A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equations

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 04-11
www.iosrjournals.org
DOI: 10.9790/5728-11130411 www.iosrjournals.org 4 |Page
A Two Grid Discretization Method For Decoupling
Time-Harmonic Maxwell’s Equations
V.F.Payne And S.E. Ubani
Department of Mathematics, University of Ibadan, Ibadan, Oyo State, Nigeria.
Abstract:In this work, we study a two grid finite element methods for solving coupled partial differential
equations of Time-Harmonic Maxwell’s equations. A brief survey of finite element methods for Maxwell’s
equation and related fundamentals, such as Sobolev spaces, elliptic regularity results, finite element methods for
Second order problems and its algorithms were reviewed. The method is based on discretization using continuous
1
H -conforming elements for decoupling systems of partial differential equations. With this method, the solution
of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid
together with the solution of decoupled equations on the fine grid.
Keywords: Time-Harmonic Maxwell’s equation, finite element methods, two-grid scheme, fine and coarser
grids.
I. Introduction
Maxwell’s equations consist of two pairs of coupled partial deferential equations relating to four fields,
two of which model the sources of electromagnetism. These equations characterized the fundamental relations
between the electric field and magnetic field as recognized by the founder [James Clark Maxwell(1831-1879)] of
the Modern theory of electromagnetism. However, the modern version of the Maxwell’s equations has two
fundamental field vector functions ),( txE and ),( txH in the classical electromagnetic field, with space
variable
3
Rx and time variable .Rt The distribution of electric charges is given by a scalar charge
density function ),( tx and the current is described by the current density function ).,( txJ
This paper is concerned with the discretization of time-harmonic Maxwell equations with finite
elements. The analysis of Maxwell’s equations for simplified cases can be reduced to the solution of the
Helmholtz equation, which in turn can be discretized using standard
1
H -conforming (continuous) elements.
II. Literature Review
Given that
}on0=,in=  unfuu  (1)
Considering the weak form for the curl-curl problem (1):
Find  ;(0 curlHu ) such that
),(=),(),( vfvuvu  (2)
for all ),;(0  curlH where (.,.) denotes the inner product of .)]([ 2
2 L Hence the space
);(0 curlH is defined as follows:
)(=:)]([={=);( 2
2
1
1
22
2
2
1












 L
xx
LcurlH




 (3)
 on0=:);({=);(0  ncurlHcurlH (4)
where n is the unit outer normal.
Remark: The  on0=n is equivalent to ,on0=  where  is the unit tangent vector
along . The curl-curl problem (2) is usually solved directly using )curl(H conforming vector
finite-elements [13,14,15,16,17]. However, this is non-elliptic when the )(0 curlH formulation is used and
hence the convergence analysis of both the numerical scheme and its solvers are more complicated. For any
);(0  curlHu due to the well-known Helmholtz decomposition[18, 16], we have the following orthogonal

A two grid discretization method for decoupling time-harmonic maxwell’s equations
decomposition:
uu = (5)
where ).(and);();( 1
00  HdivHcurlHu 
 The space );( 
divH is defined as follows:
)}(=:)]([{=);( 2
2
2
1
12
2 





 L
xx
LdivH

 (6)
0}=:);({=);(   divHdivH 
(7)
It is trivial to show that )(1
0 H satisfies
),(=),(   f (8)
for all ),(1
0 H (8) is the variational form of the Poisson problem. Many successful schemes have
been developed for solving this problem. Considering u as the weak solution of the following reduced curl-curl
problem [39], then Find );();(0  
 divHcurlHu such that
),(=),(),(  fuu   (9)
for all ).;();(0  
divHcurlH
Unlike the non-elliptic curl-curl problem (1), the reduced problem (9) is an elliptic problem. In particular, the
solution u has elliptic regularity under the assumption that ,)]([ 2
2  Lf which greatly simplifies the
analysis.
III. A Model Maxwell’s Equation
The Maxwell’s equation stated as the following equations in a region of space in
3
R occupied by the
electromagnetic field:
t
H
E


 = (10)


