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Advanced Mathematical Methods and Software Platform for Solving Multi-Physics,
Multi-Domain Problems on Modern Computer Architectures: Applications to Environmental
Engineering and Medical Problems
Work Package 2 - Task 2.2 Interface Relaxation Methods (IR)
Kick Off Meeting - 2012/07/21 – Chania, Crete, Greece
Yota Tsompanopoulou - Univ. of Thessaly
outline
Timetables - Deliverables
Scientific Description
21/7/2012 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
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Timetables - Deliverables
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Start Date: 01/01/02012 End Date: 31/09/2015
Month 2012 2013 2014 2015
jan-mar
apr-jun
jul - sep
oct - dec
jan-mar
apr-jun
jul - sep
oct - dec
jan-mar
apr-jun
jul - sep
oct - dec
jan-mar
apr-jun jul aug sep
WP Title: Research Team: List of delivarables: 1-3 4-6 7-9 9-12
13-15
16-18
19-21
22-24
25-27
28-30
31-33
34-36
37-39
40-42 43 44 45
1 WP 1 COORDINATION & PLANNING 1 (2, 3)
3 (ANNUAL)INTERMEDIATE TECHNICAL REPORTS
1 (2, 3) 1 FINAL TECHNICAL REPORT
2 WP 2
NUMERICAL & ANALYTICAL METHODS FOR DISCONTINUOUS
MULTIDOMAIN PROBLEMS (DISCONTINOUS COLLOCATION
METHODS, RELAXATION METHODS, STOCHASTIC-DETERMINISTIC HYBRID
METHODS, FOKAS METHODS)
1 2.1 DISCONTINUOUS COLLCATION: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE CODE
2 (1) 2.2 RELAXATION METHODS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE
2 2.3 STOCHASTIC-DETERMINISTIC HYBRID METHODS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE
1 (3) 2.4 FOKAS METHODS: 1 TECHNICAL REPORT / 2 PAPERS
3 WP 3
MAPPING & IMPLEMENTATION ON STATE-OF-THE-ART
COMPUTING ENVIROMENTS (CLUSTERS-GRIDS-CLOUDS-FPGAS-RECONFIGURABLE
ARRAYS)
3 (1) 3.1 CLUSTERS-GRIDS-CLOUDS: 1 TECHNICAL REPORT / 3 PAPERS / SOFTWARE
2
3.2 FPGAS-RECONFIGURABLE ARRAYS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE
4 WP 4
INTEGRATING, EVALUATING AND VALIDATING THE RESULTS ON
ENVIRONMENTAL AND MEDICAL PROBLEMS
2, 3 (1) 4.1 INTEGRATION: 1 TECHNICAL REPORT / SOFTWARE
1 (2, 3) 4.2 MEDICAL PROBLEMS: 1 TECHNICAL REPORT / 1 PAPERS
1 (2, 3) 4.3 ENVIROMENTAL PROBLEMS: 1 TECHNICAL REPORT / 1 PAPERS
5 WP 5 PROJECT EVALUATION 1 PROJECT EVALUATION BY EXTERNAL EVALUATORS
6 WP 6 DISSEMINATION OF THE RESULTS 1 (2, 3) WEB SITE, WORKSHOP: WORKSHOP PROCEEDINGS
Start End
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Timetable - Deliverables
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Timetable
Start : 1/1/2012
End : 30/6/2015
Duration : 40 months
Deliverables
1 technical report (τεχνικές εκθέσεις)
3 research papers (επιστημονικά άρθρα)
Scientific Description
Interface Relaxation Methods
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Interface Relaxation (IR) Methods for General Elliptic PDEs
1. Introduction • Basic target. • A composed Problem. • Basic Methodology • General stuff.
2. IR Methods. • AVE. • GEO. • ROB.
3. Convergence Analysis. 4. Numerical Results.
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Main Target The creation of a general methodology for the
solution of the composite PDE problems with the following properties:
