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A numerical method for solving quasilinear elliptic equations with a small parameter for the highest derivatives
118 satisfying a Lipschitz condition, one must take the different constants L,,...,L. into consideration. Otherwise, even optimal algorithms may perform badly. Some authors suggest using the substitution z~=L,z,, il, are characterizedby one constant, e.g., (j(r)1 Ilfl/GL). There is some danger that many optimal algorithms constructed for such classes of functions yield inferior resultsinpractice. I wish to thank Yu.L. Levitan for his help with the computation of Table 1. REFERENCES 1. SOBOL I.M., MultidimensionalQuadrature Formulae and Haar Functions, Moscow, Nauka, 1969. 2. NIEDERREITER H., Quasi-Monte Carlo methods for global optimization. Proceedings of the 4th Pannonian Symposium on Mathematical Statistics (Austria, 1983), Budapest, Akad. Kiad;, 251-267, 1986. 3. SUKHAREV A.G., Minimax models in the theory of numerical methods. Doctorate Dissertation, Moscow, Moscow State University, 1983. 4. SOBOL I.N., On functions satisfying a Lipschitz condition in multidimensionalproblems of numerical mathematics. Dokl. Akad. Nauk SSSR, 293, 6, 1314-1319, 1987. 5. FAURB H., Discrepance de suites associees a une systeme de numeration (en dimension s). Acta Arithmetica, 41, 337-351, 1982. 6. SOBOL I.M., Points Uniformly Filling a MultidimensionalCube, Moscow, Znanie, 1985. 7. SOBOL I.M. and STATNIKOV R.B., Choice of Optimal Parameters in Problems with Multidimensional Criteria, Moscow, Nauka, 1981.
Translated by D.L.
U.S.S.R.
.,Vo1.28,No.2,pp.118-125,1988
Comput.Maths.Math.Phys
Printed in Great Britain
0041-5553/88 $10.00+0.00 @I989 Pergamon Press plC
A NUMERICAL METHOD FOR SOLVING QUASILINEAR ELLIPTIC EQUATIONS WITH A SMALL PARAMETER FOR THE HIGHEST DERIVATIVES*
I.P. BOGLAYEV
A method for the numerical solution of quasilinear elliptic equations in which the coefficient of the highest derivatives is a small parameter is constructed. This method provides an error estimate which is uniform with respect to the parameter. A non-linear difference scheme is constructed on the basis of the method of straight lines and the use of exact difference schemes for one-dimensionalproblems. Thecomputation grid is selected in such a way that the grid points crowd together in a special way near the boundary of the domain. An iterative algorithm, uniformly convergent with respect to the small parameter, is proposed for solving the non-linear scheme. In a rectangular domain equation
R=={(X,I/):O-=xtL, ‘O
(1)
is the Laplace operator,and F(z,y;.u) where p>O is a small real parameter, A==P/~x'+P/~g' is a sufficiently smooth function whose derivative with respect to IIis strictly positive: F,>m-const>O. Consider the Dirichlet problem
(2)
(3) u(r,&I?'0, where I? is the boundary of P. Linear equations of type (1) were considered in /l, 2/. Owing to the presence of the small parameter, the solutions have singularitiesof the boundary-layer type near r. The solution of a problem of type (l)-(3) is involved, for example, in a model describing *Zh.vychisl.Mat.mat.Fiz.,28,4,492-502,1988
119
the distribution of the potential and charge carriers in semiconductor structures /3/. The difference schemes proposed in /4, 5/ for certain linear singularly perturbed equations of elliptic type converge at a uniform rate of convergence with respect to the small parameter. In this paper a method for the numerical solution is constructed for the quasilinear problem (l)-(3), in which the error estimate is uniform with respect to p. We first construct a non-linear difference scheme, on a special computation grid, which converges at a uniform The construction is based on the straight-line method described in /6/ and the use of rate. exact difference schemes for one-dimensional problems /7/. The computation grid is selected in such a way that the grid points crowd together in a special way in the neighbourhood of the boundary. To this end we use independent estimates of the derivatives of the solution, which are of interest in themselves. An iterative algorithm, converging uniformly in ~1, is proposed for solving the non-linear difference scheme. 1. Estimates for the derivatives of the solutions of problem (l)-(3). The solutions of elliptic equations in domains whose boundaries have corner points (in the case of a rectangular domain RC",O
1.
