# A Crank-Nicolson Scheme for the Dirichlet-to-Neumann Semigroup

Authors: Rola Ali Ahmad; Toufic El Arwadi; Houssam Chrayteh; Jean-Marc Sac-Epée
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

The aim of this work is to study a semidiscrete Crank-Nicolson type scheme in order to approximate numerically the Dirichlet-to-Neumann semigroup. We construct an approximating family of operators for the Dirichlet-to-Neumann semigroup, which satisfies the assumptions of  Chernoff’s product formula, and consequently the Crank-Nicolson scheme converges to the exact solution. Finally, we write aP
1 finite element scheme for the problem, and we illustrate this convergence by means of a FreeFem++ implementation.


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## Body

## 1. Introduction

LetΩ be a bounded smooth domain Ω
⊂
R
n and let γ
(
x
)
=
[
γ
i
,
j
(
x
)
]
i
,
j
=
1
n be a real-valued matrix function, which is known as the electrical conductivity matrix. The matrix γ
(
x
) is symmetric and smooth; that is, γ
i
j
=
γ
j
i
∈
C
∞
(
Ω
¯
). Its positive eigenvalues are uniformly bounded; that is, there exists 0
<
c
1
<
c
2 such that, for all (
x
,
ξ
)
∈
Ω
×
R
n, one has c
1
ξ
2
≤
ξ
T
γ
(
x
)
ξ
≤
c
2
ξ
2, where · is the Euclidean norm of R
n.In this paper, we denote byX
=
L
2
(
Ω
), with boundary space ∂
X
=
L
2
(
∂
Ω
). It is well known that, for any f
∈
H
1
/
2
(
∂
Ω
), the following Dirichlet problem,(1)
div
⁡
γ
∇
u
=
0
,
in
Ω
,
u
=
f
,
on
∂
Ω
,has a unique solution u
=
L
γ
f in H
1
(
Ω
), which is called the γ-harmonic lifting of f; see [1] for more details.Define the action of the Dirichlet-to-Neumann operator onf as the outward normal derivative of u on the boundary; that is, (2)
Λ
γ
f
≔
ν
·
γ
∇
u
∂
Ω
,where ν is the outer normal vector to ∂
Ω at x
∈
∂
Ω.This operator is defined on its domain:(3)
D
Λ
γ
≔
f
∈
H
1
/
2
∂
Ω
;
Λ
γ
f
∈
L
2
∂
Ω
.The Dirichlet-to-Neumann semigroup is the trace of the following boundary value problem: (4)
BVP
div
⁡
γ
∇
u
t
,
·
=
0
,
in
Ω
,
∂
t
u
+
ν
·
γ
∇
u
=
0
,
on
∂
Ω
,
u
0
,
·
=
f
,
on
∂
Ω
.In [1], Lax has introduced a semigroup in ∂
X, where Ω is the unit ball B
(
0,1
) and γ is the identity matrix. Let v be the solution of the problem:(5)
Δ
v
=
0
,
in
Ω
,
L
v
=
f
,
on
∂
Ω
,where L is the trace operator on ∂
Ω. The Lax semigroup is defined by (6)
S
t
=
L
T
t
L
0
,where L
0 is the lifting operator and T
(
t
)
v
(
x
)
:
=
v
(
e
-
t
x
).Therefore, iff
∈
∂
X, then, for any w
∈
∂
Ω, (7)
S
t
f
w
=
v
e
-
t
w
.It has been shown that the Lax semigroup is actually the Dirichlet-to-Neumann semigroup.In [2], it is shown that Lax’s representation cannot be generalized if Ω is not the unit ball. This was a motivation for many authors to use Chernoff’s formula in order to approximate the solution.In [3], Emamirad and Shariftabar proved that the Euler Explicit scheme (8)
div
⁡
γ
∇
u
m
+
1
=
0
,
in
Ω
,
1
Δ
t
u
m
+
1
-
u
m
+
∂
u
m
∂
ν
γ
=
0
,
on
∂
Ω
,
u
0
=
f
,
on
∂
Ωconverges to the solution of the (BVP).Later, Cherif et al. proposed, in [4], the following Implicit Euler Scheme: (9)
div
⁡
γ
∇
u
m
+
1
=
0
,
in
Ω
,
1
Δ
t
u
m
+
1
-
u
m
+
∂
u
m
+
1
∂
ν
γ
=
0
,
on
∂
Ω
,
u
0
=
f
,
on
∂
Ω
,which converges to the solution of the (BVP).After studying the implicit and explicit schemes, by the same strategy, we are interested in studying the convergence of the Crank-Nicolson type scheme. This paper is divided into 4 sections, as follows.First, in Section2, we introduce the Crank-Nicolson type scheme for evolution equations.In Section3, we prove the convergence of a Crank-Nicholson type scheme for the (BVP), by constructing an approximating family, and use Chernoff’s product formula in order to prove the convergence in L
2
(
∂
Ω
).Finally, in Section4, we present a numerical implementation of the Crank-Nicolson type scheme in the computational framework of the finite element method.