1. Consists of a set of collaboration solvers for local (simple) problems.
2. Allow the choice of the most appropriate discretization method of the local problem.
3. Simplifies the geometry and the physics of the global problem.
4. Software reuse. 5. Wide applicability and high efficiency. 6. Increased adaptivity and inherent parallelism.
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A Composed PDE Problem
Uxx + Uyy = 0
U = 1.5Un
Uxx+Uyy= -1.0 Uxx+Uyy + yUx = -1.0
Uxx+Uyy + xUx = -1.0
Heat Radiaton Region
Mounting Region
Heat Producing Region
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Basic Methodology
1. Split the problem in the PDE level.
2. Use solution’s properties to set conditions on the common boundaries (interfaces)
3. Give initial values on the interfaces.
4. Solve each PDE locally.
5. Use relaxation method to smooth the solution/derivatives or/and operators on the interfaces.
6. Check convergence criteria and go back to 4.
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Basic Methodology
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, in
, on b
Lu f Ω\ Ω
u u
Initial global composed PDE problem
Basic Methodology
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, in
0 , on \
, on
i i i i
ij i i
b
i i i
L u f Ω
G u i j
u u
Set of simple PDE subproblems
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Basic Methodology
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Overview in IR Methods Primitive Relaxers: About 10 proposed/implemented/analysed.
Smooth values and normal derivatives in various ways
Advanced Relaxers: Smooth additional operators (continuity of mass,
temperature, conservation of energy/momentum, equilibrium conditions, Lagrange multipliers, Steklov-Poincare operators, etc.)
Differences in convergence and applicability.
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AVE
For k=0,1,2,…
Define:
Solve the Neumann problems:
1,...,1,12
1
2
pidx
du)β(
dx
duβg
ii xx
k)(
ii
xx
k)(
iii
,1
)12(
11 fuL k
00
)12(
1
xx
ku
1
)12(
1
1
gdx
du
xx
k
στο Ω1 ,)12(
p
k
pp fuL
1
)12(
1
p
xx
k
pg
dx
du
p
0)12(
pxx
k
pu
στο Ωp ,)12(
i
k
ii fuL
1
)12(
1
i
xx
k
i gdx
du
i
i
xx
k
i gdx
du
i
)12(
στο Ωi
i=2,…,p-1
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AVE (cont’d)
Define:
Solve the Dirichlet problems:
1...,11 12
1
12
p,,i)uα(uαhii xx
)k(
iixx
)k(
iii
,1
)22(
11 fuL k
00
)22(
1
xx
ku
1
)22(
11
huxx
k
στο Ω1 ,)22(
p
k
pp fuL
1
)22(
1
pxx
k
p hup
0)22(
pxx
k
pu
στο Ωp ,)22(
i
k
ii fuL
1
)22(
1
ixx
k
i hui
ixx
k
i hui
)22(
στο Ωi
i=2,…,p-1
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GEO
For k=0,1,2,…
Define:
Solve the Dirichlet problems:
1,...,1,2
)(
1
)()(
1
)(
pidx
du
dx
duρ
uug
ii xx
k
i
k
ii
xx
k
i
k
ii
,1
)1(
11 fuL k
00
)1(
1
xx
ku
1
)1(
11
guxx
k
στο Ω1 ,)1(
p
k
pp fuL
1
)1(
1
pxx
k
p gup
0)1(
pxx
k
pu
στο Ωp ,)1(
i
k
ii fuL
1
)1(
1
ixx
k
i gui
ixx
k
i gui
)1(
στο Ωi
i=2,…,p-1
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ROB
For k=0,1,2,…
Define:
Solve the problems:
1,...,11
11
pi
uλdx
dug
uλdx
dug
i
i
i
i
i
i
xx
(k)
ii
xx
(k)
ii
xx
(k)
ii
xx
(k)
ii
,1
)1(
11 fuL k
00
)1(
1
xx
ku
στο Ω1
1
1
1
11
1
1
11
guλdx
du
xx
)(k
xx
)(k
,)1(
i
k
ii fuL
i
i
xx
)(k
ii
xx
)(k
i guλdx
du
ii
1
1
στο Ωi
i=2,…,p-1
i
i
xx
)(k
ii
xx
)(k
i guλdx
du
ii
1
1
1
1
11
,)1(
p
k
pp fuL
0)1(
pxx
k
pu
στο Ωp
p
p
xx
)(k
pp
xx
)(k
pguλ
dx
du
pp
1
1
1
1
11
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Convergence Analysis
The model problem.
Convergence conditions and “optimum” values for the relaxation parameters of AVE.
Optimum values for the relaxation parameters of ROB.
Convergence conditions for GEO.
