An aggregation-based domain decomposition preconditioner for groundwater flow

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GROUNDWATER FLOW

E. W. JENKINS

, C. T. KELLEY

, C. T. MILLER

,AND C. E. KEES

Abstract. In this paper we give a convergence analysis of a two-level additive Schwarz method in which the coarse mesh basis is constructed with aggregation, a method common in algebraic multigrid. This coarse mesh does not need geometric information about the subdomains and can readily be used on unstructured spatial meshes. We illustrate the performance with several computational examples.

Key words. Domain decomposition, Newton-Krylov-Schwarz methods, Richards’ equation, Multiphase flow, nonlinear equations,

AMS subject classifications. 65H10, 65N55, 76S05, 76T05, 86A05

1. Introduction. This paper provides a convergence theory for the two-level Schwarz pre-conditioner first described in [8–10]. The prepre-conditioner is a two-level additive Schwarz method [4, 5, 18]. Its novel feature is the use of aggregation ideas from algebraic multigrid [1, 22] and bal-ancing Neumann-Neumann methods [13–15] to construct the coarse mesh. The use of aggregation arose from necessity. In the applications reported in [8–10] the subdomains were irregular, and a coarse mesh based on “hat functions” over the subdomains was impractical. For the same reason, we needed minimal overlap between subdomains. Unlike the method from [3], we do not need to create a coarse mesh geometry or use geometric information about the subdomains.

While our theoretical results allow for overlap, and some of the experimental results in

3 are for subdomains with substantial overlap, the application of the preconditioner in practice has been for minimally overlapping subdomains.

Our analysis uses the standard finite element framework from [18,24]. The preconditioner also works well in the context of finite differences, however, as some of the examples in

3 illustrate. This preconditioner was developed for use in the Adaptive Hydrology model (ADH) [19], a production code being developed at the US Army Engineer Research and Development Center. Because ADH is a three-dimensional unstructured mesh code, detailed grid refinement studies are too costly to perform; therefore, in this paper we illustrate the performance of the preconditioner with computational examples in two space dimensions, a setting in which grid-refinement studies and investigations of the effects of overlap can more readily be done.

1.1. Problem Formulation. We begin with the weak form of an elliptic boundary value prob-lem on a domain with boundary . The goal is to find such that

for all

(1.1)

Version of May 15, 2000. This research was supported in part by Army Research Office contract DAAD19-99-1-0186, US Army contract DACA39-95-K-0098, National Science Foundation grant DMS-9700569, a Cray Research Corporation Fellowship, and a Department of Education GAANN fellowship. Computing activity was partially sup-ported by an allocation from the North Carolina Supercomputing Center.

North Carolina State University, Center for Research in Scientific Computation and Department of Mathematics, Box 8205, Raleigh, N. C. 27695-8205, USA ([email protected], Tim [email protected],).

Department of Environmental Sciences and Engineering, 104 Rosenau Hall, University of North Carolina, Chapel Hill, NC 27599-7400 (casey [email protected], chris [email protected]).

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where is a linear functional on , and is an appropriate function space. In our examples the

problems will be second order with Dirichlet or Neumann boundary conditions and "!$#&%('*),+ .

We take approximating spaces of continuous functions .-/!0 . The approximating problem at

level1 is to find23-.4 5- such that

6

'72 -98:

+;<='

:

+ for all

:

4> -@?

(1.2)

The problem, (1.2), is equivalent to a linear system

A

2

-;CB

(1.3)

on .- where

6

'*2 8:

+;D'

A

2

8:

+ for all 2 8:

4& .- . Here 'FE

8

EG+ is the HIJ'K),+ inner product.

Schwarz preconditioners are designed to accelerate Krylov space iterative methods for the solution of (1.3).

1.2. One-level Method. We begin with the one-level additive Schwarz preconditioner. We divide) into subdomainsLJ)NMPORQMTS

%

with an overlap width of at leastU and assume thatV

Q

MTS

%

)W,;<) .

