MULTIGRID METHODSFORINTERFACEPROBLEMS
LOYCE ADAMS y
AND ZHILIN LI z
Abstract. New multigridmethodsare developed forthe maximumpreservingimmersed
in-terfacemethodappliedtoellipticandparabolicinterfaceproblems. Forellipticinterfaceproblems,
the multigridsolverdevelopedinthispaperworkswhilesomeothermultigridsolversdonot. For
diusion and reaction equations, wehave developed the second ordermaximumpreservingnite
dierenceschemeinthispaper. WeusetheCrank-Nicolsonschemetodealwiththediusionpart,
andanexplicitschemeforthereactionpart.Numericalexamplesarealsopresented.
Keywords. interfaceproblems,diusionandreactionequation,multigridmethod,
discontin-uouscoeÆcients,quadraticoptimization.
AMSsubject classications.65N06,65N50
1. Introduction. In[6],themaximumprinciplepreservingschemeisproposed
for ellipticinterfaceproblems. The resultinglinearsystem ofequations from the
-nitedierence scheme haspositiveornegativedenite symmetricpartbut maynot
besymmetric. Available multigrid solversmay notwork eÆciently to solvethe
re-sultinglinearsystem. Inthispaper,weproposeamultigridmethodwhichisdesigned
particularlyforinterfaceproblems.
Inthispaper,wealsodevelopamaximumprinciplepreservingschemefordiusion
and reaction equations with a xed and closed smooth interface in the solution
domain,seeFig.1foranillustration. Thegoverningequationsare
u
t
+a(x;t)ru=r( ru)+f; x2= +
[ ; (1.1)
[u]j =w(s); [u
n
]j =v(s) (1.2)
u(x;0)=u
0
(x); u(x;t)j
@
=g(t); (1.3)
where thecoeÆcients (x;t)
min
>0, a(x;t;u) =(
1 ;
2
), f(x;t) are piecewise
continuousbut may haveanite jumpacross , ands isthearc-length
parameteri-zationoftheinterface . Thejump isdened asthedierenceof thelimitingvalues
oftwodierentsidesoftheinterface,forexample
[u]
jX2
= lim
x!X;X2 +
u(x;t) lim
x!X;X2
u(x;t)=u +
u :
For simplicity of the discussion, we assume a Dirichlet boundary condition for the
solutionandarectangulardomain. Thenaturaljumpconditionacrosstheinterface
is [u] = 0, and [u
n
] = 0 as in the example of heat conduction in a composite
materialwithoutasource/sinkalongtheinterface. However,wemakeourdiscussion
moreexiblebyallowingnon-homogeneousjump conditions.
Thesolutionto theinterface problemis in H 1
() but notin H 2
(). Formany
applications, for example, piecewisecoeÆcients, it is reasonableto assume that the
The rstauthor was supportedin part bya DOE grant DE-FG03-96ER25292. The second
authorwassupportedinpartbyanAROgrant,39676-MA,andanNSFgrant,DMS-96-26703
y
Department of Applied Mathematics, University of Washington, Seattle, WA 98195.
z
Center forResearchinScienticComputation&DepartmentofMathematics, NorthCarolina
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
+
=
+
[
+
=()
Fig.1.1. (a). Adiagramof arectangulardomain = +
[ withanimmersed interface
. ThecoeÆcientssuchas(x)etc. mayhaveajumpacrosstheinterface. (b). Adiagramofthe
local coordinatesin thenormal and tangentialdirections,where is theanglebetween thex-axis
andthenormaldirection.
solutionispiecewise smoothexcludingtheinterface . Typically,there isajump in
thenormalderivativeofthesolution.
Whilea niteelementmethod withabody-tted gridcanbe usedto solvethe
problem,weproposeanumericalmethodthatusesCartesiangrids. Wereferthe
read-erstothereferencesin[6]foranincompleteoverviewofdierentnumericalmethods
forellipticinterfaceproblems. ThepotentialadvantageofCartesiangridmethodsis
toavoidthemeshgenerationprocesswithoutsacricingaccuracyand stability. Our
goalistodevelopsecondordermethodsincontrasttotherstorderghostuidsharp
interfacemethodin[7].
2. Themaximumprinciplepreservingschemefordiusionandreaction
equationswithaninterface. Themaximumprinciplepreservingschemeforelliptic
interfaceproblems isproposedin [6]and will notberepeatedhere. Wewill explain
thealgorithmforthediusion andreactionequation.
Weassumethedomainis[a; b][c; d],anduseauniformgrid
x
i
=a+ih; i=0;1;;M; y
j
=c+jh; j=0;1;;N; (2.1)
anddenotethetimestepsizebyt.
