A Dual Mesh Method for a Non-Local Thermistor Problem
We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem.
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nasplib_isofts_kiev_ua-123456789-1461142025-02-09T13:36:18Z A Dual Mesh Method for a Non-Local Thermistor Problem Abderrahmane El Hachimi Moulay Rchid Sidi Ammi Delfim F.M. Torres We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem. The support of the Portuguese Foundation for Science and Technology (FCT) and post-doc fellowship SFRH/BPD/20934/2004 are gratefully acknowledged. We would like to thank two anonymous referees for valuable comments and suggestions. 2006 Article A Dual Mesh Method for a Non-Local Thermistor Problem / Abderrahmane El Hachimi, Moulay Rchid Sidi Ammi, Delfim F.M. Torres // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35K55; 65N15; 65N50 https://nasplib.isofts.kiev.ua/handle/123456789/146114 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem. |
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Article |
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Abderrahmane El Hachimi Moulay Rchid Sidi Ammi Delfim F.M. Torres |
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Abderrahmane El Hachimi Moulay Rchid Sidi Ammi Delfim F.M. Torres A Dual Mesh Method for a Non-Local Thermistor Problem Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Abderrahmane El Hachimi Moulay Rchid Sidi Ammi Delfim F.M. Torres |
| author_sort |
Abderrahmane El Hachimi |
| title |
A Dual Mesh Method for a Non-Local Thermistor Problem |
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A Dual Mesh Method for a Non-Local Thermistor Problem |
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A Dual Mesh Method for a Non-Local Thermistor Problem |
| title_fullStr |
A Dual Mesh Method for a Non-Local Thermistor Problem |
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A Dual Mesh Method for a Non-Local Thermistor Problem |
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dual mesh method for a non-local thermistor problem |
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Інститут математики НАН України |
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2006 |
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https://nasplib.isofts.kiev.ua/handle/123456789/146114 |
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A Dual Mesh Method for a Non-Local Thermistor Problem / Abderrahmane El Hachimi, Moulay Rchid Sidi Ammi, Delfim F.M. Torres // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 18 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT abderrahmaneelhachimi adualmeshmethodforanonlocalthermistorproblem AT moulayrchidsidiammi adualmeshmethodforanonlocalthermistorproblem AT delfimfmtorres adualmeshmethodforanonlocalthermistorproblem AT abderrahmaneelhachimi dualmeshmethodforanonlocalthermistorproblem AT moulayrchidsidiammi dualmeshmethodforanonlocalthermistorproblem AT delfimfmtorres dualmeshmethodforanonlocalthermistorproblem |
| first_indexed |
2025-11-26T06:39:20Z |
| last_indexed |
2025-11-26T06:39:20Z |
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1849833991083917312 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 058, 10 pages
A Dual Mesh Method for a Non-Local
Thermistor Problem
Abderrahmane EL HACHIMI †, Moulay Rchid SIDI AMMI ‡ and Delf im F.M. TORRES ‡
† UFR: Applied and Industrial Mathematics, University of Chouaib Doukkali, El Jadida, Maroc
E-mail: elhachimi@ucd.ac.ma
‡ Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
E-mail: sidiammi@mat.ua.pt, delfim@mat.ua.pt
URL: http://www.mat.ua.pt/delfim
Received December 20, 2005, in final form May 08, 2006; Published online June 02, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper058/
Abstract. We use a dual mesh numerical method to study a non-local parabolic problem
arising from the well-known thermistor problem.