If
~(5, y)~C(sl)llC'(Q) is a solution
The proof follows satisfies the equation
from known results
of problem
for linear problems,
F.O is evaluated at some intermediate it follows from the conditions
point.
if one notes that
s(s,bl)
y; 0),
L,(u) -~zA~--F.o~-F(z, where L,(u),
(l)-(3), then
(1.1)
But in the case of a linear operator
L,(u)<0(aO)inQ and ulr>O (GO) that
n&O
(GO)
in 51. Introducing
the auxiliary
function
~=rn-~ ( rnu~ap( F (5,y; 011, x,
we have L,(w*u)90 in Q, wf~[rZO. This implies the truth of the lemma. estimates for the derivatives of the solution of We shall formulate and prove a priori Our method goes back to the work of Bernshtein /lo/. We first derive an problem (l)-(3). a priori estimate for the derivatives on the boundary of the domain, using for that purpose certain auxiliary functions, proceeding then to estimate the derivatives in the interior of the domain. Lemma 2.
Let
u(x,~)EC"(S~)M?'+~(Q)
[
Here and below
be a solution
g++++w]P
of problem
(l)-(3).
Then
(1.2)
($19Kn[I++iTI(Y)].
,(x)=exp(-_)+exp[--1,
O
‘l(Y)‘exp(-~)+exp[-~],
n=1,2,3.
K, and G, denote constants independent of
‘p.
the We will first prove the estimate for the first derivatives on I'. Number Proof. sides of the rectangle rr,i-1, 2, 3, 4, described in the positive direction; let r,-((5, Y): ~~xsgZ,, y-4). Consider, say, a point PO-(x0, O)EI?,, O
4 (P)=
b
O,
~[l - 27-'(pa-8')s 8-a], 8
8 is a fairly small but fixed number. Let us assume where p'-(Z-G%)*, and 09x628 or neighbourhood of 20 of diameter 48 is contained in r, (the case In the domain @(P,)=(O
w-G,(s-'4).
C-Gi(Y+vq)h
that
the
L-28sk,
120 Evaluate L.(w), where L. is as in (1.1): ’ Q.*-G,p-‘qp+GI*pL-‘1 -F,“Go(e-t-i). L, ( ID)-p’Goe-c[Ga’~It follows from (2) that L,(w)>mG,+e-(G,(G,‘+G,‘q~-~G,q,-m).
In view of the fact that in Q,(p,) IGl~GP,
Iq=&'G~t
we see that for sufficiently large G, L,(w)ZmG,
NOW choose Go so large that
in
Q~(Po),
L,(~~ortu)h~~G~*tF(s, y; O)>O On the boundary of -rnr lu(s, y)l for
Q,(Pt) y=q
in
Qd(Po).
we have wztuG0, because w&O
and u-=0 for and large Go. By the maximum principle, w*uco
p0, in
while w6 Qa(Po).
Since
for g=O and pG8, it follows that at these points
W*U-0
$-(wiu)
(1.3)
Let Ps-(a,O) be such that 0Cz,6215.Then the proof of (1.3) in the domain Q@0)={OC~C28, O-=%=q(p)) proceeds similarly, provided one remembers that w~u(O on the boundary of Qd(PO) and that for z=O we have, by (3), WGO, u=o. We have thus proved estimates for the first derivatives of the solution on the boundary: n~R,l~. laa/azl~~r.~KJ~L,la~~/a~l~,,
(1.4)
We will now establish estimates_(1.2) for n=l in the interior of a. Differentiate Eq.(l) with respect to x. Putting p(r, y)=au/ax, we have a boundary-valueproblem L,(p) -p*Ap-F,p=F.,
where K, and
K.
IPbl~KlCL,
Pk%
(4.5)
are evaluated on the solution ~(5, P). Define an auxiliary function n,(X)=KJ1+N'e(s)l,
where the constants K, and x will be determined below. Since L,(l-I&:p)--xX- (F,-x’)II,fF., it follows that &,(lJ,-&p),
for x*.Sm
and K,>x*~IF,I.
K,>R,, then ll,zttp>Oon F. If K, is sufficiently large, it follows from By (1.4), if the maximum principle for L, that (pIGI,, proving (1.2) with n=l for ada2; Similarly, the introduction of the auxiliary function Q,(P)-K,[l+p-'q(g)] yields estimate n=l, for atday. (1.2), The a priori estimates (1.2) for the second derivatives are established along the same lines. One first proves estimates on the boundary of 52. For example, for acta , one considers problem (1.5). Extending the function Qi=Wrpsmoothly to P((p~cpl~l=~R,) and putting p”-=P(P-9) I we have a problem L,(p)-Fu@+fl.-$A@,
fib==0,
analogous to (l)-(3). Consequently,we have estimatesof type (1.41, i.e., ~aP/aX~~,.+ZR,Ip'. To obtain estimates for a%&'x in the interior of P, we differentiate the equation in (1.5) with respect to x and consider the following boundary-valueproblem for r(x,y)==a'u/&r': L,(r) -Cl*Ar-F.r~F,,p’+2F,pfF,, rlr
fixed,
Irlrl(RJp'.
