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390

transmission of the disease. Building upon this idea, we next consider a more complex

habitat with several separated urban areas fJJ,

• • • ,

0£ surrounded by a rural environment

—

U;=i ^X- Finally, we are lead to the consideration of a complex region consisting of

repeated fragmented local structure. Such a scenario would be exhibited by regions found

in developped areas like western Europe, Japan or perhaps New England where one en-

counters a highly heterogeneous landscape consisting of repetitively interspersed fields,

forests and small villages. At first glance, it appears as though this is covered by case

above. However, from a practical point of view we are more interested in how the complex

microstructure of the region affects the general macroscopic progression of the infection

through the region. If we formulate this question in mathematiccd terms we can obtain a

partial answer by implementing homogenization arguments.

Ordinary differential equations have been used to model the circulation of disease

through populations begining with the 19th century. Murray [18], and Busenberg and

Cooke [3], have excellent monographs detailing the state of the art of this activity.

Dif-

fusive epidemic models have become widely structured in the past two decades. The

interested reader is referred to [4], [8] and the references contained therein. Differen-

tial equations involving diffusion operators with discontinuous coefficients were originally

termed diffractive and were used to model neutron transport in reactor engineering. The

literature on this subject is incomplete but the reader is referred to [16]; see also [4], [20]

[22],

[23], [24]. Homogenization techniques were introduced in the late seventies and they

have proved useful in the modeling of composite materials, in macroscopic properties of

crystaline or polymer structures, and the description of aqueous and hydrocarbons flow

through porous materials (cf. [2] and [15]).

We set the stage by describing the basic underlying epidemiological kinetics. We

use the state variables u and v to denote susceptible and infective classes of individuals.

We model the transfer from the susceptibles to the infectives with an incidence term

f(u,

v) = kuv where k > 0 is a positive constant. The resulting system of differential

equations has the form

u = —kuv, v = kuv; w(0) = u

, v(0) =

VQ.

(1)

Standard techniques yield (cf [8]) the following result.

Theorem 1.1 Ifu

and

are nonnegative initial data then

lim u(t) = 0 and lim v(t) = w

+ v

t—*oo

Thus we totally answer questions of wellposedness, characterize forward invariant re-

gions and define the long term behavior of solutions. From the viewpoint of qualitative

dynamics this is certainly an unsurprising result, and from an epidemiological view point

it simply says that the whole population contracts a non fatal disease. There certainly

are much more complex models with much more complicated and exotic qualitative be-

havior. However, our purpose at hand is not to describe complex realistic disease kinetics

but rather to describe how spatial considerations may be introduced to epidemiological

modeling.

391

2 Diffusive epidemic modeling

Here we describe a population dispersing over but remaining confined to a bounded do-

main (or habitat) in R

. For technical reasons we assume that Q is a Lipschitz domain

with piecewise smooth boundary dQ such that fi lies locally on one side of dQ. Our state

variables become population densities u(x, t) and v(x, t) for x e O and t > 0 representing

the spatial density of the susceptible and infective populations.

The dispersion mechanism is assumed to be the standard Fickian approximation for

Brownian motion. However we shall specify distinct diffusivities di and a\ for the suscep-

tible and infective classes. Additionally, we require that

rfiGC°°(n)nC(n) fori = lor2,

mi{di(x),

x G Q | i = 1 or 2} > d > 0 for some d > 0.

(2)

The following reaction diffusion type system describes the spread of the disease through

-— = V

•

di(z)Vu - kuv, ie!l, t > 0,

§1

(

)

— = V

•

(x)Vv + kuv, x € Q, t > 0,

with no-flux boundary conditions

f)ii f)n

{

)^-=

{

)—

= 0, xedtt, t>o, (4)

on Or]

(where

is the outward unit normal vector to O on dQ) and initial conditions

u(x,0) =

UQ(X)

and v(x,0) = v

(x), x £ fi. (5)

Let

denote the spatial average

(m J

tp(x)dx. It readily follows from meth-

ods established in [11], see also [4] and [8]

Theorem 2.1 Assume u

, v

e C(S7) and are nonnegative on fi. Then system (3)-(4)-(5)

admits a continuously differentiate solution pair (u,v) with u(x,t) > 0 and v(x,t) > 0

for x e 0, t > 0 and u, v € L°°(fi x R+). T/ien hm^^ \\u(.,t)

-"(4)11^

= 0, and

lim

|K.,t)-«(t)lloo^

= 0

Finally, limt-,,^ u(t) = 0, and limt_,

v(t) =

= uo +

t>o-

Thus the diffusive model mimics the longtime behavior of the spatially homogeneous

one.