Key words: non-local thermistor problem; joule heating; box scheme method
2000 Mathematics Subject Classification: 35K55; 65N15; 65N50
1 Introduction
In this work we propose a dual mesh numerical scheme for analysis of the following non-local
parabolic problem coming from conservation law of electric charges:
∂u
∂t
−∇ · (k(u)∇u) = λ
f(u)(∫
Ω f(u) dx
)2 in Ω×]0;T [,
u = 0 on ∂Ω×]0;T [, u/t=0 = u0 in Ω, (1)
where ∇ denotes the gradient with respect to the x-variables. The nonlinear problem (1) is
obtained, under some simplificative conditions, by reducing the well-known thermistor problem
(cf., e.g., [13, 14, 15]), which consists of the heat equation, with joule heating as a source, and
subject to current conservation:
ut = ∇ · (k(u)∇u) + σ(u) |∇ϕ|2 , ∇ · (σ(u)∇ϕ) = 0, (2)
where the domain Ω ⊂ R2 occupied by the thermistor is a bounded convex polygonal; ϕ =
ϕ(x, t) and u = u(x, t) are, respectively, the distributions of the electric potential and the
temperature in Ω; σ(u) and k(u) are, respectively, the temperature-dependant electrical and
thermal conductivities; σ(u) |∇ϕ|2 is the joule heating. The literature on problem (2) is vast
(see e.g. [2, 6, 7, 8, 9, 10, 11, 16, 17]). With respect to numerical approximation results to
problem (2) we are aware of [1, 11, 12, 18]: in [18] a numerical analysis of the non-steady
thermistor problem by a finite element method is discussed; in [12] the authors study a spatially
and completely discrete finite element model; in [11] a semi-discretization by the backward Euler
scheme is given for the special case k = Id; in [1] a box approximation scheme is presented and
analyzed. A completely discrete scheme based on the backward Euler method with semi-implicit
linearization to (2) is presented in [12] for the special case k(u) = 1. Existence and uniqueness
of solutions to the problem (1) were proved in [10].
mailto:elhachimi@ucd.ac.ma
mailto:sidiammi@mat.ua.pt
mailto:delfim@mat.ua.pt
http://www.mat.ua.pt/delfim
http://www.emis.de/journals/SIGMA/2006/Paper058/
2 A. El Hachimi, M.R. Sidi Ammi and D.F.M. Torres
Finite volume methods emerged recently and seem to have a significant role on concrete
applications, because they have very interesting properties in view of the subjacent physical
problems: in particular in conservation of flows. An equation coming from a conservation law
has a good chance to be correctly discretized by the finite volume method. We also recall that
these schemes have been widely used to approximate solutions of the heat linear equation, semi-
linear or parabolic equations. Since we consider data f with lack of regularity when compared to
previous work, we need a new way to discretize (1). We present a dual mesh method capable of
handling the non local term λf(u)
(
∫
Ω f(u) dx)2 which is a noticeable feature of (1), by generalizing the
results of [1]. A box approximation scheme for discretizing (1) with the case k being different
from the identity is obtained. Speed of convergence is directly related with regularity of the
continuous problem. When one increases regularity of the second term and data, the solution
see its regularity increasing in parallel, and precise speed of convergence can be established.
In the existing literature (see e.g. [5, 12]) the error estimates for both the finite element or
volume element method are usually derived for solutions that are sufficiently smooth. Because
the domain is polygonal, special attention has to be paid to regularity of the exact solution.
We give sufficient conditions in terms of data and the solution u that yield error estimates (see
hypothesis (H1) below).
The text is organized as follows. In Section 2 we set up the notation and the functional
spaces used throughout the paper. Section 3 introduces a box scheme model for problem (1),
and existence and uniqueness of the solution of the approximating problem (12) is obtained from
the fixed point theorem and equivalence of norms in the finite dimensional space S0
h. Finally,
in Section 4, under some regularity assumptions, we prove error estimates.
2 Notation and functional spaces
Let (·, ·) and ‖·‖ denote the inner product and norm in L2(Ω); H1
0 (Ω) =
{
u ∈ H1(Ω), u/∂Ω = 0
}
;
‖ · ‖s, ‖ · ‖s,p denote the Hs(Ω) and the W s,p(Ω) norm respectively; Th denote a triangulation
of Ω; T h
v be the set of vertices of a quasi-uniform triangulation Th; and
{
S0
h
}
h>0
be the family
of approximating subspaces of H1
0 (Ω) defined by
S0
h =
{
v ∈ H1
0 (Ω) : v/e is a linear function for all e ∈ Th
}
.
