5.4 Uniform Coarse Mesh
5.4.1 Uniform Coarse Mesh with 2 Mesh Points
We consider a global mesh on⌦= (0,1), where⌦=⌦1[⌦2, such that it is fine on
⌦1 with the mesh size h, and coarse on ⌦\⌦1 with only two mesh points which are uniformly distributed with mesh size h1, see Figure 5.1. Using the finite di↵erence method, we discretize this problem and we find the system in (5.1.6), where fromK1 in (5.1.7) we have the blocks
Ae2 =
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 69
0 20 40 60 80 100
2 4 6 8 10 12 14 16
p∗12 ˆ p12
Figure 5.2: The Bank-Jimack Robin parameter ˆp12versus the OSM Robin parameter p⇤12 for = 5.05e 1.
Now we compare the last entry of the matrix in (5.4.1) with the last entry of the matrix in OSM of⌘ in (5.2.4) and we obtain
2
h2 +⌘ E12= 1 h2 +⌘
2 +p12
h . Hence we find the Robin parameter
p12= 1 h +⌘h
2 hE12.
We study the behavior ofp12 for the case whenh, the fine mesh size, tends to zero.
Considering the mesh in Figure 5.1 (middle row) we obtain h1 = 1 2 h. Inserting h1 intoE12, we find the limit of p12 when h!0, which we call ˆp12,
ˆ p12:= 1
4
(1 )4⌘2+ 16(1 )2⌘+ 32
(1 )3⌘+ 8(1 ) . (5.4.3)
We see that the Robin parameter is a rational function of⌘. In Figure 5.2 we compare the Robin parameter of the Bank-Jimack method ˆp12 with the Robin parameter of the OSM which we find by fixingC11 in (5.2.9) equal to zero, and we call it p⇤12.
p⇤12=p⌘coth(p⌘(1 )). (5.4.4) We observe that for ⌘ < 10 the Robin parameter of both methods are quite close, whereas for⌘ >10 the Robin parameters are becoming di↵erent. Next step, we find the termC11 in the convergence factor of the Bank-Jimack method,
C11= N1
D1, (5.4.5)
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD the mesh sizeh, and coarse on⌦\⌦2 with only two mesh points which are uniformly distributed with mesh sizeh2, see Figure 5.1 (last row). Using the finite di↵erence method we discretize this problem and we find the matrix system in (5.1.8), and from K2 in (4.3.10) we find the blocks We compare the first entry of the matrix in (5.4.7) with the first entry of the matrix in the OSM of the⌘ in (5.2.5) and we obtain
Hence we find the Robin parameter of the Bank-Jimack method p21= 1
h +⌘h
2 hE21, (5.4.10)
and like before we find the limit of p21 when h tends to zero. From Figure 5.1 (last row), we findh2= ↵2h. Insertingh2 into the formula ofE21, we find the limit ofp21
Again we see that the Robin parameter is a rational function of ⌘. In Figure 5.3 we compare the Robin parameter of the Bank-Jimack method ˆp21 with the Robin
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 71
0 20 40 60 80 100
2 4 6 8 10 12 14 16
p∗21 ˆ p21
Figure 5.3: The Bank-Jimack Robin parameter ˆp21versus the OSM Robin parameter p⇤21 for↵= 5.05e 1.
parameter of the OSM which we find by fixingC22 in (5.2.10) equal to zero, and we call itp⇤21.
p⇤21=p⌘coth(p⌘↵), (5.4.12)
We should notice that in the case of having symmetry for the subdomains ⌦1 and
⌦2, which means that ↵ = 1 , the Robin parameters in (5.4.3) and (5.4.11) are identical. For the termC22 in the convergence factor of the Bank-Jimack method we have
C22= N2
D2, (5.4.13)
where we findN2 and D2 by inserting ˆp21 into (5.2.10)
N2:=p
⌘cosh(p
⌘↵) 1 4
↵4⌘2+ 16↵2⌘+ 32
↵3⌘+ 8↵ sinh(p
⌘↵), D2:=p
⌘cosh(p
⌘(1 ↵)) +1 4
↵4⌘2+ 16↵2⌘+ 32
↵3⌘+ 8↵ sinh(p
⌘(1 ↵)).
Now we find the convergence factor of the Bank-Jimack method for a uniform coarse mesh with 2 points
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 72
⇢:=C11C22=
p⌘cosh(p⌘(1 )) 14(1 (1)4⌘2+16(1)3⌘+8(1 )2⌘+32) sinh(p⌘(1 )) p⌘cosh(p⌘ ) +14(1 (1)4⌘2+16(1)3⌘+8(1 )2⌘+32) sinh(p⌘ )
(5.4.14) p⌘cosh(p⌘↵) 14↵4⌘2↵+16↵3⌘+8↵2⌘+32sinh(p⌘↵)
p⌘cosh(p⌘(1 ↵)) +14↵4⌘2↵+16↵3⌘+8↵2⌘+32sinh(p⌘(1 ↵)).
Now our goal is to observe the behavior of the convergence factor⇢for the case when we have a uniform coarse mesh with 2 points. We consider the convergence factor that we found in (5.4.14). We consider =↵+L, whereL is the size of the overlap, and we find the convergence factor⇢depending on⌘for di↵erent sizes of the overlap, see Figure 5.4.
We notice that using MATLAB we are limited for calculating the hyperbolic functions sinh and cosh in the convergence factor. For example for ⌘ 103, using MATLAB we find
sinh(⌘) =1, cosh(⌘) =1.