3 Models with fragmented habitat

Here we take up the fragmented situation described in the introduction, first with two

compartments. The surrounding region Q satisfies the conditions of section 2 as does Q*.

392

Moreover

assume that dQ*

= 0. We

assume

the

diffusivities

to be

discontinuous

across

the

boundary

dQ*,

namely

[ "i2(^J

for x G

S2 — £2 *,

where (d^,

)

satisfy

(2) on

and 0

—

respectively.

further assume that

the

disease

may be

transfered from infectives

susceptibles only

in the

region

Q*.

This

can

be modeled

introducing

incidence term having

the

form f(x,u,v)

k(x)uv where

k(x)

k\(x),k

> 0 and

characteristic function

Q*. These considerations

produce

system

partial differential equations with possibly discontinuous coefficients

—

= V

•

dAx)Vu

k(x)uv,

x e Q, x 4

dQ*,

t > 0,

(6)

-H

= V

•

(z)V«;

fc(a:)w,

re e ft, a; 0

<9ft*,

i > 0.

We need

specify what happens

at the

interface between Q*

and the

surrounding

region.

We let

[ra]an. denote

the

saltus,

jump,

of a

function

w on

dQ*

and

require that

Man*

= Man- = 0, and

drj

•

0. (7)

an*

We require boundary

and

initial conditions (4)-(5).

We point

that

the

first part

of (7)

insures continuity

of the

state variables across

dQ*

and the

second part

of (7)

requires continuity

of the

flux across dQ*.

engineering

applications such

condition stipulates that matter

conserved during transport across

dQ*.

moment's reflection will convince

the

reader that

discontinuity

diffusitivi-

ties across

dQ*

produces

discontinuity

of the

normal derivatives across

the

interface.

This discontinuity

typical

diffractive diffusion

and it can be

observed even

in the

computation

eigenvalues associated with

the one

dimensional steady state case

of the

differential operator,

[14].

classical solution

(6)-(7)

and

(4)-(5)

mean

pair

continuous functions

u(x,t),

v(x,t)

for x 6 fi, t > 0

which

are

twice continuously differentiable

in x and

continuously differentiable

in t on

open subsets

and O

—

f2*

and

satisfy

the

partial

differential equations

(6) on

and f2

—

and the

boundary

and

interface conditions

(7)

and (4).

Finally

require that

Jim

|K,t) -

«o|L,n

J™

H-.*) ~ "OIL* = °- (

)

Theorem

3.1 If

the foregoing conditions are satisfied there exists globally defined classi-

cal, nonnegative solutions

(6)-(7)-(8)

and (4).

There exists

constant

> 0

which

is upon Hwollooft, Ikolloon

< d =

inf{dj(x)

| i = 1

or2,x

€ Q}

such that

8up{|K.,t)IL,n.ll«(-.*)ILfl}<-Mo.

(9)

393

Proof.

We begin with the observation that we use the differential operators V •

di(x)Vw = 0, x £

—

8Q* together with saltus and boundary conditions similar to

(7) and (4), to define generators Ai(p) of analytic semigroups {Tf(t) \ t > 0} in

(Q),

p > 1 and C(Q). Thus we define

^•flw^l;:!

: k(x)uv,

••

—h(x)uv,

and locally solve our problem using the standard variation of parameters formulations

[19]

«(.,t)=7T(t)«o+ / Tf(t-3)F

(u{.,8)v(.,3))d8,

J p

«(.,*)= If (t)«o + f T?(t-s)F

(u{.,s)v{.,8))ds.

J 0

The nonnegativity of u(x,t),v(x,t) is established in [14] and [21] and follows from a

maximum principle argument. By virtue of the k(x)u(x,t)v(x,t) > 0 and hence du/dt <

V • di(x)Vu, we may now apply a maximum principle to observe that llu^^U^jj <

ll^olloofj = ki and hence that dv/dt < V • d

(a:)Vr> + kik(x)v. In addition, we can

integrate our equations over Q and sum the results to obtain a uniform L

(f2) a priori

estimate for

v(-,t).