## 2. The Crank-Nicolson Scheme

In numerical analysis, the Crank-Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid-20th century.LetA be a m-accretive operator on the Hilbert space H defined as A
:
D
(
A
)
→
H.Consider the evolution problem:(10)
∂
u
∂
t
+
A
u
=
0
,
for
0
<
t
<
T
,
u
0
,
x
=
u
0
x
.We approximate the time derivative by using the finite difference method as follows: (11)
u
n
Δ
t
≈
u
n
∂
u
∂
t
n
Δ
t
=
1
Δ
t
u
n
+
1
Δ
t
-
u
n
Δ
t
≈
1
Δ
t
u
n
+
1
-
u
n
,
for
n
=
0
,
…
,
T
Δ
t
.Crank-Nicolson introduced the scheme: (12)
u
n
+
1
-
u
n
Δ
t
+
1
2
A
u
n
+
1
2
A
u
n
+
1
=
0
,
for
n
=
0
,
…
,
T
Δ
t
,
u
0
given
.The Chernoff product formula holds not only in the case of semigroup of contractions but also if the semigroup is bounded, so that the Crank-Nicholson scheme for the (BVP) can be written as (13)
(CNS)
div
⁡
γ
∇
u
m
+
1
+
div
⁡
γ
∇
u
m
=
0
,
in
Ω
,
1
Δ
t
u
m
+
1
-
u
m
+
1
2
∂
u
m
∂
ν
γ
+
∂
u
m
+
1
∂
ν
γ
=
0
,
on
⁡
∂
Ω
,
u
0
=
f
,
on
⁡
∂
Ω
.