LetXYW be the restriction map from an element of .- to the subspace ZW of 5- with support on

)W . Let

A

W[;XNW

A

X]\

W

be the subdomain operator. We assume that

A

W is nonsingular on ZW and define

^

W,;X]\

W&_ Aa`

%

W

XYW

8

where _

A

W is an approximation of

A

W . The one-level additive Schwarz preconditioner is

b

;

Q

c

WdS

%

^

W ?

(1.4)

1.3. Two-level Method. The two-level additive Schwarz method adds a coarse mesh term

^fe

;X \

e

_

A `

%

e

X

e

to the one-level preconditioner. Here _

A

e

is an approximation of

A

e

. We let .g denote the space of

coarse mesh basis functions. If the coarse mesh restriction map X

e

and the coarse mesh operator

A e

are well designed, the condition number ofb A

is significantly reduced.

One way to define a coarse mesh problem is to discretize the continuous problem on a coarser mesh. There are a few difficulties associated with forming the coarse problem this way. For unstructured meshes, such as the ones considered in [8–10], the interpolation operators between the fine mesh and the coarse mesh are difficult to define. Second, a coarse mesh must be generated, stored, and parallelized. Finally, the PDE must be discretized and recomputed on the coarse mesh. Alternatively, the discretization of the coarse mesh operator may be defined in terms of the existing fine mesh discretization. A Galerkin or variational coarse grid correction uses the fine grid matrix to obtain the coarse grid operator as

A e

;X e A

X

\

e

, whereX

e

is the interpolation operator to the coarse mesh function space, andX

\

e

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coarse mesh basis functions are obtained from the fine mesh basis functions, then the coarse mesh spacehji is contained in the fine grid spaceh5k .

We use the aggregation-based basis from [8–10] in this paper, where one coarse mesh ba-sis function is defined for each subdomain as the sum of the fine mesh baba-sis functions for that subdomain.

To set the notation that we will need in l 2, let the expansion of a functionmonhak in the finite

element basis be

mqprts>m svuws

(1.5)

where the

uxs

’s are the basis functions for the fine mesh, and a functionmy&n>h i can be represented

on the coarse mesh space as

myop r{z my@|x}~

(1.6)

where the }

z

’s are the finite element basis functions for the coarse mesh space. Since h i h k ,

}

z

can be written as

}

z

p r

z ut

(1.7)

The index represents the subdomain number. For our coarse mesh basis functions, the value of

z

is either or . If the fine mesh basis function u

is contained in subdomain , then

z

p ;

otherwise, z

p .

Further expanding the representation ofmy gives

my p

z

my@|}

z

p

z

my@|

z u

p

z

my@|

z u

p

Y

] my

u

Thus ] is the operator which interpolates from the coarse mesh to the fine mesh. Any function

on the coarse mesh can be represented solely in terms of the already existing fine mesh functions, making the formulation of a separate coarse mesh unnecessary.

1.4. Condition Estimate. In Assumption 1.1 we make precise the idea that is the

charac-teristic diameter of a subdomain. In Assumption 1.2 we make precise the overlap between the

subdomains and the properties of the coarse mesh basis functions. ASSUMPTION 1.1.

1. There is such that diam 7

([

. for allp$¡ ¢¢

¤£ .

2. There is¥ such that for all¦on

there exists¨§© such that¦n

and

ª@«¬¤®

¦ ¯

¤°

¯

§¥

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3. There are±[²³±]´µ³±Y¶]·¸ such that for all¹oº>» the ball

¼¾½

¹³±[²¿>ÀÁ$ÂtÃ¨º>»ÄZÅ@ÆÇ¤È

½

Ã³d¹ÀNÉ±[²¿ËÊ

intersects at most±]´wÌ±aÍ

¶ subdomains

»Î (i.e., an object of diameter Ï

½

¿>À intersects at

mostÏ ½=Ð

À subdomains»NÑ).

4. Ò

½

»Î(ÀYÓ±5¿/Í ,ÔÕÁ

Ð

³¤Ö¢Ö¢Ö³¤×ØÖ

In Assumption 1.1,Ò denotes Lebesgue measure.