These latter observations allow us to apply Moser-Alikakos iterative

arguments to guarantee the existence of Mo > 0 so that

IIKC^H,*,

n < Mo. •

In [1] these techniques are carefully outlined for standard diffusion; it has been ob-

served in [6] that these arguments immediately carry over to the diffractive case because

the interface compatibility conditions insure the contributions resulting from boundary

integrals in integration by parts cancel. Given the existence of a priori estimates on u and

v our local results can be extended globally via standard analytic semigroup continuation

arguments [13].

Theorem 3.2 If (u,v) is the solution pair to (6)-(7)-(8) and (4) then the conclusion of

Theorem 2.1 still holds.

Proof.

We begin with the first equation. If we integrate on fi x (0,t) we obtain

(-.

)lli

o+ / k(x)u{x,t)v{x,t)dxdt=\\u

Jo Jn

Use of the existing uniform a priori estimates on ||«(-,*)lloo,n

d ll

(-!*)lloo,n yields

kuv G L

((0, oo) x fi), for all p

(1,

oo). Multiplying (6) by u and du/dt yields udu/dt =

•

di(x)Vu

—

k(x)u

v, and

Standard applications of integration by parts and Young's inequalities produce the

following

f)ij

|Vu|eL

((0,oo)xn), — eL

((r,oo)xn), r > 0, (10)

394

weLoc((0,oo),Z,

(n)), IVuleLooCCr.oo),^^)), r>0. (11)

Analogous arguments produce the same results for v,dv/dt and |Vu|. We may now

apply Meyers Lemma for parabolic equations with discontinuous coefficients, [17] and [2],

to guarantee that u,v€

1<p

((0,

oo) x fi) for some p > 2 and hence that there exists a p >

2sothat |Vu|,

|VD|

€

((T,OO)XQ.)

forsomep > 2 andr > 0. If

differentiate (6) with

respect to t and set du/dt = w we have dw/dt = V• di(a:) Vw

—

k(x)udv/dt

—

k(x)vdu/dt.

Similarly for dv/dt = z we have <9z/d£ = V-d,2(x)Vz+k(x)vdu/dt+k(x)udv/dt. Working

with the nonlinearity we observe that k(x)udv/dt + k(x)vdu/dt e

£OO((T, OO),

(^))- We

now use regularity and embedding results for analytic semigroups, [13], to argue that there

is ap> 2 so that |Vw| = \Vdu/dt\, |Vz| = \Vdv/dt\ e L«,((r, oo);L

(n)). This together

with previously obtained estimates and the analogous results for |Vv\ will insure that there

exists a p > 2 so that

lim|||Vu(.

)

t)|||

]im|||V«(.,t)|||

= 0.

£—>oo ^'

(—too

A standard application of the Ritz Lemma guarantees that there exists a constant M

(obtained from the positive eigenvalues of the stationary diffractive diffusion problem) so

that

l|w(.,*)-«(*)lk

<W3|||V«(.,t)|||

2in

H;t)-v(t)\\2,n<M

\\\Vv(.,t)\\kn-

Therefore there will exist a constant M

so that

\H,t) -u(t)\\

,n< M

\\\Vu(.,t)\\\

IK,<)-^)IU<M

|||V<,t)|||

P]n

The Sobolev embedding theorem now insures the existence of a C

so that

IK,<)-«Wlloo,n<C

|||Vu(.,t)|||

P:n

\H,t)-v{t)\\ooV<C

\\\Vv{.,t)\\\

Returning to (6) we observe that elementary analysis yields lim^^ ||u(.,t)||

= 0,

and Umt_

ti(t) = 0. Because ||u(.,i)||

+ |K.,t)||

1]n

= ||«olli,

+ Iklli.n

limt^oo v(t) = co. •

For future reference we point out that our estimates are dependent upon the initial

data and the maximum and minimum values of the diffusion coefficients di, but not upon

the modulus of continuity of di on fi* or Q

—

fi*. This observation will be important

for the next section. We also point out that a priori estimates upon |||Vu(.,£)|||

and

lll^"(-i*)lllpn

f°

r some

P > 2 will be sufficient to guarantee the precompactness of our

solution trajectories.