## 3. Convergence of the Crank-Nicholson Type Scheme

In order to show the convergence of the (CNS), let us recall the following Chernoff theorem.Theorem 1 (Chernoff’s product formula).
Let X be a Banach space, real or complex, and letV
(
t
)
;
t
≥
0 be a family of contractions on X with V
(
0
)
=
I.
Assume that there existsA
:
D
(
A
)
⊆
X
→
X which generates a C
0-semigroup of contractions S
(
t
)
;
t
≥
0 and that for all x
∈
D
(
A
) there exists (14)
lim
h
→
0
⁡
1
h
V
h
-
I
x
=
A
x
.Then, for each x in X, one has l
i
m
n
→
∞
V
n
(
t
/
n
)
x
=
S
(
t
)
x.Remark 2.
Note that Chernoff’s product formula holds not only in the case of semigroup of contractions but also if the semigroup is bounded (see [5]).This theorem was used to prove the convergence of both implicit and explicit schemes.Now, in order to prove the convergence of the Crank-Nicolson scheme, we will show that the operatorZ
(
t
), defined below, satisfies the 3 conditions of Chernoff’s theorem.For anyx
∈
∂
Ω, we have(15)
∂
u
m
+
1
∂
ν
γ
=
1
Δ
x
u
m
+
1
x
-
u
m
+
1
x
-
Δ
x
γ
x
ν
,
∂
u
m
∂
ν
γ
=
1
Δ
x
u
m
x
-
u
m
x
-
Δ
x
γ
x
ν
.Using in (CNS) the approximation given in (15), we get(16)
1
+
1
2
Δ
t
Δ
x
u
m
+
1
x
-
1
-
1
2
Δ
t
Δ
x
u
m
x
-
1
2
Δ
t
Δ
x
u
m
+
1
x
-
Δ
x
γ
x
ν
-
1
2
Δ
t
Δ
x
u
m
+
1
x
-
Δ
x
γ
ν
=
0
.Taking α
=
Δ
t
/
Δ
x, we get(17)
1
+
α
2
u
m
+
1
x
-
α
2
u
m
+
1
x
-
Δ
x
γ
x
ν
=
1
-
α
2
u
m
x
+
α
2
u
m
+
1
x
-
Δ
x
γ
x
νwhich can be written as (18)
W
Δ
t
u
m
+
1
=
V
Δ
t
u
m
,where (19)
W
Δ
t
f
=
1
+
α
2
u
-
α
2
u
x
-
α
-
1
t
γ
x
ν
,
V
Δ
t
f
=
1
-
α
2
u
+
α
2
u
x
-
α
-
1
t
γ
x
ν
.Now, we define the family Q
(
t
) as the inverse of W
(
t
) in the following sense: (20)
W
Δ
t
Q
Δ
t
f
x
=
f
x
,and we show that Z
(
t
)
=
Q
(
Δ
t
)
V
(
Δ
t
) satisfies all the assumptions of Chernoff’s product formula.(i)Z
(
0
)
f
=
f. For any x
∈
∂
Ω, we have(21)
W
0
f
x
=
1
+
α
2
u
x
-
α
2
u
x
=
u
x
=
f
xso, f
(
x
)
=
W
(
0
)
[
Q
(
0
)
f
(
x
)
]
=
Q
(
0
)
f
(
x
).Moreover,(22)
V
0
f
x
=
1
-
α
2
u
x
+
α
2
u
x
=
u
x
=
f
xso,(23)
V
0
f
x
=
f
x
.Using (6) and (7), we get (24)
W
0
f
x
=
V
0
f
x
.Therefore, f
(
x
)
=
Q
(
0
)
V
(
0
)
f
(
x
)
=
Z
(
0
)
f
(
x
).(ii)Z
′
(
0
)
f
=
-
Λ
γ
f. The derivative of W
(
t
) with respect to t is (25)
W
′
t
f
=
-
α
2
-
α
-
1
γ
ν
·
∇
u
v
-
α
-
1
t
γ
ν
.At point t
=
0, we have W
′
(
0
)
f
=
1
/
2
Λ
γ
f.Similarly, the derivative ofV
(
t
) with respect to t is (26)
V
′
t
f
=
α
2
-
α
-
1
γ
ν
·
∇
u
v
-
α
-
1
t
γ
ν
.At point t
=
0, we have V
′
(
0
)
f
=
-
1
/
2
Λ
γ
f.ButW
(
t
)
Q
(
t
)
f
=
f, so W
′
(
t
)
Q
(
0
)
f
+
W
0
Q
′
(
0
)
f
=
0, and then Q
′
(
0
)
f
=
-
1
/
2
Λ
γ
f.Now, the derivative ofZ
(
t
) with respect to t is (27)
Z
′
t
f
=
Q
′
t
V
t
f
+
Q
t
V
′
t
f
.At t
=
0, Z
′
0
f
=
Q
′
(
0
)
V
(
0
)
f
+
Q
0
V
′
(
0
)
f
=
-
Λ
γ
f.Therefore,(28)
Z
′
0
f
=
Λ
γ
f
.(iii)Z
(
t
)
Is a Contraction. In fact,(29)
W
t
f
L
2
∂
Ω
2
=
∫
∂
Ω
u
x
+
α
2
u
x
-
u
x
-
α
-
1
t
γ
ν
2
d
σbut u
(
x
)
-
u
(
x
-
α
-
1
t
γ
ν
)
/
α
-
1 is an upper bound of the normal derivative ∂
u
/
∂
ν
γ, so that this term is positive according to Hopf’s lemma, and, consequently, (30)
∫
∂
Ω
u
x
+
α
2
u
x
-
u
x
-
α
-
1
t
γ
ν
2
d
σ
≥
∫
∂
Ω
u
x
2
d
σ
=
f
L
2
∂
Ω
2which implies that (31)
W
t
f
L
2
∂
Ω
≥
f
L
2
∂
Ω
.Now (32)
f
L
2
∂
Ω
=
W
t
Q
t
f
L
2
∂
Ω
≥
Q
t
f
L
2
∂
Ω
.Thus, Q
(
t
) is of contraction.SimilarlyV
(
t
) is also of contraction.Finally,(33)
Z
t
f
L
2
∂
Ω
=
Q
t
V
t
f
L
2
∂
Ω
≤
Q
t
f
L
2
∂
Ω
≤
f
L
2
∂
Ω
.Therefore, Z
(
t
) is of contraction.So, we have proved thatZ
(
t
) satisfies all the assumptions of Chernoff’s product formula, and, consequently, the Crank-Nicholson scheme converges to its exact solution.Theorem 3.
The operatorZ
(
t
) satisfies the Chernoff conditions; hence, the Crank-Nicolson method is convergent.