Define the auxiliary function then it follows that for x' Rr. By the maximum proving estimates (1.2) for n-2. principle, IrlGI,, The proof of estimates (1.2) for n-3 is similar. 2.
Non-linear difference scheme.
1. Construction of the difference scheme and error estimate. On the set a, introduce
121 a computation grid IIJ~==~~("XG)~(*), where 7Gijh("=(~<, i4, 1,...,NC'),ZFO, XN(~~), iihjh(*'=@/,, i-0, 1,..., Nf'),yo==o, yrw=b}. Put rh-C%nr,Oh-WlP. Henceforth we shall use the method of straight lines and exact difference schemes for one-dimensionalproblems /7/ to construct a difference scheme on L. In /l/, considering the one-dimensionalproblem @"(0‘fV,
f,Wn-const>o
w(o)=w(z)=o,
m),
we constructed a difference scheme, uniformly convergent with respect to k,based on the method of exact integral relationships. To this end one first isolates the principal part of the differential operator LL(P)-~'wN(t)-gw(t)=~(f. m), Pf-P, where ~-%onst>O is a parameter, and then establishes integral identities for the solution : m(t) on a given grid {tcI-o,l,...,N,to-O,tn=Z}
s
%+1
t1
%I@)0 (6 w) ds19
w,-w(tO,
i=i,2,...,N-
Y=Pc19
w=y SW y&-i), (yAt,) I,
w=ytctn (yAt,-J+cth where
1, wO=~w~;=O,
Atc;=tt+,-tr,
[k,tt+s1 of the homogeneous problems
gJ?(t),cp~n(t) are solutions on
JL(Wr)-O, cpc'(k)--l,cp:(toJ-o, JL((PP)==O, cpP(t,)=o, qP(tol)--l. One then approximate the right-hand side of (2.1) by replacing the function [&&+,I by its value at t,;thisgives a difference scheme (T*W),
5 &
[a+1 (Wt+1-
1=4,2,...,N-4,
EL B
WI) -
-
WC-,)I =
f h WA
(2.2a)
Wo-Wn-0, h
$1 (4 ds +
+
4 w,
cP(Cm(t)) on
ti
B,YW]= CI -
(ai+1+ 4.
The integral identities (2.1) will be used to construct a difference scheme, based on the straight-linemethod, for problem (l)-(3). On the straight line y,.j=1, 2,...,N(*)-1, identities (2.1) become
(T’“u),, = aPL.1,I - cl%,,+ d&4+1,j = 1 1
P
xrp:fi”’
tS
Xi+1 (s) $ypp’ (s) ds +
$’ (‘)(4 I$*’(4 ds],
j
xCl
1
e=1,2,..., N(l)- 1, uo,= uN(q = 0, $p' (8)= F (s,VJ;
where
u
(8,
1/1))
-
fi(‘)u
(St
YJ)
-
pa
bu&‘J)
,
On the straight line 21,i--l,2,...,N(')-1, we have
~"'-censt>O, I&j-u(zc, &).
1 1
P
f=1,2
tS y
&“’
"lC1 (s) qi” (s) ds +
"j-1
,...,
N@)-I,
5 cpj”(*)(s)cp?’(8) dsj
,
"I U,O=U*N(~)=O,
91”(s) = F (XI, s; u (51,s)) -. p’u (x,, s) fi(*) = const ;zO.
The coefficients of the operators T"' T"' and the right-hand sides of the identities are determined by analogy with the formula: in (2.1). Substituting $"'(s),+"'(s) in the form
122 we rewrite the integral identities as follows ((&II&Q) :
Tg’u = F (z, 8; u) - p’ g TP’u=F(2,y;u)-pa-gg where the operators notation
(2.3a)
+ J(1)(z, y; cp”‘),
a*u f .m (5, y; lp’) ,
(2.3b)
Th"),!fh")are defined by the formulae in (2.2) and we have used
the
Adding the two identities of (2.3) together for (Z,Y)E% and using the fact that n(s,EJ) is a solution of Es.(l), we obtain the following relationships for solutions of problem Cl)(3): TP’u+ TP’U = F (2, &!; u) + J(1) + J(r), (2, y) E Wh, uIq$=O. (2.4) As a difference scheme for (l)-(3) we take the following non-linear system of algebraic equations for the grid function v(s,y): Lhu = Tf’Uf Tf'U = F (x, I/: U), (x, g) E (oh, UIrh=O. where u is a 'solutionof problem (l)Lemma 3. The error Z~U(X, y)-U(s, g), (2, ~)Eo~, (3) and U a solution of (2.5), satisfies the estimate
ma= lz 6~ rdlb + k UEZ* Proof.
