iterative and finite difference methods for solving a

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~10~ ~lW]_ 1992!rj:. l~

~ ~ M JO URN AL OF APPLIED SCIENCES

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f4

Vol. 10, No.

1

t···· -·

January, 1992

ITERATIVE AND FINITE DIFFERENCE METHODS FOR SOLVING A SYSTEM OF NONLINEAR ELLIPTIC EQUATIONS .

I

Guo

BENYU

(Shanghai Unive1·sity of Science and Technology) JOHN J.H. MILLER

(Numerical Analysis G1·oup, Trinity College, Dublin 2, freland)

Abstract Iterative and finite difference methods are proposed for a system of nonliear elliptic -equations. The existence of solutions and error Lestimations of approximate solutions are proved. The numerical results support the .analysis.

Key words: Nonlinear elliptic equations, iterative and fiuite difference methods.

I. Introduction In studying problems arising in electromagietio fields, semioonductor devices, biology and other topios, we have to consider systems of_nonlinear elliptic equations and their numerioal solutions. As we know, the properties of systems of elliptio €quations are very different from those of single equations, and so many difficulties arise. During the past two decades many scientists have paid more and more .attention to such problems, and :a lot of literature concerning the solutions and -their properties has appeared, e.g., Aronson and Weinberger [1], Fife and Tang ;[2, 3], Grindrod and Sleeman [4], Rabinowitz [5] and Mock [6]. In this paper we consider a system of nonlinear elliptio equations and its numerical solutions. It is not difficult to prove the existence of solutions following the work of [1-5]. We prefer to give a new const1·uctive proof in Section II,which also provides an iterative method to calculate them. The main idea is to construct ~equences of supersolutions and subsolutions, the limits of whioh are the solutions •of the original problem. The sequences given in this paper are slightly different from the usual ones used in [4], and they are :more suitable for computations. In Section III, we discretize the problem to obtain a system of nonlinear di:fferenm,. 1991 ~a R 22

s ®Jlt

)

10

2

~

equations, which may be regarded as a disorete model of some physical and1 biological problems. We prove the existenoe of solutions by sequenoes of disorete, supersolutions and disorete subsolutions, whioh may be used to solve this system. This work is an improvement of the results of [7]. In that paper, Huy, MoKenna, and Walter oonsidered the numerical solutions of the problem -

2 " a ui +ft(a; __

~

i=l

ax,

' U1 ' U2

••• J

J

U ) m

=0 1,o;;;;;i,o;;;;;m J

(1.1} J

and proved the existenoe of solutions of the oorresponding disorete problem by sequences of supersolutions and subsolutions with the oonditions that

aj; I< u Bf; --o("-L .) . nx raim ,:::, i ""'s.LY.l 1 1 ,:::, ""' 1, -r J , ln ~4 U--. •

I uU;

(1.2}

UUj

1

These conditions are too strict in practical oases. For instanoe, for Fisher equation, we have 2

- 8ua; ,: :, u.. + f(u) =0, .

f(u) = -u+u2 •

Clearly :~ is not uniformly bounded above in D x !RI"'.

Indeed, the numerical

results in [7] also showed that condition (1.2) is not neoessary. We get the similar results as in [7], but with more relaxed assumptions. Thus the proofs of theorems are more difficult. In faot, this is a disorete analogy to the oontinuous version of Grindrod and Sleeman [4]. We also estimate the errors between the exact solutions and approximate ones, in Section IV. Finally we give some numerical results in Seotion V. They show that if the problems satisfy the assumptions proposed in this -papet·, then we still get the oonvergenoe and the monotonioity of sequenoes of supersolutions and subsolutions.

II. Iterative method Let Q be an open convex bounded domain in ~n with Lipsohitz continuous boundary I' where I'=I' 1 UI' 2 and mEs (I' 1 )>0. We denote by D the olosure of D. Let a;= (ro1 , x 2 , ···, a,n)TEIRn and u= (u1 , u2 , ···, u,,.)T be a vector funotion of a;, and let the give? veotor funotion f(a;, u) E [0 1 (Dx/R"')]"' have components fi(a,, u)! Furthermore, let Z=diag(Z1 , Z2 ,

···,

Z,,.)with

Z,u, = -V • (a;'vu,) + b,1 ~u. , 1-O, xEI'2, thenu>Ofatfa;ED. Jr,,pairticuZatf, iJ1,u>OfoirxEDO'fu>OfOffI'1, then it>O fair w

ED.

Lemma 2.2 If uE [0 2 (D)

n0 1 (D)]m ar,,rl, Zu,O,

xE D,

u,O,

xE I'1,

{ OU or,,

,o,

xEI'2,

then u,O fair xE .Q. Jr,, pamcuZar,if 1,u' MID z,z,

·1f0· aJ. 8UJ

k k+ld Z.JZ,. a,

1-a1M1 2 +I z,kl2H'(D) ) • --,,,--2- Cl z,k+1 1H'(D)

Substituting the previous estimates into (2 .16), we obtain

(2Ao-Boo-Bo01a:(o)-B101rn,-01Mm) jzk+1lir,(D) ,01M1( m+})lz"lho> from which the conclusion follows. If b;;dsn;;,.O fer wEI' 2 or mes (I'2 ) =0 then (2.14) holds, but with the ~nvergenoe rate 7'=

If in addition, ~~'.