2.1. The time discretization. The nite dierence scheme is based on the
prediction-correctionCrank-Nicolsondiscretization
u n+1
u n
t
+(ar
h u)
n+ 1
2
= 1
2
(r
h r
h u)
n
+(r
h r
h u)
n+1
+f n+
1
2
;
(2.2)
where
(ar
h u)
n+ 1
2
= 3
2 (ar
h u)
n 1
2 ( ar
h u)
n 1
; (2.3)
r
h
isthediscretegradient,andt n+
1
2
=t n
+t=2. Thisdiscretizationissecondorder
accurate in time. The diusion term is discretized implicitly so that we can take
largetimesteps,whilethereactiontermisdiscretizedexplicitlysothatsecondorder
i j
points in the centered ve-point stencil are on the same side of the interface, the
discreizationis thestandardone
r
h U
ij =
U
i+1;j U
i 1;j
2h ;
U
i;j+1 U
i;j 1
2h
; (2.4)
(r
h r
h )U
ij =
1
h
i+1=2;j (U
i+1;j U
ij )
h
i 1=2;j (U
i;j U
i 1;j )
h
(2.5)
+ 1
h
i;j+1=2 (U
i;j+1 U
i;j )
h
i;j 1=2 (U
i;j U
i;j 1 )
h
;
wherewehaveomittedthetimeindexforsimplicity.
At an irregular grid point where the centered ve-point stencilconsists of grid
pointsfromdierentsidesoftheinterface ,thediscretizationisdoneusingamethod
ofdeterminedcoeÆcients. Welet
(ar
h U
ij )=
n
s
2
X
k =1
k U
i+ik;j+jk Q
ij
; (2.6)
(r
h r
h )U
ij =
n
s
1
X
k =1
k U
i+ik;j+jk C
ij
; (2.7)
where n
s
1 and n
s
2
arethenumberof gridpoints in thenite dierencestencil,and
U
ij
is the approximation to the solution u(x;y) in (1.1)-(1.3) at (x
i ;y
j
). The sum
overkinvolvesanitenumberofpointsneighboring(x
i ;y
j
). Soeachi
k ;j
k
willtake
valuesinthesetf0;1;2g. ThecoeÆcients
k
andtheindicesi
k ;j
k
willdepend
on(i;j),sotheseshould reallybelabeled
ijk
;etc.,butforsimplicityofnotationwe
willconcentrateonasinglegridpoint(x
i ;y
j
)anddropthese indices.
Our criteria is to choose the coeÆcients to minimize the truncation error and
maintain the stability. The procedure to set the system of equations for the
un-determinedcoeÆcientsisdiscussedinthefollowingsubsections.
2.2.1. The local coordinates and interface relations. It is reasonableto
assume that the solution is piecewise smooth. The discontinuities in the solution
or/andthederivativesoccuralongtheinterface .
Inorder todetermine thenite dierencecoeÆcients
k , and
k
atan irregular
grid point (x
i ;y
j
), we choose the orthogonal projection (x
i ;y
j ) of (x
i ;y
j
) on the
interface ,oranypoint(x
i ;y
j
)2 that isclose to(x
i ;y
j
). Thelocal coordinates
inthenormalandthetangentialdirectionsare
= (x x
i
)cos+(y y
j )sin;
= (x x
i
)sin+(y y
j )cos:
(2.8)
Inaneighborhoodofthepoint(x
i ;y
j
),theinterface canbeparameterizedas
=(); with (0)=0; 0
(0)=0: (2.9)
Thecurvatureoftheinterfaceat(x
i ;y
j )is
00
(0).
Assume that theirregulargridpoint(x
i ;y
j
)2 . Inourmethod, the discrete
formof(2.2)-(2.3)istheapproximationtotheoriginaldierentialequationat(x
i ;y
insteadof (x
i ;y
j
). For example,f n+ 2 is ff g n+ 1 2 = lim (x;y)!(x i ;y j );(x;y)2
f(x;y;t n
+t=2):
Bydierentiating thejump conditions(1.2) along the interfacewith respect to
thearc-lengths,whichis inthelocal coordinates,togetherwith(2.9),wehavethe
followingjump relations(see[4]forthederivation),
u +
=u +w;
u +
=u
+ v + ; u + =u +w s ; (2.10) u + =u
+(1 ) 00 u 00 +
v+w
ss ;
u +
=u
+ + + u
+(1 ) 00 u + ( + ) 2 v+ 00 w s + v s + ;
where = = +
, w
s and w
ss
arethe rst and second order derivativesof w with
respecttothearc-lengthsalongtheinterface. Likewise,v
s
istherstderivativeofv
withrespectto thearclengthparameters. With the localcoordinates,the original
dierentialequationcanbewrittenas
u t +~ 1 u +~ 2 u =(u +u )+ u + u
+f; (2.11)
where ~ 1 = 1
cos+
2
sin; ~
2
=
1
sin+
2
cos; (2.12)
andu
andu
arethedirectionalderivativeofuin thenormalandtangential
direc-tions. Therefore [u t +~ 1 u +~ 2 u
]=[(u
+u )+ u + u
+f]: (2.13)
Usingthejumprelationsin (2.10)andaftersomemanipulations,weget
u +
=u
+( 1)u
+ ( 1)
00 + + +~ + 1 ~ 1 + ! u + [~ 2 ] + u + ~ + 2 + + w s w ss (2.14) + 1 + w t + 00 + + ~ + 1 + ( + ) 2 v [f] + :
Thuswehaveexpressedallthequantitiesofthesolutionuuptosecondorder
deriva-tivesfrom+sideintermsofthosefromthe sideoftheinterface .