ASSUMPTION 1.2. Assume the basis functionsÙfÑ of the coarse space satisfy

1. ÚÛÙfÑÚ ¶ ÜÝÞàßâá

É± ÜwãFä

Ý

å

æ

ÙfÑ

æ

¶ç9è

É±.¿oÍ

2. There is a domain» ÑGé(êë

» such thatì

Ñ

ÙÑ

½

¹ÀíÁ

Ð

for every¹>º» ÑGé¤ê

and dist½

¹³î3»,ÀïÉ

±.ð for every¹oº>»,ñ¡» ÑGé¤ê

. 3. Ç¢ò@ó¡ó

½

ÙfÑ7À ë"ô

»NÑ.

Parts 1 and 2 of Assumption 1.2 differ from similar assumptions in [1] in the ¿

´

semi-norm of the coarse mesh basis functions, and in the distance from the space»

ÑGé(ê

to î3» .

In õ 2 we prove

THEOREM 1.1. Let ö5÷¨Á ìùø Îdúüû

öZÎ

ë

±

½

ô

»ïÀ and let Assumptions 1.1 and 1.2 hold. Assume

that there isýÓ

Ð

such that

½*þ

³

þ

ÀNÉ

½ÿ

´

Î

þ

³

þ

ÀNÉý ½*þ

³

þ

À

for allþ

ºöa÷ and¸ É ÔÉC× . Let

Á

ø

Îúüû

¼

Î

(1.8)

then there is±D·¸ , independent of¿ and such that

½

À,É±ý ½=Ð

Ì

½

¿ 3À

¶

ÀÖ

(1.9)

2. Convergence Theory. We base our analysis on the result from [23, 24]. THEOREM 2.1. Let û be a positive constant so that, for any

þ

º ö5÷ , there exists a

decom-positionþ

ÁCì ø

ÑTúüû

þ

Ñ such that

þ

ÑwºöüÑ and

ø

ÑTúüû

½

Ñ

þ

Ñ ³

þ

Ñ ÀYÉû

½ þ

³

þ

ÀÖ

(2.1)

Let

q´ Á

´RÎ

ø

ÑTú´ ÑÛÎ ³

(2.2)

where, for Ð

ÉÆd³ÔÉC× ,

ÑÛÎ[Á<¸ if wÑÎ[Á¸ (i.e., iföüÑ.öâÎ ),

ÑÛÎ[Á

Ð

otherwise. Then

½

ÀYÉý û

½=Ð

Ìq´À³

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where !"#

$&%('%*),+.-0/13254 6798

'

6

';:=<

We assume that the energy norm is equivalent to the>

8

seminorm, and we can therefore replace (2.1) by

)

?@BA

$DCE

@

CF GIHJLKNMDOP

$

CEIC

F

GIHQJRKSM

<

(2.4)

Our estimate for

P

$

will be based on (2.4). The value of

P

8 is an indicator of the number of subdomains that contain any given point in

T ;

we assume our partition is such that

P

8

VU WYX[Z

. We solve the subdomain problems exactly in our numerical results, so

\]X

in ^ 3. Thus our condition number estimate is based on the estimate

of

P

$

, which we obtain using Lemmas 2.2 and 2.3. Our analysis uses some of the techniques developed in [1], but we do not smooth the coarse mesh basis. Our estimates in Lemma 2.2 are thus different from those in [1].

We define the coarse mesh operator _a`bdcfe b

G by

_

E

)

?@BA

8Sg @hf@Qi

g

@

g

@

W

E

ZI

X

j

W

T

@

Zlk

Km

E

WnoZ.pn

i

where

Erq

bc . Using the Cauchy-Schwarz inequality and Assumption 1.1 we get

s

k

KNm

E

WtnoZupvn

s

Owyx;Ezx|{} >~Y

F

<

Hence

C

g

@

COw > ~Y

F

>

~

x;Ezx { }

w

>

7

~Y

F

x;EIx { }

<

The value of

P

$

depends on the bound of the energy of_

E

and on the

F

bound of the error in the coarse mesh operator, i.e.,

E

_

E

. These bounds are provided in Lemma 2.2. Here, as in the remainder of this section,

w

is a constant that is independent of > and .

w

may increase as the analysis progresses.