Our work on this two compartment model immediately generalizes to a more com-

plex situation. For example we could consider a situation where we have k subregions

{Ql,

• •

•

}

such that fi* C U with

f| W

= 0

The

boundaries dQ and

dftj,

• •

• ,

are assumed to satisfy the same conditions as before. In this case, the generalization to a

395

system of partial differential equations of the form (6), (4) and (5) is straightforward. We

require the diffusitivities to be strictly positive, i.e. maXx^^d^x) \ i = 1, 2} > d > 0 for

some d > 0, but allow them to be distinct and possibly discontinuous across the interface

of fi* = fij U

•

• U

with the ambient region. For i = 1 or 2 we specify

4(*)

= f

}°,

> ^-^

---

^)>

(12)

' \

dij(x)

x e 0.*,

where d

and dy £ C

—

(fi* U

• • •

U f2£) and C

(H^) respectively. We allow a more

general incidence term f(x,u,v) = k(x)uv, where k(x) is uniformly bounded on

and

uniformly continuous on

Q*j

(j = 1 to k) and 0

—

(O* U

• • •

U fi£) and nonnegative on H.

The interface compatibility conditions are a straightforward adaptation of (7).

Time dependent diffusivities could be treated using the theory of linear evolution

equations rather than semigroup theory.

4 Complex Dynamics with repeated microstructure

Many habitats consist of repetitively interspersed fragmented subpatches. In certain

cases we can isolate a subdomain 0, and consider fi as being produced by a periodic

reproduction of Q where the structure of © is small by comparison with the size fL

Despite the disparity in scale this microstructure can have profound effects upon the long

term dynamics of the spread of the disease through Q, [12]. Theoretically, this situation

can be described by methods of the preceding section however from a more practical

computational view the microstructure of our domain is far too fine to track and it becomes

reasonable to implement homogenization technique to in essence average the effects of the

microstructure over the whole domain and obtain a system which approximates the global

dynamics over Q.

We introduce a basic cell 0 =

j=1

[0, Y

}

°]

C R

. A function

(p :

-> R is said to 6

periodic if it admits a period Y? in the direction yj. We let di(y) and (^(y) be functions

which satisfy (12) with subregions 0!

•

• •

in 9. We assume that di(y) and d

(y) satisfy

the conditions outlined at the end of section

We can also find the appropriate conditions

for the coefficient of the incidence term f(y,u,v) = k(y)uv on the fundamental cell 0.

We extend di, d

and k to all of R

by periodicity, producing periodic functions d

, d

and k. We let n*

• • •

fj* be the distinguished regions of the periodically reproduced copy

For e > 0 we define ip

(x) = tp{x/e) for x 6

and ip =

d\,d\,k.

For small e the

following system of partial differential equations on Q x (0, oo)

3li

-^ = V

•

d^(z)Vu

- k

(x)u

OV,

— = V

•

€

(x)Vv

+ k

{x)u

(13)

will have rapidly oscillating coefficients. We impose homogeneous Neumann exterior

boundary conditions similar to (4) and initial conditions (5).

396

We shall need compatibility conditions on the interfaces between the Oj and 0 and

their respective reproductions ©^ (J =

• • •

k, I =

• • •

n) we require that

Mae*,

MeSj,

= 0. Ki^du/dr,]^ = [^(^dv/dr,]^ = °- (

)

We are guaranted the existence of solutions to (13)-(14), (4) and (5) for sufficiently

small e > 0 and the question now becomes what happens as e J. 0.

Let

M{<p)

denote the spatial average

M(<p)

= ^sie) Je

*P(y)dy-

Theorem 4.1 For each e > 0 and T > 0 there exists a

globally

defined classical, nonneg-

ative solution of (13)-(14), (4) ".nd (5),

Moreover there exists positive definite,

symmetric matrices with constant coefficients Z)'

' and D^ representing homogenizations

of di and d

respectively and a constant k = M(k), such that (u

(.,t),v

(.,t)) converges

strongly in L

((0,T)) to a nonnegative classical pair (w,z) solution to

~ = V -D^Vw-kwz, xeQ,t>0,

& - (

)

— = V -D^Vz + kwz, x(=Q,t>0,

with an initial condition (5) and standard no flux boundary conditions

J2^f^«*fa.XJ)

^^.

cos(

i'**•) = °'

x€an

''>°-

Proof.