2.2.2. Determining the discretization of the diusion term using the
maximum principle preserving scheme. As we stated earlier in this paper, if
the irregular grid point (x
i ;y
j
(x
i ;y
j
)insteadof(x
i ;y
j
). Thetruncationerrorofthenitedierenceapproximation
(2.7)tothediusiontermin(1.1)is
T ij = n s 1 X k =1 k u(x i+i k ;y j+j k ;t) C
ij
( r(ru)) : (2.15)
The truncation error is evaluated at (x
i ;y
j
). The grid points involved are from
outside, +
,andinside, ,oftheclosedinterface . UsingtheTaylorexpansionat
(x i ;y j
),wecanwrite
u(x
i+ik ;y
j+jk
;t)=u(
k ;
k ;t)=u
+ k u + k u + 1 2 2 k u + k k u + 1 2 2 k u +O(h 3 );
wherethe+or signischosendependingonwhether(
k ;
k
)liesonthe+or side
of . Thereforethetruncationerrorcanbewritten as
T
ij =a
1 u +a
2 u + +a 3 u +a 4 u + +a 5 u +a 6 u + +a 7 u +a 8 u + +a 9 u +a 10 u + +a 11 u +a 12 u + C ij +
r(ru)
:
Thequantities u
,u
, , arethe limitingvaluesof thefunctions at (x i ;y j )from
+sideor sideof theinterface. ThecoeÆcientsa
j
depend onlyonthepositionof
thestencilrelativetothe interface. Theyareindependent ofthefunctions uandf.
IfwedenetheindexsetsK +
andK by
K
=fk: (
k ;
k
)isonthesideof g;
thenthea
j
aregivenby
a 1 = X k 2K k ; a 2 = X k 2K + k ; a 3 = X k 2K k k ; a 4 = X k 2K + k k ; a 5 = X k 2K k k ; a 6 = X k 2K + k k ; a 7 = 1 2 X k 2K 2 k k ; a 8 = 1 2 X k 2K + 2 k k ; a 9 = 1 2 X k 2K 2 k k ; a 10 = 1 2 X k 2K + 2 k k ; a 11 = X k 2K k k k ; a 12 = X k 2K + k k k : (2.16)
Usingthe interfacerelations(2.10) and(2.14), weeliminate thequantitiesfrom one
T ij = (a 1 +a 2 )u +
( a 3 +a 4 +a 8 ( 1) 00 + + +~ + 1 ~ 1 + ! +a 10 (1 ) 00 +a 12 + + u + a 5 +a 6 +a 8 [~ 2 ] + +a 12 (1 ) 00 u + a 7 +a 8 u + a 9 +a 10 +a 8
( 1) u
+fa 11 +a 12 gu +( ^ T ij C ij )+; (2.17) where ^ T ij =a 2 w+ a 6 +a 8 ~ + 2 + + w s +(a 10 a 8 )w ss + a 8 + w t + 1 + a 4 +a 8 00 + ~ + 1 + + a 10 00 a 12 + + v (2.18) +a 12 1 + v s a 8 [f] + :
Tominimizethemagnitudeofthetruncationerror,weshould set
a 1 +a 2 = 0 a 3 +a 4 +a 8 ( 1) 00 + + +~ + 1 ~ 1 + ! +a 10 (1 ) 00 +a 12 + + = a 5 +a 6 +a 8 [~ 2 ] + +a 12 (1 ) 00 = a 7 +a 8 = a 9 +a 10 +a 8
( 1) =
a 11 +a 12 = 0: (2.19)
If we can nd f
k
gsuch that the linear system of equations is satised,wehavea
consistentdiscretization for thediusion term but we cannotguaranteethe
stabil-ity. The stability condition can be guaranteed, however, by enforcing the discrete
maximumprinciple
k
0 if (i
k ;j
k
)6=(0;0);
k
<0 if (i
k ;j
k
)=(0;0):
(2.20)
In order to solve (2.19)-(2.20), we set-up the following quadratic optimization
problemtodeterminethediscretizationforthediusionterm
min 1 2 k gk 2 2
; s:t: (2.21)
A=b;
k
0; if (i
k ;j
k
)6=(0;0)
k
<0; if (i
k ;j
k
where g 2R 1
, and A =b is the systemoflinear equations(2.19). Naturallywe
wanttochoose
k
in asuch awaythattheybecomethecoeÆcientsofthestandard
ve-pointcentral dierencescheme if +
= is aconstant. This canbedone by
selectingthevectorg as
g
k =
i+ik;j+jk
h 2
; if (i
k ;j
k
)2f( 1;0);(1;0);(0; 1);(0;1)g;
g
k =
4
i;j
h 2
; if (i
k ;j
k
)=(0;0); g
k
=0; otherwise:
(2.23)
Wechoosen
s1
=9whichseemstoworkverywell. In[6],wenumericallydemonstrated
thatthesolutiontotheoptimizationproblemhasasolutionifhissuÆcientlysmall.