01M1(m+ ~) .

Olearly such an estimate is also valid even if mes (I\) =0. · We can also estimate the error between'!!!." and Yl in the same way.

1~

11

ITERATIVE AND FINITE DIFFFRENCE METHODS FOR ...

III. Finite difference method There are two ways to solve problem (1.1) numerically. The first is to use an iterative method as shown in Section II. Then we use any known numerical methods (e. g., finite element method, fl.nit~ difference method, spectral method and others) to solve the linear system to get the value of

w~ or'!!!.~ step by step.

The

second way is to discretize problem (1.1) directly to get a nonlinear system. Then we apply iterations to solve the dis3rete nonlinear system. _Since some problems arising in biology, physics and other related topics are originally discrete, more and more people are paying attention to such discrete systems (e.g., see [8, 9]). In this section, we consider finite difference methods. For simplicity we suppose that

O0, a, E t:2,., u,.;;;.o, xEI'1,,., { u,.,,.;;;.o, xE I'2,11, Definition 3.2 y,. is a subsolutior,, for (3 .2), if Z,.u,."0, xE Q.,., fu"O, xEI'1,11, { !fll,,."O, xEI'2,11• In general a subsolution ~ can be greater than a ·supersolution u71 • But if m=!_ an d

af au >O for xE !RI and u71 E iitC:ib, u- 11),, then we have !h."u- 71 • Indeed in this case'"

we have

Z,.(u,.-fu) =L,.ii,.-L~-f(a, ,u,.) +f(a;, y,.) =Lu,.-Lu,.-. 81 (w, e,.)(u,.-'M,.) 8u where 071 lies between Y:,. and u71 • So by the maximum principle, u71 71 • We have no definitive result concerning the existence of snpersolutions and·

°'u

subsolutions. But if m=1 and

~ >O for wE Q and uE ~. then (3.2) certainly has

supersolutions and subsolutions; To see this, let v11 be any discrete function. satisfying

v,.;;;.O, ·x EI'1,11, {v,.,,.;;;.o, wEI'2, , 11

and w 71 ba the solution of the problem

1 Wl

13

ITERATIVE AND FINITE DIFFERENCE METHODS FOR. ..

Z,.w= ILv,.I, {

w,.=O, w,.,,.=0,

wED.,., wEI'1,1,, wEI'2,1,,

iObviously w,.;;;.o. Let ii,.=v,.+w,., then

•Furthermore L,.ii,.=Z,.v,.+Z,.w,.+f(a;,

ii,.) =Z,.v,.+ ILiv,., +f(a;,

u,.)

=L,.v,.+ ILiv,.I +f(a;, iii)-f(a;, v,.);;;. :~ (a;, Oi)wi>O

-where e,. lies between ii1, and v,.. Thus ii,. is a supersolution for (3 .2). We ·can also prove the existence of a subsolution '!h, for (3 .2) in a similar way.

If m>1, we have no general results. But if.f (w, u7.) ;;;.o for some constant vector ,u:;;;.o, then u: is a supersolution. On the other hand, if f(a;, u,.,.)O. By the maximum principle, we have that wt;;;,,y,_,. and so w~+1 EK(y,_,.,

L,.w~+ =z,.w~+.

u,.). Moreover

+f(a1, w~+ ) = -M1i;.+ +f(a1, w~+ )-f(a1, wD = -Mlz~+1 +F(a1, ()~+ 1 )zz+1 1 • 'Thus ()~+l E K (w., u,.) and then conditions (2) where ()~+l lies between w~ and and (3) imply that L,.wf+1 -;;;;,.o. So w~+ 1 is also a supersolution. The above arguments tell us that there is a function EK (fh., u,.) such that 1

1

1

1

1

w~+

w,.

fun

Tc~oo

w:=w,..

By letting k~oo in (3.3), we know that second conclusion in a similar way.

wis a solution of (3.2).

We can prove the

The proof of Theorem 2 .1 also provides an iterative method for solvinJl (3. 2). Since for each k, (3 .3) is a linear system for the unknown function w:+1 , we can solve it easily by using the usual difference methods. If m=1 and aubsolution solution.

y,_-,,,

o..;;;:

..;;;M for a;ED and u,.EIRIN, then problem (3.2) always has a

and a supersolution

u,. such that fh.O, Boo+ Bo02,8(0) -B102n-20~ 1m-Boh-B1oh-B1h,8(0) 0 for w1 =1-h, 1 and b;;,O for w;=O, h, or I' 2 ,,. is empty, then(3 .2) has a

unique solution in ~(u,., *' u:)provided that

B102rn,+202M1m+ Bok+ B16h+B1hf3(«3)
02M1( m+ ~) 0. If m..=1 and

f!Ju ~O for all wE.0 and uE~, then by the maximum principle,

we can get an error estimate in the maximum norm(see[12]).

V. Numerical Results This section is devoted to numerical results. We first consider the logistic model

ITERATIVE AND FINITE DIFFERENCE METHODS OFR ...

d 2u - da; 2 +f(u) =0, O

-42. We next consider the following system

- ~:~ + f(v, -

d 2w da;2

+ g(v,

w) =0,

O