## 4. Numerical Results

In the last section of this paper, we present the resolution of (CNS) in the particular case whereγ is the unit matrix and Ω is the open unit disk. The point in dealing with this case is that the exact solution is known to be v
(
e
-
t
x
,
e
-
t
y
), where v is the solution of the Laplace equation Δ
v
=
0, with v
=
u
0 on the boundary (u
0 is any regular function). Thus, we can compare this exact solution with the approximate solution obtained via our Crank-Nicolson scheme. For this test, we choose u
0
(
x
,
y
)
=
0.5
(
x
2
-
y
2
)
+
y
+
0.5. The variational formulation of the (CNS) can be derived by multiplying both sides of the problem by a test function v: (34)
∫
Ω
γ
Δ
u
m
+
1
v
d
x
+
∫
Ω
γ
Δ
u
m
v
d
x
=
0
.Using Green’s formula, we get (35)
-
∫
Ω
γ
∇
u
m
+
1
∇
v
d
x
-
∫
∂
Ω
2
Δ
t
u
m
+
1
-
u
m
v
d
σ
-
∫
Ω
γ
∇
u
m
∇
v
d
x
=
0
.Multiplying by Δ
t, we get (36)
∫
Ω
Δ
t
γ
∇
u
m
+
1
∇
v
d
x
+
2
∫
∂
Ω
u
m
+
1
v
d
σ
=
-
∫
Ω
Δ
t
γ
∇
u
m
∇
v
d
x
+
2
∫
∂
Ω
u
m
v
d
σwhich is of the form (37)
a
u
m
+
1
,
v
=
l
v
,where the bilinear form (38)
a
u
m
+
1
,
v
=
∫
Ω
Δ
t
γ
∇
u
m
+
1
∇
v
d
x
+
2
∫
∂
Ω
u
m
+
1
v
d
σ
,
l
v
=
-
∫
Ω
Δ
t
γ
∇
u
m
∇
v
d
x
+
2
∫
∂
Ω
u
m
v
d
σis the linear form.Using this variational formulation, the problem can be approximate viaP
1-finite elements implemented in Freefem++.The actual algorithm is as follows.Choose an initialu
0 and, for a given Δ
t, where t
<
N
Δ
t, we have the following:fort
=
0
;
t
<
N
Δ
t do(i)
u
n
+
1
=
Z
(
u
n
)
(ii)
computeu
n
+
1 via the variational problem
(iii)
n
←
n
+
1end do.As an illustration, let us represent the exact (Figure1) and the approximate solution (Figure 2), the L
2-error between the exact and the approximate solutions (Figure 3), and the decrement of this L
2-error according to the fineness of the meshing (Figure 4).Figure 1
The exact solution.Figure 2
The numerical solution.Figure 3
The difference between the exact and the numerical solutions.Figure 4
TheL
2-error according to the fineness of the mesh.


## 5. Conclusion

As a conclusion, we have proved the convergence of the (CNS) by showing that the constructed family of operators satisfies the assumptions of Chernoff’s product formula and consequently approximated numerically the Dirichlet-to-Neumann semigroup.



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*Source: 429641-2015-07-09.xml*