In the remainder of this paper we denote by c various constants that may depend on the data
of the problem, and that are not necessarily the same at each occurrence. We assume that the
family of triangulations is such that the following estimates [4] hold for all v ∈ S0
h:
‖v‖β,q ≤ chr−β−2max{0,1/p−1/q}‖v‖r,p, 0 ≤ r ≤ β ≤ 1, 1 ≤ p, q ≤ ∞,
‖v‖0,∞ ≤ c| lnh|
1
2 ‖v‖1. (3)
Let Ph : L2(Ω) → S0
h be the standard L2-projection. One has [4]:
‖v − Phv‖+ h‖v − Phv‖1 ≤ ch2‖v‖2,
‖v − Phv‖0,∞ ≤ ch‖v‖2, ‖Phv‖1,∞ ≤ c‖v‖1,∞. (4)
We construct the box scheme Bh (dual mesh) employed in the discretization as follows. From
a given triangle e ∈ Th, we choose a point q ∈ e as the intersection of the perpendicular bisectors
of the three edges of e. Then, we connect q by straight-line segments to the edge midpoints of e.
To each vertex p ∈ T h
v , we associate the box bp ∈ Bh, consisting of the union of subregions which
have p as a corner (see Fig. 1). For the piecewise constant interpolation operator Ih, defined by
Ih : C(Ω) → L2(Ω), Ihv = v(p), on bp ∈ Bh, ∀ p ∈ T h
v ,
A Dual Mesh Method for a Non-Local Thermistor Problem 3
Figure 1. Construction of the dual mesh.
we have the following standard error estimates [1, 3]:
c−1‖v‖ ≤ ‖Ihv‖ ≤ c‖v‖, ∀ v ∈ Sh
0 ,
‖v − Ihv‖ ≤ ch‖v‖1, ∀ v ∈ Sh
0 . (5)
We denote by Nh(p) the set of the neighboring vertices of p ∈ T h
v , ∂b =
⊔
p∈T h
v
∂bp, ∂bp =⊔
p∗∈Nh(p){Γpp∗}, where Γpp∗ = ∂bp
⋂
∂bp∗ (see Fig. 1). Let l∂b : ∂b → R+ be defined as follows:
for p ∈ T h
v and bp ∈ Bh,
l∂b/Γpp∗ = |p− p ∗ | for p∗ ∈ Nh(p).
For b ∈ Bh, we denote the jump in w across ∂b at x by [w]∂b(x) = w(x + 0)− w(x− 0), where
w(x± 0) are the outside and inside limit values of w(x) along the normal directions for ∂b.
We now collect from the literature [1, 3] some important lemmas and trace results, that are
needed in the sequel.
Lemma 1. Assume that Bh is a dual mesh. If v is a piecewise linear function, and x is not
a vertex, then
[Ihv]/∂bp(x) =
∂v
∂n
l∂b/Γpp∗, x ∈ Γpp∗, ∀ b ∈ Bh,
where n is the unit outward normal vector on ∂b.
The h-dependent norms are defined as follows:
‖v‖1,h =
(∑
l∈∂b
|[Ihv]l|2
) 1
2
and ‖v‖0,h = ‖Ihv‖ .
Lemma 2. There exists a constant c > 0 such that
c−1 ‖∇v‖ ≤ ‖v‖1,h ≤ c‖∇v‖, ∀ v ∈ S0
h,
c−1‖v‖ ≤ ‖v‖0,h ≤ c‖v‖, ∀ v ∈ S0
h.