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Model problem (1-dim)
0xa 1x1ix ix 1px bxp
1 ip
(a,b)x
0)()( buau
,2 fuγuxx
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It is known that:
The solution of -uxx + γ2u = 0 ,
with
is given by
(a,b)x
,υu(a)c(a)uc 121
243 υu(b)c(b)uc
1α)γ(b
4321
α)γ(b
4321
x)γ(b
43
x)γ(b
43 υ)ecγc)(cγ(c)ecγ)(ccγc(
)ecγc()ecγ(c
.υ)ecγc)(cγ(c)ecγ)(ccγc(
)ecγ(c)ecγc(2α)γ(b
4321
α)γ(b
4321
α)γ(x
21
α)γ(x
21
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D.E. for the error functions (AVE) i=1,…,p , k=0,1,…
Neumann steps (similar for the Dirichlet steps):
1...,112
1
2
p,,idx
dε)β(
dx
dεβgε
ii xx
k)(
ii
xx
k)(
iii
,012
11 )k(εL
00
12
1
xx
)k(ε
1
12
1
1
gεdx
dε
xx
)k(
in Ω1 ,εL )k(
pp 012
1
)12(
1
p
xx
k
pg
dx
d
p
012
pxx
)k(
pε
in Ωp ,εL )k(
ii 012
1
12
1
i
xx
)k(
i gεdx
dε
i
i
xx
)k(
i gεdx
dε
i
12
in Ωi
i=2,…,p-1
,uuε i
(k)
i
(k)
i
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23
Error functions (AVE)
,1
1
2
1
2
sinh
1cosh
2
11
2
111sinh
cosh12
k)(
,iii
k)(
i,ii)ii
(γiγ
))i
x(xi
(γ
k)(
i,ii
k)(
,iii)ii
(γiγ
x))i
(xi
(γ)k(
i
)dεβ(dεβ
)dεβ(dεβ(x)ε
12 ,...,pi
Neumann:
Dirichlet:
,1
1
12
1
12
)sinh(
1sinh
12
11
12
111sinh
sinh22
)k(
,iii
)k(
i,iiii
γ
))i
x(xi
(γ
)k(
i,ii
)k(
,iii)ii
(γ
x))i
(xi
(γ)k(
i
)εα(εα
)εα(εα(x)ε
12 ,...,pi
,
2
2
dx
)(xdεdε
j
k)(
ik)(
i,j where .1212 )(xεε j
)k(
i
)k(
i,j
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Relation between two consecutive errors (AVE)
,
112211
T,...,ε,εεε (k)
,pp
(k)
,
(k)
,
(k)
,112211
T,...,dε,dεdεdε (k)
,pp
(k)
,
(k)
,
(k)
,12)22( )k(Dk
dεMε
k)(Nk
εMdε2)12(
,2
1
ii
iD
,iiγm
αM
,1
11
1
ii
ii
ii
iiD
i,iγm
)nα(
γm
nαM
,12
11
1
ii
iD
i,iγm
)α(M ,12 ,...,pi,2,1 ...,pi
22 ,...,pi
,2
1
i
iiN
,iim
γβM
,1
1
11
i
iii
i
iiiN
i,im
γ)nβ(
m
γnβM
,12
1
11
i
iiN
i,im
)γβ(M ,12 ,...,pi,2,1 ...,pi
22 ,...,pi
•
•
•
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Convergence, optimum values (AVE)
Theorem: Let of length
and γi=γ, i=1,...,p. If then
is minimized by
,...,pii 1 ,
DN MM
iiii
iii
nmnm
nmα
11
1
.22
11
1 ,...,p-i,nmnm
nmβ
iiii
iii
and
,1 ,21ln
,...,piγ
)(
i
iii
p
i
i ,xx, ΩΩΩ 1
1
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D.E. for the error functions (GEO)
1,...,1,2
)(
1
)()(
1
)(
pidx
dε
dx
dερ
εεgε
ii xx
k
i
k
ii
xx
k
i
k
ii
,0)1(
11 kεL
00
)1(
1
xx
kε
1
)1(
11
gεεxx
k
στο Ω1 ,0)1( k
ppεL
1
)1(
1
pxx
k
p gεεp
0)1(
pxx
k
pε
στο Ωp ,0)1( k
iiεL
1
)1(
1
ixx
k
i gεεi
ixx
k
i gεεi
)1(
στο Ωi
i=2,…,p-1
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27
Error functions (GEO)
1,...,2 ,2
2
1
1
1111
1111
11
pi(k)
,ii
(k)
i,ii
(k)
,ii
(k)
i,i
γγ
)x(xγ)x(xγ
(k)
i,i
(k)
,iii
(k)
i,i
(k)
,ii
γγ
x)(xγx)(xγ)(k
i
dεdερεε
ee
ee
dεdερεε
ee
ee(x)ε
iiii
iiii
iiii
iiii
p=3, T,dεdεdεdεεε,εεε (k)
,
(k)
,
(k)
,
(k)
,
(k)
,
(k)
,
(k)
,
(k)
,
(k)
2322121123221211 ,,,,,
,,1,0 ,)1(
kεMε(k)k ,88RM
,
tanh
1
ii
iγ
A
2,1 ,sinh
1 i
γB
ii
i
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Structure of iteration matrix (GEO)
3323323333
22222222122122222222
22222222122122222222
1111111111
22
22
11
11
0022
00
2222
2222
000022
002
1
2
100
002
1
2
100
00002
1
2
1
00002
1
2
1
γAργAργAγA
γAργAργBργBργAγAγBγB