I(rmygmh I JC1) I + &xh
I m II*
By (2.4), (2.5), we conclude that ~(s,v), (3,Y)E?& Lhz- F,,% = - (J(1) + J(‘)), (2, y) E oh,
w-9
is a solution of the problem
zIrf$=O,
where we have used the mean-value theorem, and PyO=Fy(x,II; 6(z,y)),Clbeing an intermediate point of the interval whose endpoints are u and U. The truth of (2.6) now follows from (2) and the maximum principle for difference equations (see, e.g., /12, p.248, Theorem 3/. 2. Computation grid. We now introduce a special grid tjh'which takes the qualitative behaviour of the solution of problem (l)-(3) into consideration. It follows from (1.2) that in the neighbourhood of the sides rr,rr boundary layers form along the x-axis, and in the neighbourhood of r,,I's - along the y-axis, the dimensions of the boundary layer soy being O(&), &=t.l.llIl~]/x. 'l,'x'jjJh'l" The grid ZL'=G)* is defined as follows Z'E [O,h,]: zi=-pl[l 5Np=he,
x0=0, zt E(h,,
I, -h,)
-(I
-p)&j,
i=o,i,.
. .,Np,
&l/No”),
: m:x(x',,-s')=I#,
izS_NP'+l, . . ., N:‘,
=I, + -$ In(I- (1- p)6(N'l) - i)], 21FII,-'h,,z,]:z, i=Nf'+1
(2.7e)
(2.7b) (2.79
,...,N"',
x&i -1,-h,?
Z*W-=l,,
N"'=,'$'OW+N,'~'+l
The grid points in 8~~'~)' are defined by similar formulae in the intervals [O,h,], (&I,-&), N,@l,ho),iv'*).The total number of grid points Nlkh,, L I, using the notation N,'*)==N,"), N"'N'" is independent of p. The grid points in the boundary layer are so chosen that at two neighbouring grid points the variation of the solution is bounded uniformly in P by a quantity O(6). We stipulate furthermore that N"JI=N'~'. 3. Estimate of the convergence for scheme (2.5) on the grid Gi,,’ : Theorem 1.
of the difference
scheme.
The following theorem holds
The rate of convergence of a solution of the non-linear scheme (2.5) on the
12.3 grid
(2.71 to a solution
where
the constant
Proof. now estimate
K
of problem
(l)-(3) satisfies
the following
estimate
and the total number of grid points N are independent
We have already established an estimate (2.6) for the error ](I)J@' on the grid o,,'.Rewrite I"',I(') in (2.3) as
J(Z) (?p’)
=
J(2)
($P’) +
~ro,l;_pwu,
In /7/ we established
estimates
(
J(S)
-
pB
q+l)~F_pwU,
for terms of type
uniformly
in
I(:
of p. z-u-u.
We shall
g) , ’
li"(qc')), ~"'($"') on the grid
(2.7) :
(2.8) We claim that
(2.9) By (2.3), with
yj fixed, we have
(
J(1) xiv YJ;
-
1=
p’a @lJ(SlYJ) ayr
we have the estimate
(2.10) Indeed, differentiating
(1) with respect
to r, we obtain
a ac P’-ax ayl
ac
au
( >=--Ir’,,+F,,+F,.
Estimating the right-hand side by means of (1.2) for points in (2.7) are so chosen that IarP(--r~)if and
erY(--~+l)l<6
n=l,3,
we verify
The
grid
m;r(mi+i- 4 <6/x,
+(,z,+~E[O,I~~]U[Z~-~~,I,]. Hence, by (2.10), we obtain (2.9). (2.9) and using Lemma 3 , we complete
(2.10).
Combining the proof of the theorem.
the estimates
(2.8)
3. An iterative algorithm for solving the non-linear difference scheme. 1. The iterative algorithm. We adopt the additional assumption that F(z, y; u) (see (1)) satisfies
a two-sided
estimate
m
M=const.