Lemma 3. For any a ∈ C(Ω) there exists a positive constant c such that∣∣∣∣∣∣−
∑
b∈Bh
∫
∂b
a
∂u
∂n
Ihv
∣∣∣∣∣∣ ≤ c‖u‖1‖v‖1, ∀ u, v ∈ S0
h. (6)
Moreover, if there exists a constant a0 > 0 such that a ≥ a0 in Ω, then
c−1‖v‖2
1 ≤ −
∑
b∈Bh
∫
∂b
a
∂v
∂n
Ihv, ∀ v ∈ S0
h. (7)
4 A. El Hachimi, M.R. Sidi Ammi and D.F.M. Torres
Let Qh : H2(Ω) → S0
h be defined by Qhu− ihu ∈ S0
h, and
−
∑
b∈Bh
∫
∂b
a
∂(u−Qhu)
∂n
Ihv = 0, ∀ v ∈ S0
h, (8)
where ih : C(Ω) → S0
h is the Lagrangian interpolation operator and u ∈ H2(Ω).
Lemma 4. Assume that a ∈ L∞(Ω), with a ≥ a0 for some constant a0 > 0. Then, there exists
c > 0 such that for u ∈ H2(Ω)
‖u−Qhu‖1 ≤ ch‖u‖2. (9)
Moreover, if u ∈ H2(Ω)
⋂
W 1,∞(Ω), then
‖Qhu‖1,∞ ≤ c (‖u‖1,∞ + ‖u‖2) . (10)
Lemma 5. For each b ∈ Bh one has
h
1
2 ‖v‖L2(∂b) ≤ c
(
‖v‖L2(b) + h‖v‖H1(b)
)
, ∀ v ∈ H1(b).
Throughout this work, we assume that the following hypotheses on the solution and data of
problem (1) are satisfied:
(H1) u ∈ L∞(H1
0 (Ω)
⋂
H2(Ω)), ut ∈ L2(H1(Ω));
(H2) c−1 ≤ k(s) ≤ c;
(H3) there exist positive constants c1, c2 and ν, such that ν ≤ f(ξ) ≤ c1|ξ|+ c2 for all ξ ∈ R;
(H4) |f(ξ)− f(ξ′)|+ |k(ξ)− k(ξ′)| ≤ c |ξ − ξ′|.
3 Existence and uniqueness result for the box scheme method
Let u be the solution of (1). Integrating over an element b in Bh we obtain:∫
b
ut −
∫
∂b
k(u)
∂u
∂n
=
λ(∫
f(u) dx
)2 ∫
b
f(u), ∀ b ∈ Bh. (11)
We consider a box scheme defined as follows: find uh ∈ S0
h such that
(
Ihuh
t , Ihv
)
−
∑
b∈Bh
∫
∂b
k
(
uh
)∂uh
∂n
Ihv =
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
, ∀ v ∈ S0
h, (12)
where uh(0) = Phu0 and Ih is the interpolation operator.
Theorem 1. Let (H1)–(H4) be satisfied. Then, for each h > 0, there exists t0(h) such that (12)
possesses a unique solution uh for 0 ≤ t ≤ t0(h).
Proof. We begin by proving existence of solution. We define a nonlinear operator G from S0
h
to S0
h as follows. For each uh ∈ S0
h, wh = G(uh) is obtained as the unique solution of the
following problem:
(
Ihwh
t , Ihv
)
−
∑
b∈Bh
∫
∂b
k
(
uh
)∂wh
∂n
Ihv =
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
, ∀ v ∈ S0
h. (13)
A Dual Mesh Method for a Non-Local Thermistor Problem 5
We remark that G is well defined. Using v = wh as a test function in (13), hypotheses (H2)
and (H3), and Holder’s inequality, we can write:
1
2
d
dt
‖Ihwh‖2 + c‖wh‖2
1 ≤ c
(
f
(
uh
)
, Ihwh
)
≤ c
∫ (
|uh|+ 1
)
|Ihwh|
≤ c‖uh‖L2‖Ihwh‖L2 + c‖Ihwh‖ ≤ c‖uh‖1‖Ihwh‖1 + c‖Ihwh‖
≤ c
2
‖Ihwh‖2
1 + c‖uh‖2
1 + c.