LEMMA 2.2. If Assumptions 1.1 and 1.2 hold, then

x;E

_

EIx F{}3Ow

>

F

CEIC FG H

(2.5)

C

_

EzCFGzH Ow

>

CEICFGIH

<

(2.6)

Proof. Let

TuT @

[

`T @*

\*

d

@

T

@

.

¢¡.£

[pv¤¦¥§Wn i¨

T

Z

$

©n

T\` pv¤¦¥ªWn

i¨

T

Z

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If«¬¯® , then°v±²¦³ª´«0µ¶¸·u¹3º»d¼ by Assumption 1.2. Thus

®½.¾\¿lÀvÁ·5À,Â¶¸·5À*Ã¶¸·aÄ¾\Å*ÆÈÇ

Therefore, we have that°v±²¦³É¢Ê¢Ë9Ìv´«0µ¶¸·u¹3ºy»dÍ since°±ÎvÏa´Ð·5ÀÐ¹Ñº»dÍ for all±. HenceÒ ºy»Í

and using the Poincar´e inequality [6] we get

Ó|ÔzÓ;Õ×ÖYØ ËÚÙ º Ó;ÔIÓÕ×ÖYØ ËNÛYÙ º»dÍÝÜ Ô ÜÞIß Ø ËNÛàÙ Ç (2.7)

LetáâÀo¾ãÁåäæÂ·IçIÃ·5ÀÄ¾\Å*Æ and let èÎvé(°o´LárÀ¹ denote the cardinality ofáâÀ. We have

ÓåêÔIÓëÕ Ö Ø

Ë¸Ù ¾]ì&í ÀÊîuï ÀS´ Ô ¹.ðfÀN´ Ô ¹oµí ç;Êî3ï çl´ Ô ¹.ð3çl´ Ô ¹ñ ºVí ÀÊ¦î í ç;Ê&òDóBô©î Ü ï ÀS´ Ô ¹[Ü0Ü ï çl´ Ô ¹[Ü Ó ðfÀ ÓÕ Ö Ó ðuç Ó Õ Ö ºÝõ ë í ÀÊî í ç;Ê&òDóöô©î ì ï ë À ´ Ô ¹ø÷ ï ë ç ´ Ô

¹àñÈ»dÍù sinceÎ

ë

÷ûú ëýüyþ

Îú

ºy»Í ù¸ÿ ÁèåÎé(°Ú´

À¹;Æ í ÀÊ¦î ï ë À ´ Ô ¹5º» í ÀÊî

Ó;ÔIÓ ëÕ Ö Ø

óÙ º»

Ó;ÔIÓ ëÕ Ö Ø

ËSÛàÙ

Ç

Therefore

ÓÔ ê ÔIÓÕ

Ö Ø Ë¸Ù º Ó;ÔzÓ|Õ Ö Ø ËÚÙ ÷ Ó;ê ÔzÓ;Õ Ö Ø Ë¸Ù º» Ó|ÔzÓ;Õ Ö Ø Ë¸Ù ºy»ÍÝÜ Ô ÜÞ ß Ø ËÚÙ Ç (2.8)

Since ÜðfÀ&Ü

ë Þ ß º» Þ ß we have Ü ê Ô Ü ë Þ ß Ø Ë¸Ù º» Þ ß ÿ ÁèåÎvé×°f´LáâÀ¹Ælí ÀÊîuï ë À ´ Ô ¹ º» õ Þ í ÀÊ¦î

Ó;ÔIÓ ëÕ Ö Ø

óöÙ

ºy» õ

Þ

Ó;ÔIÓ ëÕ Ö Ø

ËSÛàÙ º» õ Þ Í ë Ü Ô Ü ë Þ ß Ø Ë Û Ù ¾\» Þ Ü Ô Ü ë Þ ß Ø Ë Û Ù Ç (2.9)

The estimates above are for the set ® around the edge of the domain. We complete the proof

by making similar estimates in· À

. Let

Ô

be an extension of

Ô

such that

Ô

¾

Ô

on· and

Ü Ô ÜÞ ß ´ ¹

º» ´Ð·u¹ Ü

Ô ÜÞ ß Ø Ù Ç Let ® ½À ¾ çÊ|ò ó

·Iç , and let ®ýÀ be the ball around ®

½À. Then by Assumption 1(c), we have that

°v±QÎvÏ ´ ®ýÀ¹3º»Í .