(3.1)
Construction of the iterative algorithm is based on a discrete analogue of the method upper and lower functions developed for one-dimensional schemes in /3, 7/. The iteration sequence {u(n)(r,Y), (5, Y)a,,) is defined by the recurrence formulae
,l’JC
fixed,
U(n+*),U(n)+Z(n+l)
cP)(Q=O'
n=O,l,...,
(3.2a)
,
Lh~(n+l)_M~("+I),-[Lbl(")-F(2, y;VW)], fj(') = b'"= M.
of
(I,Y)=a%
(3.2b) (3.2~)
124 Theorem 2.
If the initial approximation U to'in (3.2) satisfies the conditions L,.?Y'-F(t, y; u'O')Go (>O),
(3.3a)
(2,b()EOh,
(3.3b)
u@?lrh=0,
then,the functions U'*',n-O,&..., of (3.2) are upper (lower) functions of (2.5) and (3.11, where V(s, g) is a solution of problem (2.5). The sequence (U'"') i.e., u’GU(“’ (GP’) is monotone decreasidg (increasing)and converges to u' at the rate of a geometric progression with qd-mM-l. Proof. We prove the theorem for upper functions. By the maximum principle, if ~*l'+"-_Mz("+l',O,2("+"(,-0, then z'"+"
for (2,Y)=OA. Using the mean-value theorem, we conclude from (3.2) that ,*U'"+"-_F(r, y; U(n+r')=(M-F"O)z("+", (I,b()EOh. (3.4)
If U'"' in (3.2) is an upper function, then by the maximum principle Z'WO and by induction, using (3.1) and (3.4), we obtain z~"+~'GO, n=O, I,..., i.e., (Vn') is a monotone decreasing sequence. By (3.2) and (3.4),
The maximum principle yields the estimate (1 z(n+l'I( E max 1~(~+r'(z,y)l <
Hence the following estimate for the convergence of the iteration sequence: where
s("
z(n+"~<4n[jP'll, is determined by the system ' Lh2(l’ -
Jf$+” = -
[L’JP - F (U”)l,
(2, e) E %r
(3.5a) a(‘) ‘rh =o.
It follows from (3.4) that {U("'} converges to a solution of problem (2.5), (3.1). theorem is proved.
(3.5b) The
2. Choice of initial approximations. we describe one method of constructing U"' the basis of (3.3). Define V(z, u) on %I,+so that VIm=O. Solve the linear system RIfh=O. IL,,V-F(V)I, (%~)=@hl LhR-mR=Then
on (3.6)
U'O)=Vi-R is an upper function. Indeed,
Lh(V+R)-_((V+R)=mR-_Lhv-_((V)I+thV-_((V)F,,OR<(m- F,O)R
(GV)~a,,,
V+Rlr,=O,
where we have used the mean-value theorem, the non-negativity of R and the bounds (3.1). Solving the system &,S--8=(&V-F((V)Ir (r,#)Emh, S[r*=O, instead of (3.6), we obtain a lower function U@'-VSS. Note that if
GlO' and
0'0' are upper and lower functions, respectively, it follows
(ci'"') are upper and from Theorem 2 that all the terms of the respective sequences (8tn'), lower functions, with PO'<. . . < o(n’.$O(n+l’ <. , . < Uf < . , . < tP+l’<
i7’“‘g . . .
Parallel computation of the upper and lower functions enables one a posteriori to improve the zone in which problem (2.5) is solved. For example, if V==O,(z,y)=Z&, then the maximum principle yields the following estimate for W-R:
IIUco)II
Hence, using the maximum principle for (3.5b), we obtain
1’z”“‘< AI-1
max ImU IX, Y)EQl*
- IF@, y;O)I - F(x, y;U@))I< K,
125 i.e., the sequence (zcn)}of (3.5&, and hence also
(W)}, converges at a rate uniform inn.
Remarks. 1. The exact solution of problem (l)-(3) on expressed as
[zs+(+I] with YJ fixed may
be
I) is the Green's function of the operator L,(ur)=~'m"-fi~')w where G,('l(r, (for the definition see /7/). Define the following function on the straight line yJ.j=l,Z,...,N(*)-i: up' (2,y,) = u (I*, y.) q?. (1)(4 + CJ (Q' 11
=[z,,W,!.
Uj) qp"'
(4.
(3.7)
i=o,1 ,...,NJ')-i,
where u(r,v),(I,JJ)EB~is a solution of the difference scheme (2.5). Then the following estimates hold throughout the interval .[O, Jr] : &zl I'(+* bJj)u~'(z*Yj)16 +?
j-1.2,...,NC*) -1.