Thus, we have
d
dt
‖Ihwh‖2 + c‖wh‖2
1 ≤ c‖uh‖2
1 + c . (14)
Integrating (14) with respect to t and using the equivalency of ‖Ih · ‖ and ‖ · ‖ in S0
h (see (5))
yields
‖wh‖2 + c
∫ t
0
‖wh‖2
1 ≤ c‖IhPhu0‖2 + c
∫ t
0
‖uh‖2
1 dx + ct
≤ c‖u0‖2 + c
∫ t
0
‖uh‖2
1 dx + ct.
Define now the following set
D =
{
uh ∈ S0
h, ‖uh‖2 + c
∫ t
0
‖uh‖2
1 ≤ c
(
‖u0‖2 + 1
)}
.
We can easily see that D is closed subset of L∞(0, t, L2(Ω)) with its natural norm. We conclude
that there exists t > 0 such that G(D) ⊂ D. To obtain that G has a fixed point wh = G(wh),
we prove that G is a contraction. Conclusion follows from Banach’s fixed point theorem. For
this purpose, let uh
1 and uh
2 ∈ S0
h × S0
h such that Guh
1 = wh
1 and Guh
2 = wh
2 . We have, from the
equation (13) verified by wh
1 and wh
2 , that
(
Ih
(
wh
1t − wh
2t
)
, Ihv
)
−
∑
b∈Bh
∫
∂b
k
(
uh
1
)∂wh
1
∂n
Ihv +
∑
b∈Bh
∫
∂b
k
(
uh
2
)∂wh
2
∂n
Ihv
=
λ(∫
Ω f(uh
1) dx
)2 (f(uh
1
)
, Ihv
)
− λ(∫
Ω f(uh
2) dx
)2 (f(uh
2
)
, Ihv
)
.
On the other hand, one has
−
∑
b∈Bh
∫
∂b
k
(
uh
1
)∂wh
1
∂n
Ihv +
∑
b∈Bh
∫
∂b
k
(
uh
2
)∂wh
2
∂n
Ihv
= −
∑
b∈Bh
∫
∂b
k(uh
1)
∂
(
wh
1 − wh
2
)
∂n
Ihv +
∑
b∈Bh
∫
∂b
(
k
(
uh
2
)
− k
(
uh
1
))∂wh
2
∂n
Ihv.
Schwartz inequality implies that
∑
b∈Bh
∫
∂b
(
k
(
uh
1
)
− k
(
uh
2
))∂wh
2
∂n
Ihv ≥ −c‖wh
2‖1,∞
∑
b∈Bh
‖uh
1 − uh
2‖0,∂b
‖v‖.
By Lemma 5, we have
h
1
2
∑
b∈Bh
‖uh
1 − uh
2‖0,∂b ≤ c
∑
b∈Bh
(
‖uh
1 − uh
2‖L2(b) + h‖uh
1 − uh
2‖H1(b)
)
≤ c‖uh
1 − uh
2‖1.