To solve this problem we use the definition of the hyperbolic functions and we fac-torize the terms which become too large. Then we get the convergence factor
⇢= exp( 2p⌘L)N1
D1, (5.4.15)
where the numeratorN1 and the denominator D1 are N1 :=
✓p
⌘⇣e 2p⌘(1 )+ 1 2
⌘
+1 4
(1 )4⌘2+ 16(1 )2⌘+ 32 (1 )3⌘+ 8(1 )
⇣e 2p⌘(1 ) 1 2
⌘◆
✓p⌘⇣e 2p⌘↵+ 1 2
⌘
1 4
↵4⌘2+ 16↵2⌘+ 32
↵3⌘+ 8↵
⇣1 e 2p⌘↵
2
⌘◆,
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 73
D1:=
✓p⌘⇣e 2p⌘ + 1 2
⌘
+1 4
(1 )4⌘2+ 16(1 )2⌘+ 32 (1 )3⌘+ 8(1 )
⇣1 e 2p⌘ 2
⌘◆
✓p
⌘⇣e 2p⌘(1 ↵)+ 1 2
⌘
1 4
↵4⌘2+ 16↵2⌘+ 32
↵3⌘+ 8↵
⇣e 2p⌘(1 ↵) 1 2
⌘◆.
The MATLAB functionuniformrhoCoarse2is the implementation of computing the convergence factor.
f u n c t i o n y = u n i f o r m r h o C o a r s e 2 ( L )
% u n i f o r m r h o C o a r s e 2 is the i m p l e m e n t a t i o n of the
% c o n v e r g e n c e factor of the Bank - Jimack method
% c o n s i d e r i n g a uniform coarse mesh with 2 mesh
% points . hh = h1 is the vector of the coarse mesh
% size . Output of u n i f o r m r h o C o a r s e 2 is the maximum
% of the c o n v e r g e n c e factor
global alpha ; h1 = alpha /2;
h2 = h1 ;
beta = alpha + L ;
etamax = 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; N = 1 0 0 0 0 0 ;
eta =[0:1/ N :1 l o g s p a c e ( log10 (1) , log10 ( etamax ) ,N ) ];
s = sqrt ( eta ) ;
N1 =( s .*( exp ( -2.* s .*(1 - beta ) ) +1) . / 2 + ( 1 . / 2 ) .*(( eta .^2.* h1 .^4+4.* eta .* h1 .^2+2)...
./(( eta .* h1 .^2+2) .* h1 ) ) .*( exp ( -2.* s .*(1 - beta ) ) -1) ./2) ...
.*( s .*( exp ( -2.* s .* alpha ) +1) ./2 -(1/2) .*(( alpha ^2.* eta .^2.* h2 ^2 -...
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 74
alpha .* eta .^2.* h2 ^3+2.* alpha ^2.* eta +4) ./( alpha .*( alpha .* eta .* h2 - eta .* h2 ^2+...
2) ) ) .*( - exp ( -2.* s .* alpha ) +1) ./2) ;
D1 =( s .*( exp ( -2.* s .* beta ) +1) . / 2 + ( 1 . / 2 ) .*(( eta .^2.* h1 .^4+4.*
eta .* h1 .^2+2) ./...
(( eta .* h1 .^2+2) .* h1 ) ) .*( - exp ( -2.* s .* beta ) +1) ./2)...
.*( s .*( exp ( -2.* s .*(1 - alpha ) ) +1) ./2 -(1./2) .*(( alpha ^2.*
eta .^2.* h2 ^2 -...
alpha .* eta .^2.* h2 ^3+2.* alpha ^2.* eta +4) ./( alpha .*( alpha .* eta .* h2 - eta .* h2 ^2+...
2) ) ) .*( exp ( -2.* s .*(1 - alpha ) ) -1) ./2) ; r =( exp ( -2.* s .* L ) ) .*( N1 ./ D1 ) ;
s e m i l o g x ( eta ,r ,'-') ; y = max ( r ) ;
pause drawnow end
For simplicity we suppose thath1=h2, and to satisfy this condition we fix↵= 12L and = 1+L2 . Using the uniformrhoCoarse2in MATLAB we obtain the maximum of max⌘(⇢). After plotting the convergence factor, we find the frequency ⌘1 where the convergence factor is maximized, see Figure 5.4.
In Table 5.1, we present the convergence factor ⇢ for di↵erent sizes of L, and the corresponding frequency⌘1. In Figure 5.4 we observe the behavior of the convergence factor ⇢ versus the frequency ⌘ for di↵erent amounts of L and we see that there is always one bump. In Figure 5.5 we observe the dependency of the convergence factor
⇢to the size of overlap and we see that the relation betweenL and⇢ for the 2 point case is⇢= 1 O(L12). In 2006 Gander proved in [19] that the convergence factor of the OSM in the case of having an overlap of the sizeL behaves as 1 ⇢OSM ⇠L13. Hence, the OSM is better than the Bank-Jimack method with a uniform coarse mesh with 2 points. In Figures 5.6 we observe the behavior of the frequency ⌘1 versus L and we see that the relation between them is⌘1 =O(L 1).
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
Figure 5.4: Convergence factor of the Bank-Jimack method for a uniform coarse mesh with 2 mesh points for di↵erent sizes of the overlap.
L ⇢ ⌘1
10 9 9.995e 1 1.60e10
Table 5.1: Convergence of the Bank-Jimack method for a uniform coarse mesh with 2 mesh points for di↵erent sizes of overlap, and the corresponding frequency⌘1.
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
IN1D FOR THE ⌘ EQUATION 76
10-8 10-6 10-4 10-2
10-4 10-3 10-2 10-1 100
1-ρ L1/2
Figure 5.5: 1 ⇢ versus L
10-8 10-6 10-4 10-2
102 104 106 108 1010 1012
η1 L-1
Figure 5.6: ⌘1 versusL
CHAPTER 5. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD
Figure 5.7: Global fine mesh, and two partially uniform coarse meshes with 3 mesh points.