9].
?
?
σ∈EK
m(DK,σ)Jσ(uT,φT) =
?
σ∈E
m(Dσ)Jσ(uT,φT),
dm(σ)dσ, and using Assumption 1, one then
α
?φT?1,2,T.

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12A. Bradji and R. Herbin
5.3Passage to the Limit on the L1Term
Lemma 2. Under Assumption 1, let (Tn)n∈Nbe a sequence of admissible meshes
in the sense of Definition 1. Let (φn,un) be a solution of the system (16)-(20)
for T = Tn, and let Jn(un,φn) ∈ X(Tn) be defined by (20). Assume that hn=
max{diam(K),K ∈ Tn} → 0, as n → ∞, and that there exists ζ > 0, not
depending on n, such that (22) holds. Assume that
1. unconverges to some u ∈ ∩q<
2. φnconverges to some φ ∈ H1
Let ψ ∈ C1
d
d−1W1,q
0(Ω), in L2(Ω), as n → ∞.
0(Ω) in L2(Ω), as n → ∞.
c(Ω,R), and let ψn∈ X(Tn) be defined by:
ψn(x) = ψ(xK), for a.e. x ∈ K, ∀K ∈ Tn.
Then:
?
Ω
Jn(un,φn)(x)ψn(x)dx →
?
Ω
κ(x,u(x))|∇φ|2(x)ψ(x)dx as n → +∞.(33)
Thus, Jn(un,φn) → |∇φ|2as n → +∞ for the weak ? topology of (C(Ω))
C(Ω)) denotes the set of continuous functions with compact support on Ω.
?, where
Proof. Assume that n is large enough so that the function ψ (which has a com-
pact support in Ω) vanishes on the cells neighbouring the boundary ∂Ω. Noting
that m(DK,σ) =1
?
K∈Tn
= Tn
dm(σ)dK,σ, one has:
Ω
Jn(un,φn)(x)ψn(x)dx =
?
7+ Tn
?
8,
σ∈EK
m(σ)dK,στκ
σ(un)(Dσφn)2
dσ
ψ(xK)
where
Tn
7=
?
K∈Tn
?
σ∈EK
m(σ)dK,στκ
σ(un)(Dσφn)2
dσ
ψ(xL),
(34)
and
Tn
8=
?
K∈Tn
?
σ=K|L∈EK
m(σ)dK,στκ
σ(un)(Dσφn)2
dσ
(ψ(xK) − ψ(xL)).
Since dK,σ ≤ dσ and |ψ(xK) − ψ(xL)| ≤ 2hn?∇ψ?(L∞(Ω))d and τκ
?κ?2
αdσ
σ(un) ≤
L∞(Ω×R,R)
, we have
|Tn
8| ≤ 2
?κ?2
L∞(Ω×R,R)
α
hn?∇ψ?(L∞(Ω))d?φn?2
1,2,X(T ).
Using (28) we then obtain that:
|Tn
8| → 0, as n → +∞.

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On the Discretization of the Coupled Heat and Electrical Diffusion Problems13
We turn now to the term Tn
side of (34), we get
7, reordering the sum on the edges in the right hand
Tn
7=
?
σ=K|L∈Eint
m(σ)τκ
σ(un)(Dσφn)2
dσ
ψσ
where
ψσ=dK,σψ(xL) + dL,σψ(xK)
dσ
, σ ∈ Eintand σ = K|L
(35)
We may then decompose Tn
7= Tn
9+ Tn
10, with
Tn
9= −
?
σ=K|L∈Eint
m(σ)τκ
σ(un)(φn
L− φn
K)(φn
Kψn
K− φn
Lψn
L),
(36)
and
Tn
10= −
?
σ=K|L∈Eint
m(σ)τκ
σ(un)((φn
L− φn
K)φn
K(ψσ− ψK) − (φn
L− φn
K)φn
L(ψσ− ψL)).
Reordering the sum of the right hand side of (36) on the control volumes,
using the fact that φnis the solution of the first equation of the finite volume
scheme (16), and that un(resp.φn) converges to u (resp. φ) in L2(Ω) as n → ∞,
we get that:
Tn
9→
?
Ω
κ(x,u(x))|∇φ|2(x)ψ(x)dx +
?
Ω
κ(x,u(x))∇φ(x) · ∇ψ(x)φ(x)dx(37)
as n → +∞. Reordering the sum of Tn
using Lemma 2 in [14] (see [5] for details), we get
10on the edges of the control volumes and
Tn
10→ −
?
Ω
∇φ(x) · ∇ψ(x)φ(x)κ(x,u(x))dx as n → +∞, (38)
from which it is easy to see that
?
Ω
Jn(un,φn)(x)ψn(x) →
?
Ω
κ(x,u(x))|∇φ|2(x)ψ(x)dx,
which proves (33).
6 Conclusion and Perspectives
We proved here the convergence of a cell centred finite volume method for the
coupled heat and potential equation; the condition on the considered meshes is
such that the discrete maximum principle holds. Indeed, the technique of proof
mimics the tools used for the existence the continuous case, which requires the
monotonicity of the operator. In the case of the cell centred finite volume, the
scheme satisfies the maximum principle for any admissible mesh. These include

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14A. Bradji and R. Herbin
triangles and rectangles in two space dimensions, and Vorono¨ ı meshes in any
dimension.
In two space dimensions, the piecewise linear finite element method satisfies
the discrete maximum principle for triangular meshes under the Delaunay con-
dition. It is easy to show that under this condition, the matrix of the scheme
is identical to that of the cell-centred finite volume on the dual Vorono¨ ı mesh.
Therefore, the convergence of the finite element scheme may be obtained from
that of the finite volume scheme, as explained in [18].
In three space dimensions, there is no known way to build a Vorono¨ ı mesh
from a tetrahedral one, and therefore one must proceed directly with the finite
element interpolation operator, as in [6] in the case of a linear diffusion operator,
and [5] in the case of the present coupled problem. In the three–dimensional case,
a known sufficient condition for the maximum principle to hold on a tetrahedral
meshes is that all angles of all the faces be strictly acute. Unfortunately, there
does not seem to be an easy way to construct such meshes in practise [2], so that
convergence results for the finite element scheme in 3D remain quite academic.
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