We use the QL code developed by K. Schittkowski [8] to solve the optimization
problem.
OnceweknowthecoeÆcientsf
k
g,thecorrectiontermthenisC
ij =
^
T
ij ,where
^
T
ij
dependsonf
k
gandthejumpconditions.
2.2.3. Determinethe discretizationforthe reactionterm. Borrowingan
ideafromtheprojectionmethodfortheNavier-Stokesequations,see[3]forexample,
weuseanexplicitdiscretizationforthereactiontermsinceitinvolvesonlyrstorder
derivativesofthesolution. Thetruncationerrorofthenitedierenceapproximation
(2.6)tothereactiontermat(x
i ;y
j )is
T r
ij =
n
s
2
X
k =1
k u(x
i+i
k ;y
j+j
k ;t) Q
ij
(aru) : (2.24)
Aswementionedin[4,6],wecanrequirethetruncationerrortobeO(h)atirregular
grid pointswithout aectingsecond order accuracy. Thereforeweuse thestandard
centeredve-pointstencil(n
s
2
=5) andthelinearsystemofequationsis
a
1 +a
2 = 0
a
3 +a
4
= ~
1
a
5 +a
6
= ~
2
(2.25)
where we have neglected higher order terms of h. Note that the a
k
's are literally
dened in (2.16) with
k
being substituted with
k and n
s
1
beingsubstituted with
n
s
2
. Thesolutionto thelinear systemof equations isalso dierent. Thesystemis
anunder-determinedsystemandthesolutionisdened astheleastsquaressolution
withtheleastsquare2-norm. Thecorrectiontermthenis
Q
ij =a
2 w+a
6 w
s +
a
4
+
v; (2.26)
wherewehaveignoredhigherordertermsofh.
2.2.4. The CFL restriction. Since the reactionterm is discretized using an
explicitschemeandthemaximumdistancefrom (x
i ;y
j )to(x
i ;y
j )is
p
2h,theCFL
restrictionforthetimestepsizeis
t
h
p
2kak
2
in thissectionis consistent. Thelocaltruncation errorsareofO(h 2
)at regulargrid
points, and of O(h) at irregular grid points, or along the interface. We expect the
globalaccuracyisofO(h 2
)sincewecanallowoneorderlowerapproximationalonga
boundarywithoutaectingglobalaccuracy. Thestabilityoftheschemeisguaranteed
bythesignconstraintsofthescheme.
3. Multigrid method. In this section, we describe a multigrid method that
is appropriatefor solvingthe linear systemsthat arise from problems with internal
interfaces that have been discretized using the methods described in Section 2 for
parabolic problems from time t n
to t n+1
, and in [6] for elliptic interface problems
usingthemaximumprinciplepreservingschemes.
Wedenotethelinearsystemto besolvedonthenestgridasA h
u h
=f h
where
thelefthandsideofthisequationateachgridpointcanbepictoriallyrepresentedby
the9-pointstencilwiththenumberingschemegivenbelow:
8 5 9
o o o
1 2 3
o o o
6 4 7
o o o
Theequationfortheunknownatthecenter ofthestencilisthengivenby
1 u
1 +
2 u
2 +
3 u
3 +
4 u
4 +
5 u
5 +
6 u
6 +
7 u
7 +
8 u
8 +
9 u
9 =f
2
(3.1)
where the'sare thecoeÆcientsin therowof A h
correspondingto theunknownat
thecenterofthestencil.
Thecomponentsofthemultigridmethodare thesmoother,thechoiceof coarse
grid equations, theinterpolation operator,and therestriction operator. We discuss
eachinturnbelow.
3.1. The Smoother. A simplepointGauss-Seidel smoother isused. On each
gridlevel,thecoarsegridpointsaresmoothedrst,theverticaledgenegridpoints
aresmoothedsecond,thehorizontaledgenegridpointsaresmoothedthird,andthe
negridpointsin thecenterofacellaresmoothedlast. Thecodehastheoptionto
continuesmoothing(bothpre-andpost-smoothing)untilstallingoccursortosmooth
withaxednumberofpre-and axednumber(canbeadierentnumber)of
post-steps. For theproblems wereportin thispaper,weused 2pre-and2post-smooths.
Wefoundthepost-smoothstobecrucialtothesuccessofthemethod.