=E (11)
J
t
E
H 


 = (12)
0=H (13)
where  is the electric permitivity, and  is the magnetic permeability. Equation (10) is called
Faraday’s law and describes how the changing of magnetic field affects the electric field. The equation (12) is
referred as Ampère’s law. The divergence conditions (11) and (13) are Gauss’ laws of electric displacement and
magnetic induction respectively.
Let the radiation frequency be , such that 0,> then we can find solutions of the Maxwell’s equations of
the form
)(êxp=),( xEtxE ti
(14)
)(êxp=),( xHtxH ti
(15)
)(êxp=),( xJtxJ ti
(16)
)(êxp=),( xtx ti
 
(17)
Differentiating (15) yields

)(êxp= xHti
t
H ti
 



(18)
Substituting (14) and (18) into (10), we obtain
)(ˆ=ˆ xHiE  (19)
Also, putting (14) and (17) into (11), gives

ˆ
=Ê (20)
Further differentiation of (14), yields
)(êxp= xEiw
t
E iwt



(21)
Substituting (15),(16) and (21) into (12), we obtain
JEiH ˆˆ=ˆ   (22)
Combining (15) and (13), gives
0=ˆH (23)
It can be observed that when the charge is consumed, the divergence conditions (20) and (23) are always
satisfied, provided that the equations (19) and (22) holds. Then combining the equations (13) and (22) we have
JiEE ˆ=ˆˆ 2
 (24)
and
JHH ˆ=ˆˆ 2
  (25)
Considering equation (1) with perfectly conducting boundary condition for the curl-curl problem
(24)-(25), where
2
R is a bounded polygonal domain, R is a constant, and .)]([ 2
2  Lf The
curl-curl problem (1) appears in the semi-discretization of electric fields in the time-dependent (time-domain)
Maxwell’s equations when 0> and the time-harmonic (frequency domain) Maxwell’s equations when
0. If the vector equation is written as the scalar-valued in the real and imaginary parts respectively, we
obtain the following equivalent coupled equations and the boundary condition:
 in= 122111 fuuu  (26)
 in= 212212 fuuu  (27)
1,2,=0,= juj (28)
IV. Variational Formulation
The variational formulation can be obtained in terms of either electric field E or magnetic field .H
This depends upon the choice in which one of the equations (24)-(25) is to be satisfied weakly when discretized in
the distributional sense, and the other one strongly. Hence, in the E -field formulation, we select weak sense by
multiplying the curl-curl coupled equation (26) by a suitable test function 1 and integrate over the domain 
to obtain
dxfdxudxudxu 1112211111 =   
 (29)
Therefore, we get
dxfdxu 1111 =   
 (30)
Thus, the boundary integral varnishes since 0=1 on . Therefore, we obtain the variational
formulation of the coupled equation (26) as follows:
1111 =)(    
fu (31)
Expressing (31) in form of inner product, we get
)(=)( 1111   fu (32)
Similarly, we obtain the weak formulation of the couple equation (27) as

)(=)( 2222   fu (33)
Combining (29) and (31) and expressing in terms of inner products, yields
)(=)()()( 1112211111   fuuu (34)
In a similar manner, we get
)(=)()()( 2221222222   fuuu (35)
Combining the equations (34) and (35), we have
)()(=),(ˆ 2211 uuu   (36)
and
)()()()(=),( 212222122111   uuuuuN (37)
where
),(=)()( 2211  fff  (38)
),(),(ˆ=),(  uNuu  (39)
Hence, the variational formulation for equations (26)-(28) is derived as
);(,),(=),( 0  curlHufu  (40)
V. Theorem(Regularity Result)
Assume that
)(0),(x)(),(x)( 1
22
xVLLVLLf  
(41)
Then the variational problem (40) has a unique solution )(x)( 22
 HH and
02 PˆPPP fψ (42)
Proof. By the bilinear and linear continuous operators which implies the linearity and boundedness of a set, we
have
)(x)(,,|),(| 1
0
1
011  HHwuwuwua PPPˆP
Hence, by the Lax Milgram Lemma, the weak formulation(40) has a unique solution such that
).()( 1
0
1
01  HH Also, from the regularity theory for the second-order elliptic equations as proposed by
Evans C. Lawrence, we have
)(
)(2)(2))(2(2 