The well posedness is an immediate consequence of our results for diffrac-

tive systems. The convergence result is predicated upon the application of results in

[2] and [15]. The crucial step lies in using the result in Theorem 3.1 stating that

max

t>0

{||u

(.,i)||

oori

, ||v

(.,t)||

oot2

} < M

, M

being a constant independent of e for

0 < e < 1. Then, it follows from integration by parts that for any fixed T > 0 the

solution couples

) are bounded in the Sobolev space of order one H

(fi x

(0,

T)), in-

dependently of

for 0 < e < 1. A compactness argument yields that they lie in a relatively

compact subset of L

(0, x (0,T)). Hence there exists a sequence [u

i,v

i) converging to

some limit (u, v) strongly in L

(Q x

(0,

T)) and weakly in H

(fi x

(0,

T)) as e'

—>

0. Next it

is well known that for any function

: R

—>

E and G periodic then

—*

M.((p), as e

—•

in a weak-star L°°(fi) sense, i.e. for any ip G L

(Q) J

(x)tlj(x)dx

—»

M.(<p)

ip(x)dx,

as £

—•

0. From these two pieces of information, the convergence as e'

—>

0 of the kinetics

on the right hand sides of (13) towards the corresponding kinetics of (15) follows, weakly

in L

(Q x (0,T)) and strongly in the dual space [if

(£2

x (0,T))]'.

The goal of homogenization techniques is to handle the behavior of such quantities

as — V

•

dfVu

when e —> 0. Then one can show that there exists a positive definite

symmetric matrix D^

' depending solely on d^

', $1 and Cl such that upon extracting

further subsequences dfVu

—>•

D^'Vu weakly in L

{Q. x (0,T)) as e"

—•

0, see [2] and

[15].

Identical arguments work for the equation for v

. At this point the convergence of a

suitable subsequence of (u

, f

)o<

<i towards a solution of (15), (16) and (5) is established.

397

A uniqueness argument insures that this subsequential convergence

indeed convergence.

•

We point

out

that computation

the homogenized diffusion operators

complicated.

However

algorithm using asymptotic expansions appears

in [2] and [15], see

also

[9],

[10].

One

can

also establish

the

following result from

the

methods

Theorem

3.2.

Theorem

4.2 Let

(w,

solution pair

to the

homogenized system

(15), (16) and (5).

Then lim^o IK.,*)^

= 0

andlim^oo |k(.,i)||

00]

= c

References

[1]

Alikakos,

application of the invariance principle

reaction diffusion equations.

Diff. Eq., 33

(1979), 201-255.

[2]

Bensoussan,

J.L.

Lions

and G.

Papanicolaou, Asymptotic Analysis

for

Periodic

Structures. North Holland, Amsterdam,

1978.

[3]

Busenberg

and K. C.

Cooke, Vertically transmitted diseases. Springer-Verlag,

New

York,

1993.

[4]

Fitzgibbon,

Langlais,

Parrot

and G.

Webb,

diffusive system with

age

dependence modeling FIV. Nonlinear Analysis T.M.A.,

(1995), 975-989.

[5]

Fitzgibbon,

Hollis

and J.

Morgan., Steady state solutions

for

balanced reaction

diffusion systems

heterogeneous domains.

J. Diff. Int. Eq., to

appear.

[6]

Fitzgibbon

and J.

Morgan, Diffractive diffusion systems with locally defined

re-

actions.

In :

Evolution Equations,

J.J.

Goldstein

et al. eds, M.

Dekker,

New

York,

1994,

177-186.

[7]

Fitzgibbon

and J.

Morgan, Analysis

of a two

compartment model reaction

dif-

fusion model

on a

heterogeneous domain.

In :

Mathematical Models

Medical

and

Health Sciences,

Horn,

Simmonett

and G.

Webb

eds;

Vanderbilt University

Press,

139-144.

[8]

Fitzgibbon,

Martin

and J.

Morgan,

diffusive epidemic model with criss cross

dynamics.

Math. Anal. Appl.,

184

(1994), 399-414.

[9]

Fitzgibbon,

Langlais,

and

J.Morgan, Martin's Problem

for

Systems with Com-

partmental Diffusion.

Egyptian Mathematical Society,

(2001), 59-67.