3.2. The Coarse Grid Problem. Let u~ h
be the approximation to u h
after
pre-smoothing the problem A h
u h
= f h
. Then the error equation is A h
e h
= r h
,
wheretheresidualr h
=f h
A h
~ u h
mustbeapproximatedonagridof size2h. The
correspondingcoarsegridequationisA 2h
e 2h
=r 2h
wherethecoarsegridoperatoris
givenbytheusualGalerkinchoice,
A 2h
=R A h
I h
2h
where R isthe restrictionoperator,andI
2h
is theinterpolation operator. Likewise,
the righthand side of the coarse grid equation is givenby restrictingthe ne grid
residualtothecoarsegridby
r 2h
=R r h
: (3.3)
3.3. TheInterpolationOperator. Weusetwodierentinterpolationschemes.
Oneisfortheregulargridpointsandonefortheirregularpoints. Recallagridpoint
is termed irregular ifits nine point stencilrepresents connections to grid points on
dierentsidesoftheinternalinterface.
3.3.1. Interpolation at Regular Points. Many authors, (see for example
Dendy, [2], and Briggs, et.al. [1] and the references therein), have developed an
operatorinducedinterpolationscheme. Fortwodimensionalproblems,giventhe
val-ues of the error e 2h
at coarse grid points, we need to interpolate I h
2h e
2h
to nd an
approximationtoe h
atallnegridpoints. Forcoarsegridpoints(thecornersofgrid
cells), the value of e 2h
is simply copied to be the value of e h
. Forne grid points
that are on a vertical cell edge, the operator induced scheme would start with the
error-residualequationassociatedwith(3.1),
1 e
1 +
2 e
2 +
3 e
3 +
4 e
4 +
5 e
5 +
6 e
6 +
7 e
7 +
8 e
8 +
9 e
9 =r
2
0; (3.4)
wheretheresidual,r
2
,forthispurposeis assumedsmallrelativeto theerrorsin the
equation. This is avalidassumption for smootherror after thepre-relaxation step.
An equation for e
2
in terms ofe
4 and e
5
is desired if weare interpolating to e
2 by
using thecoarsegriderrorvaluesat e
4 ande
5
. Now,equation(3.4)could givesuch
anapproximationif alltheerrorson theleft handside couldbeestimated in terms
ofe
4 and e
5
. Ataregularpoint, alltheseneighborgridpointsare onthesameside
oftheinternalinterface,soitmakessensetouseTaylorapproximationstoexpresse
1
ande
3
intermsofe
2
,toexpresse
6 ande
7
intermsofe
4
,andtoexpresse
8 ande
9 in
termsofe
5
. Thisleadstothefollowingequationfore
2
in termsofe
4 ande
5 :
e
2 =c
4 e
4 +c
5 e
5
(3.5)
where
c
4
= (
6 +
4 +
7 )=(
1 +
2 +
3
) (3.6)
and
c
5
= (
8 +
5 +
9 )=(
1 +
2 +
3
): (3.7)
Thesamestrategyleadstoaninterpolationformulafortheerroratthecenterof
ahorizontal edgeintermsofthecoarsegridpointsoneitherside. Weuse
e
2 =c
1 e
1 +c
3 e
3
(3.8)
where
c
1
= (
6 +
1 +
8 )=(
4 +
2 +
5
0
10
20
30
40
50
60
70
0
20
40
60
80
−30
−25
−20
−15
−10
−5
0
0
10
20
30
40
50
60
70
0
20
40
60
80
0
0.5
1
1.5
2
2.5
Fig.3.1. (a). Thediscreteerror afterpre-relaxin V-cycle1. (b). Thediscrete errorafter
pre-relaxation inV-cycle2. ThePDEis giveninSection4.1with b=:1and C =0. Noticethe
jumpinthenormal derivativeof theerrorattheinterface.
and
c
3
= (
7 +
3 +
9 )=(
4 +
2 +
5
): (3.10)
Now, with the values determined at the coarse grid points (via copy) and the
centersofverticalandhorizontaledgesbytheformulasgivenabove,theerrorsatthe
centers ofeachcell canbefound bysimplysolvingequation (3.4)fore
2
in termsof
the other interpolatedvalues. This can be written like aninterpolation formula by
castingtheresultfore
2
intermsofthefourcoarsegriderrorvaluesat thecornersof
thecell.
3.3.2. Interpolation atIrregularPoints. Atirregulargridpoints,the
inter-polation formulas given in (3.5) and (3.8) are no longer valid since the error after
thepre-relaxation stepcanalso havealargejump in itsnormalderivative,and the
approximations that were made in goingfrom (3.4)to these formulasare not
suÆ-cient. This has been veried in practice for the test problems used in Section 4.1
whereweknowtheexactsolutions. InFigures3.1and3.2,weplotthediscreteerror
rightafter thepre-relax smoothingstepis completed in therst, second,and tenth
V-cycles. Theoriginalerrorhasbeensmoothedbut alargejump stillremainsatthe
interfaceandthiscontinuesthroughoutthecomputation. InFigure3.2, wealsoplot
the negativeof the nal discrete solution u h
which also has a jump in the normal
derivativeattheinterface,ajump in attheinterface, butnojump in thevalueof
u
n .