LLLH
fC PPPPPP  (43)
where the constant C depends only on the domain . This simply means that
02012 PPPˆPPP f (44)
Set =u in (40), we have
01 PˆPPP f (45)
Hence, (42) follows from (44) and (45).
In the following section, we define triangulation concept which is used in the fundamental theorems on
error analysis.
VI. Definition (Triangulation)
Let  be a quasi-uniform triangulation of the finite element space  with mesh size 0,>h and let
)(1
00  HSh
be the corresponding piecewise linear function space. It therefore means that
h
S0 is a
finite-dimensional subspace of the Hilbert space .1
0H Then the finite element approximation of the equation (42)
is defined as
Find

hh
hhhh SSufu 00),(=),(   (46)
VII. Theorem (Error Analysis Of Finite Element Discretization)
Assume that theorem (0.5) holds. Then h has the error estimate
0,1=2
2
sh s
sh PPˆPP  
 (47)
Proof. From the variational formulation (40) and (46), we obtain
hh
hhh SSuu 000=),( 
then we have
hh
hhh SSuue 000=),(  (48)
Let
hhI
SS 00  be the interpolation error bound of 1 and by (48), we have
211 PPˆPˆPPP  he I
h  (49)
In order to show uniqueness of the solution, the auxiliary problem in equations (28)- (30) are written as
 xxgxuxVxuxVxu )(=)()()()()( 122111 (50)
 xxgxuxVxuxVxu )(=)()()()()( 212212 (51)
 xjxuj 1,2=0,=)( (52)
VIII. Theorem(Uniqueness Of Solution)
Let ),(x)( 22
 LLg then there exists a unique solution
)()(x)()( 1
0
21
0
2
 HHHHu
such that
)(x)(=)( 1
0
1
0  HHww)(g,uw, (53)
and
02 PˆPPP gu (54)
Proof. The equation (54) is satisfied by (53). Also, let
hhI
SSu 00  be the interpolation of u and from (53),
we obtain
10 PPˆPP hh ehe (55)
Therefore, (47) follows from (48) and (55), which means that
),(=),( hhe   (56)
and we obtain for s=0
00 PˆPPP hh e  (57)
Then by equations (55) and(49), we have
10 PˆPPP hh e  (58)
hence we get
2
2
0 PPˆPP  hh (59)
This completes the proof of the uniqueness
IX. Definition Of Two Grid Technique
Two grid method is a discretization technique for nonlinear equations based on two grids of different
sizes. The main idea is to use a coarse grid space to produce a rough approximation of the solution of nonlinear
problems. This method involves a nonlinear problem being solved on the coarse grid with grid size H and a
linear problem on the fine grid with grid size .<< Hh
X. A New Two-Grid Finite Element Method
The finite element discretization (46) corresponds to a coupled system of equations in the general case.
Thus, to reduce the rigour another finite element space ))(( 1
000  HSS hH
defined on a coarser

quasiuniform triangulation (with mesh size hH > ) of  is proposed as the following algorithm:
XI. Algorithm A1
Step 1 Let
HH
h SS 00  such that
HH
H SSf 00),(=),(   (60)
Step 2 Let
hh
h SS 00
*
 such that
hh
hhHhh SSuuNufu 00
*
),(),(=),(ˆ   (61)
It can be observed from step 2 that the linear system is decoupled and consists of two separate Poisson
equations. However, on the coarser space it is required that a coupled system is solved in step 1, since the
smallness of the
H
S0 rather than the dimension of
h
S0 is evident. It therefore follows that
*
h can attain the
optimal accuracy in normH 1
if the coarse mesh size H is taken to be h .
XII. Theorem
Assume that (41) holds. Then
*
h has the following error estimates:
2
1
*
Hh ˆPP   (62)
Consequently,
2
1
*
Hhh  ˆPP  (63)
then,
*
 has the same accuracy as h in
1
H -norm for
2
H
Proof. Let ,=ˆand= *
hhhhh ee   it then follows that by (61) and (46) we have
hh
hhHhhh SSuuNue 000=),(),ˆ(ˆ   (64)
Putting hh eu ˆ= in the last expression, yields
00
2
1
ˆ)ˆ,ˆ(ˆˆ PPPˆPˆPP hHhhhh eeee  
and then
01
ˆ PˆPPP Hhhe   (65)
By taking (47),(44) and the inequality,
000 PPPPPP HhHh  
we get
22
0 HhHh  PP  (66)
Therefore, (62) follows from (65) and the above inequality. Also, (63) follows from (47), (62) and the following
inequality:
111
*
ˆ PPPPPP hhh e 
Algorithm A1 can be improved in a successive technique as follows:
XIII. Algorithm A2
Let 0.=0
h Assume that
hhk
h SS 00  has been obtained, then
hhk
h SS 00
1