6 A. El Hachimi, M.R. Sidi Ammi and D.F.M. Torres
Thus, from the inverse estimate (3),
∑
b∈Bh
∫
∂b
(
k
(
uh
1
)
− k
(
uh
2
))∂wh
2
∂n
Ihv ≥ −c(h)‖wh
2‖1‖uh
1 − uh
2‖1‖v‖1. (15)
On the basis of hypotheses (H1)–(H4), we have:
λ(∫
Ω f(uh
1) dx
)2 (f(uh
1
)
, Ihv
)
− λ(∫
Ω f
(
uh
2
)
dx
)2 (f(uh
2
)
, Ihv
)
=
λ(∫
Ω f
(
uh
1
)
dx
)2 (f(uh
1
)
− f
(
uh
2
)
, Ihv
)
+ λ
(
1(∫
Ω f
(
uh
1
)
dx
)2 − 1(∫
Ω f
(
uh
2
)
dx
)2
)(
f
(
uh
2
)
, Ihv
)
≤ c‖uh
1 − uh
2‖ ‖v‖+ λ
(∫
Ω f
(
uh
2
)
− f
(
uh
1
)) (∫
Ω f
(
uh
2
)
+ f
(
uh
1
))(∫
Ω f
(
uh
2
)
dx
)2 (∫
Ω f
(
uh
1
)
dx
)2 (
f
(
uh
2
)
, Ihv
)
≤ c‖uh
1 − uh
2‖ ‖v‖+ c‖Ihv‖L2(Ω)‖uh
1 − uh
2‖L1(Ω)
≤ c‖uh
1 − uh
2‖ ‖v‖ ≤ c‖uh
1 − uh
2‖1 ‖v‖1. (16)
It follows from (15) and (16) that
(
Ih
(
wh
1t − wh
2t
)
, Ihv
)
−
∑
b∈Bh
∫
∂b
k(uh
1)
∂
(
wh
1 − wh
2
)
∂n
Ihv
≤ c(h)‖uh
1 − uh
2‖1‖v‖1 −
∑
b∈Bh
∫
∂b
(
k
(
uh
1
)
− k
(
uh
2
))∂wh
2
∂n
Ihv ≤ c(h)‖uh
1 − uh
2‖1‖v‖1. (17)
Now, using v = wh
1 − wh
2 as a test function in (17), we obtain from (7):
1
2
d
dt
∥∥Ih
(
wh
1 − wh
2
)∥∥2 + c‖wh
1 − wh
2‖2
1 ≤ c(h)‖uh
1 − uh
2‖1‖wh
1 − wh
2‖1. (18)
With use of the Holder’s inequality and equivalency of ‖Ih · ‖ and ‖ · ‖, integration of (18) with
respect to time gives:
∥∥(wh
1 − wh
2
)∥∥2 ≤ c
∥∥Ih
(
wh
1 − wh
2
)∥∥2 ≤ c(h)
∫ t
0
∥∥(uh
1 − uh
2
)∥∥2
1
ds.
Thus G is a contraction. We prove now uniqueness. Following the same arguments as before,
we have
(
Ih
(
uh
1t − uh
2t
)
, Ihv
)
−
∑
b∈Bh
∫
∂b
k(uh
1)
∂(uh
1 − uh
2)
∂n
Ihv ≤ c(h)‖uh
1 − uh
2‖1‖v‖. (19)
Choosing v = uh
1 − uh
2 as test function in (19), using again (7) and integrating, we obtain
∥∥(uh
1 − uh
2
)∥∥2 ≤ c(h)
∫ t
0
∥∥(uh
1 − uh
2
)∥∥2
ds,
which gives, by Gronwall’s Lemma, uniqueness of solution. �
A Dual Mesh Method for a Non-Local Thermistor Problem 7
4 Error analysis
In this section we prove error estimates under certain assumptions on regularity of the exact
solution u.
Theorem 2. Under assumptions (H1)–(H4), if
(
u, uh
)
are solutions of (11)–(12) for 0 ≤ t ≤
t0(h), then
‖uh − u‖L∞(L2) + ‖uh − u‖L2(H1) ≤ ch.