Thestrategyweemployistodevelopaninterpolationformulaforthemidpoints
ofverticalandhorizontaledgesbyusinginformationabouttheinterfaceandmaking
thefollowingassumptionsaboutthediscreteerror.
1. The jump in the discrete errorat the internal interfaceafter pre-relaxation
is 0. That is [e h
] =0. This is reasonablesince the relaxationwill smooth
enoughforthis tobetrue. This givestherelationshipe +
=e .
2. [e h
n
]=0attheinternalinterface. Intheproblemswearesolving,thevalue
of[u
n
]isgiven,andthisinformationisbuiltintotheequationsforu h
,soas
theiterationproceedsthediscretesolutionwillbeanorderh 2
approximation
to u and will approximate this jump as well. This gives the relationship
e +
=
+
e
0
10
20
30
40
50
60
70
0
20
40
60
80
0
1
2
3
4
5
x 10
−4
0
10
20
30
40
50
60
70
0
20
40
60
80
−40
−35
−30
−25
−20
−15
−10
−5
0
Fig.3.2. (a). Thediscreteerror afterpre-relaxin V-cycle10. (b). Thevalue of ~
u h
after
V-cycle10.ThePDEisgiveninSection4.1withb=:1andC=0. Noticethejumpinthenormal
derivativeoftheerrorattheinterfaceandthejumpinthesolutionattheinterface.
3. [e h
]=0. Thisisalsoobservedinpractice,andthecorrespondingjumpinthe
solutionofthePDE,[u
]isalso0.
4. Wecannotassumethat e h
n
willbesmoothacrosstheinterface.
Wenowderivetheinterpolationformulaforthemidpointofaverticaledgethat
cutsthroughthe interface. Welook for aformulafor e
2
of theform (3.5). We rst
expand each sideof (3.5)aboutapoint(x
;y
)on theinterface, takingcare to use
thevaluesontheappropriatesideof theinterface. Wecenterthecoordinatesystem
atthisexpansionpoint,andusethenormalandtangentialcoordinates andgiven
by(2.8). Thisleadsto
e +
2
2 e
+
2 e
=c
4 (e +
4
4 e
+
4 e
)+c
5 (e +
5
5 e
+
5 e
) (3.11)
toO(h 2
),where
i
=1ifthegridpointisinsideorontheinterfaceand
i =
+
ifthe
gridpointisoutsidetheinterface. A similarequationholdsforhorizontal midpoints
withthesubscripts4and5replacedby1and3. Sincethejumpconditionshavebeen
applied,wenowconvertbacktoxandycoordinatesusing(2.8)andtherelationships
e
=e
x
cos()+e
y
sin() (3.12)
and
e
= e
x
sin()+e
y
cos(): (3.13)
Afterdoingthis,wewouldliketobeabletosetthecoeÆcientsofe ,e
x ,ande
y
tozero. Thiswouldgive3equationsbutweonlyhavethe2unknowns(c
4 andc
5 for
verticalmidpoints, orc
1 and c
3
for horizontal midpoints). Before goingfurther, we
writedownthesethreeequations.
e :c
4 +c
5 =1
e
x :c
4 (
4
4
c
4 s)+c
5 (
5
5
c
5 s)=(
2
2
c
2 s)
e
y :c
4 (
4
4 s+
4 c)+c
5 (
5
5 s+
5 c)=(
2
2 s+
2
For midpoints of vertical edges, we use the rst and third equations; thereby
forcing the coeÆcients of e and e
y
to vanish. Note that we are hoping for the
secondequationto benearlysatisedwithoutitsenforcement. Thisisdenitely the
casewhen nointerfaceis present(all
i
=1). This givesthefollowingvaluesforc
4
andc
5 :
c
5 =
(
4 s
2
+c 2
)(y
y
4 )+(
2 s
2
+c 2
)(y
2 y
)
(
4 s
2
+c 2
)(y
y
4 )+(
5 s
2
+c 2
)(y
5 y
)
(3.15)
andc
4
=1 c
5
. Thevaluess 2
andc 2
aresin 2
()andcos 2
(),respectively.
Formidpointsofhorizontaledges,weusetherstandsecondequations;thereby
forcingthecoeÆcientsofe ande
x
tovanish. Notethat wearehopingforthethird
equation to be nearlysatised without its enforcement. This is denitely the case
whennointerfaceis present. Thevaluesobtainedforc
1 andc
3 are:
c
1 =
(
3 c
2
+s 2
)(x
x
3 )+(
2 c
2
+s 2
)(x
2 x
)
(
3 c
2
+s 2
)(x
x
3 )+(
1 c
2
+s 2
)(x
1 x
)
(3.16)
andc
3
=1 c
1
. Thevaluess 2
andc 2
aresin 2
()andcos 2
(),respectively.