[10]

Fitzgibbon,

Langlais,

and

J.Morgan,

Mathematical Model

of the

Spread

of Feline Leukemia Virus (FeLV) Through

Highly Heterogeneous Spatial Domain.

SIAM

Math. Anal.,

(2001), 570-588.

[11]

Haraux

and M.

Kirane, Estimations

pour

des

problemes paraboliques semil-

ineaires.

Ann. Fac. Sci.

Toulouse,

(1983), 265-280.

398

[12] F. Heiser, Contribution a I'analyse mathematique de deux systemes ecologiques en

environnements heterogenes. Dissertation, University of Bordeaux I, 1998.

[13] D. Henry, Geometric theory of

parabolic

equations, Springer-Verlag, Berlin, 1981.

[14] W. Horton, Global existence of solutions to reaction diffusion systems heterogeneous

domains. Dissertation, Texas A & M University, 1998.

[15] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators

and Integral Functionals. Springer Verlag, Berlin, 1994.

[16] O. Ladyzhenskaya, V. Rivkind and

Ural'ceva, On the classical solvability of diffrac-

tion problems for equations of elliptic and

parabolic

type. Translation AMS, 23, Prov-

idence, Rhode Island, 1968.

[17] N.G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic

divergence equations. Ann. Sc. Nom. Sup. Pisa, 17 (1963), 189-206.

[18] J.D. Murray, Mathematical Biology. Springer-Verlag, Berlin, 1989.

[19] A. Pazy, Semigroups of linear operators and applications to partial differential equa-

tions,. Springer-Verlag, Berlin, 1983.

[20] Z. Seftel, Estimates in L

of solutions of elliptic equations with discontinuous

coef-

ficients and satisfying general boundary conditions and conjugacy conditions. Soviet

Math, 4 (1963), 321-324.

[21] J. Smoller, Shock waves and reation diffusion equations, Springer-Verlag, Berlin, 1983.

[22] H. Stewart, Generation of analytic semigroups by strongly elliptic operators under

general boundary conditions. Trans. AMS, 259 (1980), 299-310.

[23] H. Stewart, Spectral theory of heterogeneous diffusion systems. J. Math. Anal. Appl.,

54 (1976), 59-7.

[24] H. Stewart, Generation of analytic semigroups by strongly elliptic operators. Trans.

A.M.S.,

199 (1974), 141-162.

Algebraic Multigrid for Selected PDE Systems

T. Fiillenbach, K. Stiiben

Praunhofer Institute for Algorithms and Scientific Computing (SCAI)

Schloss Birlinghoven, D-53754 Sankt Augustin, Germany

Email: tanja.fuellenbach@scai.fraunhofer.de,

klaus.stueben® scai.fraunhofer.de

Abstract

In this paper, strategies for solving systems of partial differential equations by

algebraic multigrid are discussed. In particular, a general framework for so-called

point-based strategies is introduced. For a demonstration, we have investigated

several industrial applications from semiconductor process and device simulation. It

is shown that this framework allows to construct robust and fast algebraic multigrid

approaches even for cases, where iterative solvers of the type commonly used in such

applications exhibit bad convergence or even fail.

1 Introduction

Classical algebraic multigrid (AMG) [1, 3] is known to provide very efficient and robust

solvers or preconditioners for large classes of matrix problems,

= f,

an important one being the class of (sparse) linear systems with matrices A which are

"close" to being M-matrices. Problems like this widely occur in connection with dis-

cretized scalar elliptic partial differential equations (PDEs). In such cases, classical AMG

is very mature and can handle millions of variables much more efficiently than any one-

level method. Since explicit information on the geometry (such as grid data) is not needed,

AMG is especially suited for unstructured grids both in 2D and 3D. In fact, the coarsen-

ing process is directly based on the connectivity pattern reflected by the matrix A and

interpolation is constructed based on the matrix entries.

However, extensions of classical AMG are required to efficiently solve systems of PDEs

involving two or more scalar functions (called unknowns in the following). This is because

classical AMG realizes a variable-based approach which does not distinguish between

dif-

ferent unknowns. Unless the coupling between different unknowns is very week, such an

approach cannot work efficiently for systems of PDEs where, in general, the corresponding

matrix A is far from being an M-matrix.

In the past, several ways to generalize AMG have been investigated and there is still

an ongoing rapid development of new AMG and AMG-like approaches. For a review, we

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