Proof. From (11) and (12) we obtain
(
Ih
(
uh − Phu
)
t
, Ihv
)
−
∑
b∈Bh
∫
∂b
k
(
uh
)∂ (uh
1 − Phu
)
∂n
Ihv
=
λ(∫
Ω f (uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv
)
+
∑
b∈Bh
∫
∂b
k
(
uh
)∂ (Phu− u)
∂n
Ihv
−
∑
b∈Bh
∫
∂b
(
k(u)− k
(
uh
))∂u
∂n
Ihv +
(
(I − Ph)ut, Ihv
)
+
(
(I − Ih)Phut, Ihv
)
. (20)
We now estimate, separately, the terms on the right-hand side of (20). We have from (6) and (4)
that ∣∣∣∣∣∣
∑
b∈Bh
∫
∂b
k
(
uh
)∂(Phu− u)
∂n
Ihv
∣∣∣∣∣∣ ≤ c‖Phu− u‖1‖v‖1 ≤ ch‖u‖2‖v‖1 ≤ ch‖v‖1, (21)
∣∣∣∣∣∣
∑
b∈Bh
∫
∂b
(
k(u)− k
(
uh
))∂u
∂n
Ihv
∣∣∣∣∣∣
≤ c‖v‖1
∑
b∈Bh
(∫
∂b
|u− uh|
∣∣∣∣∂u
∂n
∣∣∣∣)2
1
2
≤ ch
1
2 ‖v‖1
∑
b∈Bh
‖u− uh‖2
0,∂b
1
2
. (22)
By Lemma 5, inverse inequality (3) and (4), we have:∑
b∈Bh
‖uh − u‖2
0,∂b ≤ 2
∑
b∈Bh
‖uh − Phu‖2
0,∂b + 2
∑
b∈Bh
‖Phu− u‖2
0,∂b
≤ c
(
h2 + ‖uh − Phu‖2
)
. (23)
Consequently, we obtain from (23) that∣∣∣∣∣∣
∑
b∈Bh
∫
∂b
(
k(u)− k
(
uh
))∂u
∂n
Ihv
∣∣∣∣∣∣ ≤ c
(
h + ‖uh − Phu‖
)
‖v‖1. (24)
Based on our earlier development in (16), we also know:
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv) ≤ c‖uh − u‖‖v‖1. (25)
Let v = uh − Phu be a test function in (20). Using Lemma 3, it follows from (21)–(25) that
1
2
d
dt
‖Ih(uh − Phu)‖2 + c‖uh − Phu‖2
1
8 A. El Hachimi, M.R. Sidi Ammi and D.F.M. Torres
≤ c‖uh − u‖‖uh − Phu‖1 + ch‖uh − Phu‖1 + c
(
h + ‖uh − Phu‖
)
‖uh − Phu‖1
+ c(‖(I − Ph)ut‖+ ‖(I − Ih)Phut‖)‖uh − Phu‖
≤ c(‖(I − Ph)ut‖+ ‖(I − Ih)Phut‖)‖uh − Phu‖
+ c
(
h + ‖uh − Phu‖
)
‖uh − Phu‖1 + ‖Phu− u‖‖uh − Phu‖1.
By properties (4) and Cauchy’s inequality, it follows:
d
dt
∥∥Ih
(
uh − Phu
)∥∥2 + c‖uh − Phu‖2
1 ≤ c‖Phu− u‖1‖uh − Phu‖1
+ c
{
h + ‖(I − Ph)ut‖+ ‖(I − Ih)Phut‖+ ‖uh − Phu‖
}
‖uh − Phu‖1
≤ c
{
h2 + ‖(I − Ph)ut‖2
1 + ‖(I − Ih)Phut‖2 + ‖uh − Phu‖2
}
+
c
2
‖uh − Phu‖2
1
≤ c
{
h2 + h2‖ut‖2
2 + ch2‖Phut‖2
1
}
+ c‖uh − Phu‖2 +
c
2
‖uh − Phu‖2
1.
Hence,
d
dt
∥∥Ih
(
uh − Phu
)∥∥2 + c
∥∥uh − Phu
∥∥2
1
≤ ch2 + c‖uh − Phu‖2. (26)
Integrating (26) and applying Gronwall Lemma and using again the equivalency of ‖ · ‖ and
‖Ih · ‖, we get that
‖uh − Phu‖2 + c
∫ t
0
‖uh − Phu‖2
1 ≤ ch2.
Then, by the triangular inequality, we conclude with the intended result. �
Under more restrictive hypotheses on the data, it is possible to derive the following error
estimate.