Forä ¾µ¦Ç¢Ç¢Çåµ , defineèYçu¾

õ Ø Ë|Ù Ë Ô °« , ! Ô çÑ¾ Ô"

èYç . Assumption 1.2 and the definition

of

ê

imply that for«¬¯· Ãâ· À

,

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Therefore #%$'&)(*$+#%,-/.0132547698;: < => ?A@ #%$'&)(*$+#%,-/.01 2CB 132D4E698 F < => ?A@ #3G$ >IHKJL> &)(NM%G$ >IHOJ%>QP # ,-.701 2CB 132D4E6Q8 F < => ?A@ #3G$ > &R(SG$ > #%,-/.701 2TB 1325476Q8 since ( JL> F JL> : < => ?A@ #3G$ > #L,-.70U 2 8 H #%(VG$ > #%,-/.701 2WB 132547698YX We use #Z(VG$ > #%,-.[01 2CB 132D4E6W8\:^] ] ]`_'a%bEc 2d a MZG$ >QP e a ] ] ] ,

- . 01Y8 :gf _haLb`c 2jikd a MZG$ >9P i # e a

# - . 01l78m

, : card Mon >P _ a%bEc 2 d , a M%G$ >P # e a #L,-/.701lE8 :qp*rts _ aLb`c 2 d , a MZG$ >P

:qpur s rwv s

_ a%bEc 2 #3G$ > #%,-.[01l[8x:qpy#3G$ > #%,-.[0oU 2 8Yz

and the Poincar´e inequality to obtain

#L$&)({$+# ,-. 01 25476 8|:y} _ < > ?A@ #YG$ > # ,-. 0U 2 8x:qp*r , _ < > ?A@ i G $ > i , ~+ 0U 2 8 F p*r , _ < > ?A@ i $ i

, ~ 0oU

2 8;:ypur , _ < > ?A@ i $ i

, ~ 0U

2 8;:ypur , i $ i

, ~ 01Y8YX

This and (2.8) imply (2.6).

We complete the proof of (2.5) by combining (2.9) with the estimate

i

(*$

i

, ~ 01Y2D4E698|:

_ < > ?A@ i (*$ i

, ~ 01

2TB 1Y2D4E6W8 F _ < > ?A@ i (NM%G$ >IHKJL>P i

, ~ 01

2TB 1325476Q8 F _ < > ?A@ i (SG$ >IHOJ%> i ,~+ 01 2TB 1 2D4E68

F _ < > ?A@ i (SG$ > i ,~ 01 2CB 1 2D4E68

: _ < > ?A@ _ aLb`c 2d a MZG$ >QP e a , ~ 01Y8 : _ < > ?A@ card Mon >P _ a%bEc 2d , a MZG$ >QP i e a i ,~ 01Y8 :yp @ ~ ~ [ _ < > ?A@ _'a%bEc 2 #3G$ > # ,-/.01l78:p @ ~ _ < > ?A@ #3G$ > # ,-.[0oU 2 8 :yp @ ~ r , _ < > ?A@ i G $ > i ,~ 0U 2 8 F p ~ _ < > ?A@ i $ i ,~ 0oU 2 8x:qp ~ i $ i ,~ 01Y8IX

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LEMMA 2.3. Under Assumptions 1.1 and 1.2, for every finite element function * , there

exists a decomposition j ,

j, such that

V

o j

(2.10)

o x j

I¡¢q£

¤3¥"¦¨§©{ª

Y¡Y«

(2.11)

Proof. Define¬

to be the fine mesh operator¬ *®°¯

by

¬

²±

A³+ ´

Aµ

±9¶ ·³I¸

where ¸

´

Aµ is the finite element basis on the fine mesh, and

¶

¹

´

oAµ are the fine mesh nodal

points. Let be partitioned such that

"»º* andAy¬

¼±¹½

±

¾Rº{A³E³A¿

where

½

is a partition of unity such that

½

±Q¶ ³À

¥

for

¶

gÁ

´ÃÂ

,

½

ÄÆÅ for

¶ÈÇ

ÉÁ", and

ËÊ

½

Y¢

µ

Ì . Clearly, by construction,

o jÍy

«

The standard arguments [18] imply that

o ; j

Y¡;¢q£

¤

¥

©

ÏÎ

¾)º{

Î

Ð/Ñ Y¡

¦

h¾)º{

+ Y¡

¦

º{

+ Y¡

ª

(2.12)

We complete the proof by applying Lemma 2.2 to estimate

º*

+ I¡

and

Î

h¾Rº{

Î

ÐÑ Y¡

.