γBργBργAργAργBγBγAγA
γAργAργAγA
ρρ
ρρ
ρρ
ρρ
M
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Region of convergence (GEO)
Lemma: The non-identically zero eigenvalues of Matrix M,
are equal to the non-identically zero eigenvalues of ,
where
M~
.1
1~
33222221
22222111
)AγA(γρBγρ
Bγρ)AγA(γρM
Theorem: If of length
, GEO converges to the solution of the original
problem iff
3,21 , ,ii
)BγC(CρC
Cρρ,
Cρ
2
2
2
22112
112
1
12
220
20
where .2,1 ,11 iAγAγC iiiii
iii ,xx ΩΩΩΩΩ 1321 ,
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D.E. for the error functions (ROB)
111
11
,...,pi
ελdx
dεgε
ελdx
dεgε
i
i
i
i
i
i
xx
(k)
ii
xx
(k)
ii
xx
(k)
ii
xx
(k)
ii
,01
11 )(kεL
00
1
1
xx
)(kε
in Ω1
1
1
1
11
1
1
11
gεελdx
dε
xx
)(k
xx
)(k
,01 )(k
iiεL
i
i
xx
)(k
ii
xx
)(k
i gεελdx
dε
ii
1
1
in Ωi
i=2,…,p-1
i
i
xx
)(k
ii
xx
)(k
i gεελdx
dε
ii
1
1
1
1
11
,01 )(k
ppεL
01
pxx
)(k
pε
in Ωp
p
p
xx
)(k
pp
xx
)(k
pgεελ
dx
dε
pp
1
1
1
1
11
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31
Error functions (ROB)
•
•
1,...,2 ,))(())((
)()(
))(())((
)()(
11
11
11
11111
11
1
11
piελdεeλγλγeλγλγ
eλγeλγ
ελdεeλγλγeλγλγ
eλγeλγ(x)ε
(k)
,iii
(k)
,iiγ
iiii
γ
iiii
)x(xγ
ii
)x(xγ
ii
(k)
,iii
(k)
,iiγ
iiii
γ
iiii
x)(xγ
ii
x)(xγ
ii)(k
i
iiii
iiii
iiii
iiii
T,dε,ε,εdεdε,εεdεε (k)
p,p
(k)
p,p
(k)
,pp
(k)
,pp
(k)
,
(k)
,
(k)
,
(k)
,
(k)
11111112121111 ,,,,
,,1,0 ,)1(
kεMε(k)k
,)()(
)()(1
1
1
1)1(2),1(2
iiiiiiiiiii
iiiiiiiii
i
iinλmγλγnλmγγ
mλnγλmλnγ
dM
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Structure of iteration matrix (ROB)
000000
00000
00000
00000
00000
00000
00000
000000
11212
12112312112
12212312212
6535
6434
4313
4212
21
)(p),(p
)(p,)(p)(p,)(p
)(p,)(p)(p,)(p
,,
,,
,,
,,
,
M
MM
MM
MM
MM
MM
MM
M
M
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Optimal parameters (ROB)
where has the same structure with Μ and
,0~
0 )\Mσ(σ(M)\
,)()(~ 11
1)1(2),1(2
i
iiiiiiiiiiii
d
mλnγλmλnγλM
M~
,1,,2 pi
Theorem: Consider the decomposition of Ω into p subdomains Ωi
of length
The spectral radious of the iterative matrix is zero if the parameters λi , are selected as
,1
p
pp
pm
nγλ
.2,...,1 ,)(
1
pi
mλnγ
nλmγγλ
iiii
iiiiii
,...,pii 1 ,
Lemma: Let Μ the iteration matrix of the method, then
.11
2
iiiiiiiii nλλγmλλγd
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Numerical experiments
4 composed problems
History of convergence
Verification of “optimum” parameters
Effect of descritization of the domain
Effect of descritization of the operator
21/7/2012 34 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
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4 composed problems
PDE1: Helmholtz equations with Cartesian decomposition of domain
7,..,0, ,2 iinfuγΔu I
iii
,20 ,16 ,10 ,exp2
1 2
3
2
2
2
1
2
0 γγγγ yx
)8
exp(5 ,25
,15 ,4)(sin2
2
7
2
6
2
5
2
4
yxγγ
γπyxγ
PDE2: Helmholtz equations with general decomposition of domain
3,..,0, ,2 iinfuγΔu II
iii
,25 ,16 ,exp2
1 2
2
2
1
2
0 γγγ yx
4)8
exp(2
5)(sin2
4
yx
πyxγ
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4 composed problems PDE3: Linear operator with general decomposition of domain
IVin
yxuΔu
0
22 ,0)2(602.0
PDE4: Non-Linear operator with general decomposition of domain