Theorem 3. Assume (H1)–(H4). If k(s) = 1 and u0 ∈ H1
0 (Ω)
⋂
H2(Ω), then
‖uh − u‖L∞(H1) ≤ ch.
Proof. From equations (11) and (12), we have:
(
Ihuh
t − ut, Ihv
)
−
∑
b∈Bh
∫
∂b
∂uh
∂n
Ihv +
∑
b∈Bh
∫
∂b
∂u
∂n
Ihv
=
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv).
Using the definition (8) of Qh, we get
(
Ihuh
t − ut, Ihv
)
−
∑
b∈Bh
∫
∂b
∂
(
uh −Qhu
)
∂n
Ihv
=
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv),
and it follows that(
Ih
(
uh
t −Qhu
)
t
, Ihv
)
−
∑
b∈Bh
∫
∂b
∂(uh −Qhu)
∂n
Ihv
A Dual Mesh Method for a Non-Local Thermistor Problem 9
=
λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv)
+ ((I −Qh)ut, Ihv) + ((I − Ih)Qhut, Ihv). (27)
In order to estimate the right hand side of the last inequality, we treat both terms separately.
By similar arguments to those used in (16),∣∣∣∣∣ λ(∫
Ω f(uh) dx
)2 (f(uh
)
, Ihv
)
− λ(∫
Ω f(u) dx
)2 (f(u), Ihv)
∣∣∣∣∣ ≤ c‖uh − u‖‖v‖.
Taking a function test v = (uh −Qhu)t in (27), by (9) and (10) we have
‖Ih(uh −Qhu)t‖2 −
∑
b∈Bh
∫
∂b
∂(uh −Qhu)
∂n
Ih
(
uh −Qhu
)
t
≤ c
{
h + ‖(I − Ih)Qhut‖+ ‖(I −Qh)ut‖+ ‖Qhu− uh‖
}
‖Ih(uh −Qhu)t‖
≤ c
{
h + ch‖ut‖2 + ch‖Qhut‖1 + ‖Qhu− uh‖
}
‖Ih(uh −Qhu)t‖
≤ c
{
h + ‖Qhu− uh‖
}
‖Ih(uh −Qhu)t‖
≤ ch2 + c‖Qhu− uh‖2 +
1
2
‖Ih(uh −Qhu)t‖2. (28)
Integrating (28), we arrive to
‖uh −Qhu‖2
1 ≤ c
(
h2 +
∫ t
0
‖uh −Qhu‖2
)
≤ c
(
h2 +
∫ t
0
‖uh −Qhu‖2
1
)
= ch2 + c‖uh −Qhu‖2
L2(H1(Ω)),
and Theorem 2 gives
‖uh −Qhu‖2
1 ≤ ch2.
On the other hand, by triangular inequality, (9) and the regularity of the exact solution u, we
have
‖uh − u‖2
1 ≤ 2‖uh −Qhu‖2
1 + 2‖Qhu− u‖2
1 ≤ ch2‖u‖2
2 + ch2 ≤ ch2.
We conclude then with the desired error estimate. �
5 Conclusion
In this paper a dual mesh numerical scheme was proposed for a nonlocal thermistor problem.
We have showed the existence and uniqueness of the approximate solution via Banach’s fixed
point theorem. We have also proved H1-error bounds under minimal regularity assumptions.
We only obtain first-order estimates: higher order estimates are difficult to obtain due to the
nonstandard nonlocal term. Optimal error analysis to the present context, under appropriate
smoothness assumptions on data, can be derived by application of the techniques of [5], but this
needs further developments.
10 A. El Hachimi, M.R. Sidi Ammi and D.F.M. Torres
Acknowledgements
The support of the Portuguese Foundation for Science and Technology (FCT) and post-doc
fellowship SFRH/BPD/20934/2004 are gratefully acknowledged. We would like to thank two
anonymous referees for valuable comments and suggestions.
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1 Introduction
2 Notation and functional spaces
3 Existence and uniqueness result for the box scheme method
4 Error analysis
5 Conclusion
|