Thisinterpolationschemeallowstheerrortohavejumpsinthedirectionnormal
totheinterface. Theerrorisnotassumedtovarystrictlywithy(forvertical
interpo-lation)orstrictly withx(forhorizontalinterpolation). Thejump conditionsrelative
to and areimposed.
Theonlyremainingissueishowtointerpolatethecentersofthecells. Sincewe
nowhaveaformulafor thecellcorners(copythecoarsevalue)and theverticaland
horizontalmidpoints,wecansolve(3.4)tondthevalueofe
2
forcellcenters. Thisis
thesamestrategyadoptedforcellcentersforregularpoints. Hencetheoverallscheme
takesadvantageoftheinterfaceinformationaswellasthat ofthePDEoperator.
3.4. The Restriction Operator. By discretizing the PDEs as described in
Section2,weguaranteethattheproblemA h
onthenestgridisdiagonallydominant.
This does not however, guarantee that A 2h
is diagonally dominant. For problems
withlargejumps attheinternalinterface,thechoiceofR=(I h
2h )
T
willnotleadtoa
diagonallydominantproblem ortoaproblem that is solvedquickly withmultigrid.
Whatistrue,however,isthat simpleinjectionwillledtoadiagonallydominantA 2h
matrix. Theproofisverysimpleandwill notbepresentedhere. Wehavefoundthis
choice for R , combined with ourinterpolation scheme to be very eectivefor these
interfaceproblems.
4. Numericalresults. Wehaveperformedanumberofnumericalexperiments.
ThecomputationsaredoneusingeitheraSun'sUltra-1oraDecAlphaworkstation.
Thelinearsystemofequationsissolvedusingthemultigridmethod. Theinterfaceis
aclosed curvein thesolutiondomainandisexpressedin termsoftheperiodiccubic
splineinterpolationdevelopedin[5]. Theimplementationofthemethodsissequential
and not optimized. In this section, the following notations are used: m = n is the
numberof grid lines in thex- andy- directions; n
1
is the number ofcontrol points
used in thespline interpolationto representthe interface ; n
coarse and n
isused;n
l
isthenumberoflevelsusedforthemultigridmethod.
4.1. Numericalresultsforellipticproblems. Firstwepresentourmultigrid
resultsforellipticinterfaceproblems. Weconsidertheellipticequation
r((x;y)ru(x;y)) = f(x;y): (4.1)
Inthis example,the interfaceis the circlex 2
+y 2
= 1
4
within thecomputation
domain 1x;y1. Thevalueof is
(x;y)= (
x 2
+y 2
+1; ifx 2
+y 2
1
4 ,
b; ifx
2
+y 2
> 1
4 .
(4.2)
Thesourcetermis
f(x;y)=8(x 2
+y 2
)+4: (4.3)
Thetruesolutionis
u(x;y)= (
r 2
; ifr
1
2 ,
(1 1
8b 1
b )=4+(
r 4
2 +r
2
)=b+C log(2r)=b; ifr> 1
2 :
(4.4)
Inthisexample,wehavevariableanddiscontinuouscoeÆcients. Thejumpconditions
are
[u]=0; [u
n
]= 2C ; []=5=4 b; [f]=0: (4.5)
ThediÆcultyoftheproblemcanbecontrolledbydecreasingthevalueofbsincethe
jump inthenormalderivativeofthesolutionisgivenby
[u
n
]=(2C+5=4)=b 1: (4.6)
WeuseDirichletboundaryconditions.
Tables4.1,4.2,and4.3 showtheresultsofthemultigridsolverforvarious levels
n
l
and valuesofb. Notethat in Table4.2 whereb =0:1, boththesolutionand the
uxarecontinuous,buthasanitejump. Themaximumerroroverallgridpoints,
kE
n k
1 =max
i;j ju(x
i ;y
j ) U
ij j;
ispresented. Theorder ofconvergenceiscomputedfrom
order=
log( kE
n1 k
1 =kE
n2 k
1 )
log (n
1 =n
2 )
;
whichisthesolutionoftheequation
kE
n k
1 =Ch
order
with two dierent n's. The fourth column in the Tables gives the number of
V-cyclesto reachthe convergence criteria,and the fthcolumn givesthe average rate
ofmultigridconvergencecalculatedas
rate(k)=e log(jjr
k
jj
2 =jjr
0
jj
2 )
k
Multigridresults. Theparameters aren
coarse
=8, C =0:0, b=1:25. Thediscrete system
wassolved toa 2-normresidualtoleranceof10 6
. Constantnumberofiterationswithgridsizeis
conrmed.
n
finest n
1 n
l
V's Rate kE
n k
1
Order
32 40 3 4 .02 6:042010
3
64 80 4 5 .02 1:455410
3
2:0536
128 160 5 5 .02 3:559810
4
2:0315
256 320 6 5 .02 8:770710
5
2:0210
512 640 7 5 .02 2:220410
5
1:9819
Table4.2
Multigridresults. Theparameters arencoarse =8, C =0:0, b=0:10. Thediscrete system
wassolved toa 2-normresidualtoleranceof10 6
. Constantnumberofiterationswithgridsizeis
conrmed.