Accordingly, on the straight line z,, I-1,2,...,N(Jl-i, the function u?'(IJ* U)= u (zi* Uj)Q:*(*J (V) + U (fJ* Uf+,)~~jf”‘“(U), Ye h
1J+Il, j=O,'t ,
. . , N(‘)-i,
satisfies the estimate
The proof of, for example, the estimate for (3.7) follows from Theorem 1, the fact that the functions O< +(')‘(I), &(l) (x)6; are bounded and the explicit form of the function G,(‘)(z, :) (see /7/). 2. If the parameters PC*) and b"' in the difference scheme (2.5) tend to zero, we obtain the following traditional scheme for a non-uniform grid LhU-CI*(U?;+tl~~)-F(z,y,U), (t, JJ)ee.% Ulm-0, for which Theorems 1 and 2 remain valid. 3. All our results (Lemmas l-3 and Theorems 1 and 2)‘ which were proved for a two-dimensional singularly perturbed problem, may be generalized to equations of dimension 1>2, considered on parallelepipeds. REFERENCES lISHIK M.I. and LYUSTERNIK L.A., Regular degeneration and the boundary layer for linear differential equations with a small parameter. Uspekhi Mat. Nauk, 12, 5, 3-122, 1957. 2. 3UTUZOV V.F., Asymptotic behaviour of the solution of the equation p*Adu--Kp(z, r)u-fft, y) in a rectangular domain. Differents. Uravn., 9, 9, 1654-1660, 1973. 3. 30GLAYEV I.P., On numerical analysis of the main equations simulating the operation of semiconductor structures. Preprint, ChernogolovkaInst. Prikl. Teoret. Mat. Akad. Nauk SSSR, 1986. 4. I 3AKHVALOVN.S., On the optimization of methods for solving boundary-value problems in the presence of a boundary layer. Zh. vychisl. Mat. mat. Fiz., 9, 4, 841-859, 1969. 5. ;HISHKIN G.I., A difference scheme for elliptic equations with a small parameter as the coefficient of the highest derivatives. Dokl. Akad. Nauk SSSR, 286, 1, 57-61, 1986. 6. I .lAKARQV V.L. and SAMARSKII A.A., Application of exact difference schemes to estimating the rate of convergence of the method of straight lines. 2h. vychisl. Mat. mat. Fix., 20, 2, 371-387, 1980. 7. I 3OGLAYEV I.P., Approximate solution of a non-linear boundary-value problem with a small parameter as the coefficient of the highest derivative. 2h. vychisl. Mat. mat. Fix., 24, 11, 1649-1656, 1984. 8. JOLKOV B.A., Differential properties of solutions of boundary-valueproblems for the Laplace and Poisson equations on a rectangle. Trudy MIAN SSSR, 77, Moscow, 89-112, 1965. 9. 1LADYEHENSKAYAO.A. and URAL'TSEVA N.N., Quasilinear elliptic equations and variational problems with several independent variables. Uspekhi Mat. Nauk, 16, 1, 19-90, 1961. Izd. Akad. Nauk SSSR, 1960. 10. BERNSHTEIN S.N., Collected Papers, III, Moscow, 11. OLEINIK O.A. and RADKEVICH E.V., Second-order equations with non-negative characteristic form. Itogi Nauki. MatematicheskiiAnaliz, Moscow, VINITI, 7-252, 1969. 12. SAMARSKII A.A., Theory of Difference Schemes, Moscow, Nauka, 1977. 1.
Translated by D.L.
Translated by D.L.
U.S.S.R.
.,Vo1.28,No.2,pp.118-125,1988
Comput.Maths.Math.Phys
Printed in Great Britain
0041-5553/88 $10.00+0.00 @I989 Pergamon Press plC
A NUMERICAL METHOD FOR SOLVING QUASILINEAR ELLIPTIC EQUATIONS WITH A SMALL PARAMETER FOR THE HIGHEST DERIVATIVES*
I.P. BOGLAYEV
A method for the numerical solution of quasilinear elliptic equations in which the coefficient of the highest derivatives is a small parameter is constructed. This method provides an error estimate which is uniform with respect to the parameter. A non-linear difference scheme is constructed on the basis of the method of straight lines and the use of exact difference schemes for one-dimensionalproblems. Thecomputation grid is selected in such a way that the grid points crowd together in a special way near the boundary of the domain. An iterative algorithm, uniformly convergent with respect to the small parameter, is proposed for solving the non-linear scheme. In a rectangular domain equation
R=={(X,I/):O-=xtL, ‘O
(1)
is the Laplace operator,and F(z,y;.u) where p>O is a small real parameter, A==P/~x'+P/~g' is a sufficiently smooth function whose derivative with respect to IIis strictly positive: F,>m-const>O. Consider the Dirichlet problem
(2)
(3) u(r,&I?'0, where I? is the boundary of P. Linear equations of type (1) were considered in /l, 2/. Owing to the presence of the small parameter, the solutions have singularitiesof the boundary-layer type near r. The solution of a problem of type (l)-(3) is involved, for example, in a model describing *Zh.vychisl.Mat.mat.Fiz.,28,4,492-502,1988
119
the distribution of the potential and charge carriers in semiconductor structures /3/. The difference schemes proposed in /4, 5/ for certain linear singularly perturbed equations of elliptic type converge at a uniform rate of convergence with respect to the small parameter. In this paper a method for the numerical solution is constructed for the quasilinear problem (l)-(3), in which the error estimate is uniform with respect to p. We first construct a non-linear difference scheme, on a special computation grid, which converges at a uniform The construction is based on the straight-line method described in /6/ and the use of rate. exact difference schemes for one-dimensional problems /7/. The computation grid is selected in such a way that the grid points crowd together in a special way in the neighbourhood of the boundary. To this end we use independent estimates of the derivatives of the solution, which are of interest in themselves. An iterative algorithm, converging uniformly in ~1, is proposed for solving the non-linear difference scheme. 1. Estimates for the derivatives of the solutions of problem (l)-(3). The solutions of elliptic equations in domains whose boundaries have corner points (in the case of a rectangular domain RC",O
1.