IVIVinuΔu 31 , ,04.0 IV
y
u
x
uinuΔu 2 ,03.0)(10
IV
u
in
yxuΔu
0
22
1000
,0)2(60)1(2.0
IVIVinuΔu 31 , ,04.0
IV
u
inu
y
u
x
u
y
uΔu
2
500
1
2
2
1000
,03.0
31
21/7/2012 36 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
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History of convergence for PDE4 (ROB)
3η επαν. 2η επαν. 1η επαν.
21/7/2012 37 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
Verification of “optimum” parameters (PDE1)
ROB AVE
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Verification of “optimum” parameters (PDE3)
ROB
21/7/2012 39 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
Effect of descritization of the operator History of convergence for ROB (PDE2)
Finite Difference (5 point star)
Collocation
Bi-linear Finite Elements 21/7/2012 40 Kick Off Meeting - WP2 - Task 2.2
Yota Tsompanopoulou
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Effect of discretization of the domain
History of convergence for ROB with bi-linear finite elements
for PDE2.
h=0.10
h=0.20 21/7/2012 41 Kick Off Meeting - WP2 - Task 2.2
Yota Tsompanopoulou
Conclusions
21/7/2012 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
42
The IR methodology is suitable for composed PDE problems.
There are theoretical results for convergence, with “optimum” relaxation parameters.
Theoretical results are verified with numerical experiments (model and general problems).
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References
21/7/2012 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou
43
Analysis of an Interface Relaxation Method for Composite Elliptic
Differential Equations, P. Tsompanopoulou and E. Vavalis, Journal of
Computational and Applied Mathematics, 226(2), (Apr. 2009), pp 370-387.
An Experimental Study of Interface Relaxation Methods for Composite
Elliptic Differential Equations, P. Tsompanopoulou and E. Vavalis, Applied
Mathematical Modelling, 32, (Aug. 2008), pp 1620-1641.
Fine Tunning Interface Relaxation Methods for Elliptic Differential
Equations, J.R. Rice, P. Tsompanopoulou and E.A. Vavalis, Applied
Numerical Mathematics, 43(4), (Nov. 2002), pp 459-481.
Interface Relaxation Methods for Elliptic Differential Equations, J.R.
Rice, P. Tsompanopoulou and E.A. Vavalis, Applied Numerical Mathematics,
32 (2000), pp 219-245.
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44
Thanks!
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Possible project acronyms
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45
ACAdEMy Advanced mathematiCAl Engineering Medical
ACCEPt AdvanCed mathematiCal Engineering Problems
ADVISE ADVanced mathematIcal Software Engineering
AMALTHEA Advanced MAthematicaL meTHods EnviromentAl
AMETHyST Advanced Mathematical mETHods SofTware
ANiMATED AdvaNced MAThematical Engineering meDical
ANiMATE AdvaNced MAthematical meThods Engineering
ANAEMIA AdvaNced mAthematical Engineering MedIcAl
ANEMONE AdvaNcEd Mathematical envirOmeNtal Engineering
ACHIEVE AdvanCed matHematIcal EnViromental Engineering
CoMMENd advanCed Mathematical Methods ENgineering
CoMPLETE advanCed Mathematical PLatform EnviromenTal Engineering