n
finest n
1 n
l
V's Rate kE
n k
1
Order
64 80 4 12 .21 2:596010
2
128 160 5 12 .21 6:398110 3
2:0206
256 320 6 13 .21 1:632210 3
1:9708
512 640 7 14 .21 4:268110 4
1:9352
Table4.3
Multigridresults. Theparametersaren
coarse
=16,C =0:0, b=:005. Thediscretesystem
wassolved toa 2-normresidualtoleranceof10 6
. Constantnumberofiterationswithgridsizeis
conrmed.
n
finest n
1 n
l
V's Rate kE
n k
1
Order
64 80 3 17 .29 6:06310
1
128 160 4 21 .36 1:44910 1
2:065
256 320 5 32 .47 3:61910 2
2:002
512 640 6 42 .54 9:14810 3
usedanincompleteCholeskysmoother. Thenumberofiterationsrequiredtosolvethe
problemwasaround275forallthelevels,muchworsethantheresultsweobtained.
Also,ourmultigridsolverwasabletosolvetheproblemwhenb=10 6
in32iterations
with arate of :43 on a gridof size 256256and in 36 iterationson agrid of size
256256; whereas, the algebraic multigrid solver failed for this value of b. For a
given value of (x;y), the V-cycle ratesare constantas theproblem size increases.
OurexperimentsshowthenumberofV-cyclesisnotquiteconstantastheratio = +
increases,buttendstovarylinearlywiththelogarithmoftheratio. Thatis,V-cycles
=c
1 log
10
( =
+
)+c
2 .
4.2. Numerical results for diusion and reaction equations. Inthis
ex-ample, theinterfaceis againthe circlex 2
+y 2
= 1
4
within the computationdomain
1 x;y 1. The heat conductivity is the same as (4.2). Let g(x;y) be the
piecewisefunctiondened in(4.4)and
f(x;y)=h(t)
1 @g
@x +
2 @g
@y 8(x
2
+y 2
) 4
+h 0
(t)g; (4.8)
whereh(t)isafunction ofourchoice. Thejump conditionsare
[u]=0; [u
n
]=h(t)2C : (4.9)
Thesolutiontothediusion andreactionequations(1.1)-(1.3)is
u(x;y)=h(t)g(x;y): (4.10)
WeuseDirichletboundaryconditionsandtheinitialconditionfromtheexactsolution.
Table4.4showstheresultsofthegridrenementanalysisofourmethodfordierent
gridsizes. Thelinear systemateach timestep wassolvedusing boththemultigrid
method describedin Section 3and analgebraicmultigrid method that uses
incom-pleteCholesky smoothing. Both multigrid methods solvedthe systemsequallywell
(requiringbetween3and5V-cycles). Thesesystemsweremorediagonallydominant
thattheellipticproblemsreportedin Section4.1.
Table4.4
Thegridrenementanalysis. Theparametersareh(t)=cost, 1=100,2=1,ncoarse=9,
C=0:1,b=10,thenaltimeist=1:0. Secondorderconvergenceisconrmed.
n
finest n
1 n
l
kE
n k
1
order
32 40 3 7:560210 4
64 80 4 1:379210 4
5:48216
128 160 5 2:849610 5
2:2750
256 320 6 6:578810 6
2:1147
A remark. ThemethodhereisrecommendedformodestconvectioncoeÆcient
kak since this term was treated explicitly. If kak is very large, then the time step
is small. In this case, some may prefer to use a upwind scheme to deal with the
preservesthediscretemaximumprincipleforadiusionandreactionequation
involv-ingaxedinterfaceandpresentedamultigridmethodforsolvingthediscretesystem
ofequationsobtainedfromthemaximumprinciplepreservingschemebothforelliptic
andparabolicpartialdierentialequations.
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2000.
[2] Jr.J.E.Dendy.Blackboxmultigrid.J.Comput.Phys.,48:336{386,1982.
[3] M-C.LaiandZ.Li.Theimmersedinterfacemethodforthenavier-stokesequationswithsingular
forces. J.Comput.Phys.,inpress,2001.
[4] R.J.LeVequeandZ.Li.Theimmersedinterfacemethodforellipticequationswithdiscontinuous
coeÆcientsandsingularsources. SIAMJ.Numer.Anal.,31:1019{1044,1994.
[5] Z.Li.TheImmersedInterfaceMethod|ANumericalApproachforPartialDierential
Equa-tionswithInterfaces. PhDthesis,UniversityofWashington,1994.
[6] Z.LiandK.Ito. Maximumprinciplepreservingschemesforinterfaceproblemswith
discontin-uouscoeÆcients.SIAMJ.Sci.Comput.,inpress.
[7] X.Liu,R.Fedkiw,andM.Kang.AboundaryconditioncapturingmethodforPoisson'sequation
onirregulardomain. J.Comput.Phys.,160:151{178,2000.
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