If we let Ò

"

£

¤Y¥¼¦ §© ª

¿

we complete the proof of Theorem 1.1.

3. Numerical Results.

3.1. Laplace Equation. In this section we consider the simple test problem

Ê

ÓÅ

(3.1)

on the unit square ÔDÅÕ¿ ¥Ö×

ÔËÅÕ¿ ¥Ö

with zero Dirichlet boundary conditions. In bothØ 3.1.1 and 3.1.2

we use the function identically equal to one on the mesh as the initial iterate for a preconditioned conjugate gradient iteration. We terminated the iterations when the residual had been reduced by a factor of

¥

Å3Ù3Ú .

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3.1.1. Finite Difference Discretizations with Overlap. The experiments in this section were designed to measure the effects of overlap. We let ÛÜÞÝ3ß3à be the scale of the fine mesh and let

the overlapá be the is the nearest integer larger than

Ý

à

áãâåäKáæ

(3.2)

whereáãâ is the overlap that depends on the physical domain, andáæ is the overlap that is a constant

number of gridlines. The grid is an çèç mesh whereçwÜéÝàêäKá . In this way we can subdivide

the region intoëíì subdomains, each of size îïè°î , where

îðÜ»Ý àñßì

äKáóòNôõ

The scaleö of the subdomains isÝ3ßì .

We considered overlaps of ô (áæ"Ü÷ô/øLá/âÜ»ù ) and 1% (áæúÜ»ùÕø`áãâÜõùIô ). We discretized with

five point differences and use the function identically one on the mesh as the initial iterate. We used the MATLAB software associated with [12] for these experiments and used bilinear coarse mesh basis functions. These do not form a partition of unity as a piecewise linear set would and were used to illustrate the flexibility of the preconditioner.

In Table 3.1 we tabulate the number of conjugate gradient iterations needed for termination for several values of Û and ö , with an overlap of 1 and 1%. As the theory predicts, the iteration

count is constant asÛ andö are reduced simultaneously for the overlap of 1 and declines asö is

reduced with the overlap of 1%.

TABLE3.1

Iteration Statistics for Two-level Additive Schwarz with CG

Overlap = 1 Overlap = 1%

öûãÛ 1/32 1/64 1/128 1/256 1/32 1/64 1/128 1/256

1/4 12 16 21 28 12 16 17 20

1/8 12 15 21 27 12 15 17 19

1/16 8 12 16 21 12 13 15

1/32 8 12 15 10 11

1/64 8 12 8

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3.1.2. Finite Element Discretization with Minimal Overlap. In Table 3.2 we report similar results for a piecewise linear finite element discretization of (3.1). The overlap in this computation was minimal.

TABLE3.2

Finite Element Discretization

H ü h 1/64 1/128 1/256

1/4 37 51 68

1/8 32 44 61

1/16 26 36 49

1/32 26 37

3.2. Richards’ Equation Results. In this section we consider a test problem that models groundwater flow in a partially saturated, homogeneous soil. The model equation is the “head-based” form of Richards’ equation,ý/þ ÿ

þ

ÿ

þ

"!

(3.3)

where

is the pressure head,

ÿ

is volume fraction of the wetting phase, and

is the relative permeability of the wetting phase. The remaining terms are scalar coefficients given in Table 3.3, along with their values for the test problem.

ÿ

and

are functions of

given by,

ÿ

$#

ÿ

$%'&()+*-,

)./10324

ÿ

(3.4)

%

&()+*

,

)

./

0326587:9 &

#

)+*

,

)

.

0<;

%

&(=)*

,

)

.>/

032@?