If
~(5, y)~C(sl)llC'(Q) is a solution
The proof follows satisfies the equation
from known results
of problem
for linear problems,
F.O is evaluated at some intermediate it follows from the conditions
point.
if one notes that
s(s,bl)
y; 0),
L,(u) -~zA~--F.o~-F(z, where L,(u),
(l)-(3), then
(1.1)
But in the case of a linear operator
L,(u)<0(aO)inQ and ulr>O (GO) that
n&O
(GO)
in 51. Introducing
the auxiliary
function
~=rn-~ ( rnu~ap( F (5,y; 011, x,
we have L,(w*u)90 in Q, wf~[rZO. This implies the truth of the lemma. estimates for the derivatives of the solution of We shall formulate and prove a priori Our method goes back to the work of Bernshtein /lo/. We first derive an problem (l)-(3). a priori estimate for the derivatives on the boundary of the domain, using for that purpose certain auxiliary functions, proceeding then to estimate the derivatives in the interior of the domain. Lemma 2.
Let
u(x,~)EC"(S~)M?'+~(Q)
[
Here and below
be a solution
g++++w]P
of problem
(l)-(3).
Then
(1.2)
($19Kn[I++iTI(Y)].
,(x)=exp(-_)+exp[--1,
O
‘l(Y)‘exp(-~)+exp[-~],
n=1,2,3.
K, and G, denote constants independent of
‘p.
the We will first prove the estimate for the first derivatives on I'. Number Proof. sides of the rectangle rr,i-1, 2, 3, 4, described in the positive direction; let r,-((5, Y): ~~xsgZ,, y-4). Consider, say, a point PO-(x0, O)EI?,, O
4 (P)=
b
O,
~[l - 27-'(pa-8')s 8-a], 8
8 is a fairly small but fixed number. Let us assume where p'-(Z-G%)*, and 09x628 or neighbourhood of 20 of diameter 48 is contained in r, (the case In the domain @(P,)=(O
w-G,(s-'4).
C-Gi(Y+vq)h
that
the
L-28sk,
120 Evaluate L.(w), where L. is as in (1.1): ’ Q.*-G,p-‘qp+GI*pL-‘1 -F,“Go(e-t-i). L, ( ID)-p’Goe-c[Ga’~It follows from (2) that L,(w)>mG,+e-(G,(G,‘+G,‘q~-~G,q,-m).
In view of the fact that in Q,(p,) IGl~GP,
Iq=&'G~t
we see that for sufficiently large G, L,(w)ZmG,
NOW choose Go so large that
in
Q~(Po),
L,(~~ortu)h~~G~*tF(s, y; O)>O On the boundary of -rnr lu(s, y)l for
Q,(Pt) y=q
in
Qd(Po).
we have wztuG0, because w&O
and u-=0 for and large Go. By the maximum principle, w*uco
p0, in
while w6 Qa(Po).
Since
for g=O and pG8, it follows that at these points
W*U-0
$-(wiu)
(1.3)
Let Ps-(a,O) be such that 0Cz,6215.Then the proof of (1.3) in the domain Q@0)={OC~C28, O-=%=q(p)) proceeds similarly, provided one remembers that w~u(O on the boundary of Qd(PO) and that for z=O we have, by (3), WGO, u=o. We have thus proved estimates for the first derivatives of the solution on the boundary: n~R,l~. laa/azl~~r.~KJ~L,la~~/a~l~,,
(1.4)
We will now establish estimates_(1.2) for n=l in the interior of a. Differentiate Eq.(l) with respect to x. Putting p(r, y)=au/ax, we have a boundary-valueproblem L,(p) -p*Ap-F,p=F.,
where K, and
K.
IPbl~KlCL,
Pk%
(4.5)
are evaluated on the solution ~(5, P). Define an auxiliary function n,(X)=KJ1+N'e(s)l,
where the constants K, and x will be determined below. Since L,(l-I&:p)--xX- (F,-x’)II,fF., it follows that &,(lJ,-&p),
for x*.Sm
and K,>x*~IF,I.