 is defined as follows:
Step 3. Find
HH
H SSe 00  such that
HHk
hH SSe 00),,(),(=),(   f (67)
Step 4. Find
hhk
h SS 00
1

 such that
hh
hhH
k
hhh
k
h SSuueNuu 00
1
),,(),(=),(ˆ 
 f (68)
XIV. Theorem
Under the assumption (41),
k
h has the following error estimate:
1,1
1  
kHkk
hh ˆPP  (69)

Consequently,
1,1
1  
kHh kk
h ˆPP  (70)
where ,k
h 1,k has the same accuracy as h in
1
H -norm if .= 1
1
k
hH
Proof. From (46) and (61), we have
hh
hhH
k
hhh
k
hh SSuueNu 00
1
),),((=),(ˆ  
 (71)
which, by taking ,= 1
 k
hhhu  gives
),(ˆ 112
1
1 
 k
hh
k
hh
k
hh  ˆPP
,)( 0
1
0 PPPˆP 
 k
hhH
k
hh e 
and then
01
1
)( PˆPPP H
k
hh
k
hh e 
 (72)
It follows from (46) and (60) that
HH
H
k
hh SSe 000,=)),((   (73)
This implies that
))(),(()( 1
2
1 H
k
hhH
k
hH
k
hh eee   ˆPP
)),((= k
hhH
k
hh e  
,)( 11 PPPˆP k
hhH
k
hh e  
and then
11)( PˆPPP k
hhH
k
hh e   (74)
Let u be the solution of problem (53) with )(= H
k
hh eg   and
HHI
SSu 00  be the interpolation
of .u Then according to (54) and (73), we have
)),((=)),((=)( 2
0
I
H
k
hhH
k
hhH
k
hh uueuee   PP
11)( PPPˆP I
H
k
hh uue  
21)( PPPPˆ ueH H
k
hh  
01 )()( PPPPˆ H
k
hhH
k
hh eeH  
which implies that
10 )()( PPˆPP H
k
hhH
k
hh eHe   (75)
Therefore, from (71), (75) and (73), we have
1,1
11
1
1
1  
kHH hh
kk
hh
k
hh PPˆPPˆPP  (76)
Observe that
1
h is the solution h* obtained by Algorithm A1, thus, (68) follows from (76) and (62).
Furthermore, (70) follows from (68), (47) and the following inequality:
11 PPPPPP k
hhh
k
h  
As stated in Theorem 12.0, it is sufficient to take 1
1
= k
hH to obtain the optimal approximation in
1
h -norm. Therefore, the dimension of
H
S0 can be much smaller than the dimension of .0
h
S Finally, the
numerical examples and error estimates on the efficiency of the algorithms is demonstrated in [7] with boundary
value problem of the Schrödinger type.
XV. Comments
Using the Maxiwell equation as an illustration, we presented in this paper a new two-grid discretization
technique to decouple systems of partial differential equations. This new application of the two-grid decoupling
technique can obviously be extended in many different ways, for different discretizations such as finite volume
and finite difference methods for other types of partial differential equations.

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A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equations

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A Two Grid Discretization Method For Decoupling Time-Harmonic Maxwell’s Equations