7

(3.5)

,

ACBED

!GFH!

(3.6)

whereI

&

#

&KJ>L . These functions are derived from the van Genuchten [21] and Mualem [16]

empirical relations among pressure, saturation, and relative permeability. The test domain is the unit square

FM!&

I

ON

FM!'&

I

with boundary and initial conditions,

QP

!RF

F1STFM!

P-U

FM!&'

WV

F

QP

!'&K

F1SX&"!

P-U

&KJYM!HZJY>

WV

F

[G\

[H] QP

!'&K

#

&"STF1!

P-U

FM!&KJY"_^

Z"JYM!&` WV

F

[G\

[Ga bP

!G

F1STFM!

Pc

FM!'& Ud

F1!&'

WV

F

bP

!G

#

M!

P

!G Ud

FM!&`ON

FM!&'

F

(3.7)

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TABLE3.3

Richards’ equation parameters

Description Symbol Value

Saturated volume fraction egf hMikjMlnmolgjp<q

Residual volume fraction e'r sMikh"jtmolgjpu

Specific storage vf l"ikj"jtmolgj pw (1/m)

Hydraulic conductivity xyf z1ikj{|molgj} (m/day)

Mean pore size ~ z1i{3molgj } (1/m)

Pore size uniformity {iTtmolgj }

differential-algebraic equations, EbHR>G G"@ 6 E` e XG G> ¡ G ¢ q £¤ ¥n¦ x ¨§(© ¥ ª¨§ q ¢ «X ¢ x p © ¥ ªE ¢ p<q 8¬ ¢ q £¤ x §(© ¥ ¢ x p © ¥ ¢ q £_®¥ ¦ x

E¯§ ©

l ,²

j ii'i G± ¢

l ,³C´

E¯· ©

¥ ¹¸ xyf¯ºr E· q xyf¯ºr ER» ¶" (3.10)

The semidiscrete system was integrated in time over [0, 0.0149 days] by applying the fixed leading coefficient backward difference formulas of orders one to five [2, 11]. Order and step-size were selected via local truncation error estimates, and the local truncation error tolerance was set

to lgj¶³µ¼u .

At a given step, ¾½g§

q

, the application of the integration method yielded a nonlinear system of the form, ¸ ¾½§ q R½g§ q ¿_Q«½g§ q » À ª½g§ q @ j where¿_ªÁ

is a the backward difference formula for

Â

¶Â

. We solved the nonlinear system with an inexact Newton iteration that terminated when the 2-norm of the nonlinear residual was reduced by a factor of ljpÃ .

At each Newton iteration we obtained the Newton step, ÄKÅ

§

q , by solving the linear system,

Æ Â À Â Q Å ½g§ q 8Ç Ä Å § q ¢ À Q Å ½g§ q

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TABLE3.4

Richards’ Equation Iteration Statistics

ÈÊÉË

1/16 1/32 1/64 1/128 1/256

1/8 7 8 9 12 15

1/16 7 9 11 14

1/32 7 9 11

1/64 7 9

1/128 7

factor ofÌgÍÎ1Ï to insure that errors in the Newton step were insignificant with respect to the Newton

iteration and integration.

The preconditioner was two-level additive Schwarz with the coarse grid correction determined from the restriction and interpolation operators in Ð 1.3. The subdomains had the minimal overlap

ofÑÒÔÓÌKÕ`Ö . Table 3.4 gives the average BiCGstab iterations per Newton iteration for two-level

additive Schwarz. The iteration count is constant asÈ

and Ë

are reduced simultaneously, which is consistent with the predictions of the theory.

Acknowledgments. The authors wish to thank Marian Brezina, Jim Jones, Van Henson, David Keyes, Andrea Toselli, Mary Wheeler, Carol Woodward, and Jun Zou for many helpful discussions and Dave Klepacki, Matt Matthews, Mark Reed, Eric Sills, and Bob Walkup, for their help with the computations. We are particularly grateful to our collaborators at the US Army Engineer Research and Development Center: Charlie Berger, Jackie Hallberg, Stacy Howington, John Peters, Joe Schmidt, and Alan Stagg.

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