K,>R,, then ll,zttp>Oon F. If K, is sufficiently large, it follows from By (1.4), if the maximum principle for L, that (pIGI,, proving (1.2) with n=l for ada2; Similarly, the introduction of the auxiliary function Q,(P)-K,[l+p-'q(g)] yields estimate n=l, for atday. (1.2), The a priori estimates (1.2) for the second derivatives are established along the same lines. One first proves estimates on the boundary of 52. For example, for acta , one considers problem (1.5). Extending the function Qi=Wrpsmoothly to P((p~cpl~l=~R,) and putting p”-=P(P-9) I we have a problem L,(p)-Fu@+fl.-$A@,
fib==0,
analogous to (l)-(3). Consequently,we have estimatesof type (1.41, i.e., ~aP/aX~~,.+ZR,Ip'. To obtain estimates for a%&'x in the interior of P, we differentiate the equation in (1.5) with respect to x and consider the following boundary-valueproblem for r(x,y)==a'u/&r': L,(r) -Cl*Ar-F.r~F,,p’+2F,pfF,, rlr
fixed,
Irlrl(RJp'.
Define the auxiliary function then it follows that for x'
Non-linear difference scheme.
1. Construction of the difference scheme and error estimate. On the set a, introduce
121 a computation grid IIJ~==~~("XG)~(*), where 7Gijh("=(~<, i4, 1,...,NC'),ZFO, XN(~~), iihjh(*'=@/,, i-0, 1,..., Nf'),yo==o, yrw=b}. Put rh-C%nr,Oh-WlP. Henceforth we shall use the method of straight lines and exact difference schemes for one-dimensionalproblems /7/ to construct a difference scheme on L. In /l/, considering the one-dimensionalproblem @"(0‘fV,
f,Wn-const>o
w(o)=w(z)=o,
m),
we constructed a difference scheme, uniformly convergent with respect to k,based on the method of exact integral relationships. To this end one first isolates the principal part of the differential operator LL(P)-~'wN(t)-gw(t)=~(f. m), Pf-P, where ~-%onst>O is a parameter, and then establishes integral identities for the solution : m(t) on a given grid {tcI-o,l,...,N,to-O,tn=Z}
s
%+1
t1
%I@)0 (6 w) ds19
w,-w(tO,
i=i,2,...,N-
Y=Pc19
w=y SW y&-i), (yAt,) I,
w=ytctn (yAt,-J+cth where
1, wO=~w~;=O,
Atc;=tt+,-tr,
[k,tt+s1 of the homogeneous problems
gJ?(t),cp~n(t) are solutions on
JL(Wr)-O, cpc'(k)--l,cp:(toJ-o, JL((PP)==O, cpP(t,)=o, qP(tol)--l. One then approximate the right-hand side of (2.1) by replacing the function [&&+,I by its value at t,;thisgives a difference scheme (T*W),
5 &
[a+1 (Wt+1-
1=4,2,...,N-4,
EL B
WI) -
-
WC-,)I =
f h WA
(2.2a)
Wo-Wn-0, h
$1 (4 ds +
+
4 w,
cP(Cm(t)) on
ti
B,YW]= CI -
(ai+1+ 4.
The integral identities (2.1) will be used to construct a difference scheme, based on the straight-linemethod, for problem (l)-(3). On the straight line y,.j=1, 2,...,N(*)-1, identities (2.1) become
(T’“u),, = aPL.1,I - cl%,,+ d&4+1,j = 1 1
P
xrp:fi”’
tS
Xi+1 (s) $ypp’ (s) ds +
$’ (‘)(4 I$*’(4 ds],
j
xCl
1
e=1,2,..., N(l)- 1, uo,= uN(q = 0, $p' (8)= F (s,VJ;
where
u
(8,
1/1))
-
fi(‘)u
(St
YJ)
-
pa
bu&‘J)
,
On the straight line 21,i--l,2,...,N(')-1, we have
~"'-censt>O, I&j-u(zc, &).
1 1
P
f=1,2
tS y
&“’
"lC1 (s) qi” (s) ds +
"j-1
,...,
N@)-I,
5 cpj”(*)(s)cp?’(8) dsj
,
"I U,O=U*N(~)=O,
91”(s) = F (XI, s; u (51,s)) -. p’u (x,, s) fi(*) = const ;zO.
The coefficients of the operators T"' T"' and the right-hand sides of the identities are determined by analogy with the formula: in (2.1). Substituting $"'(s),+"'(s) in the form
122 we rewrite the integral identities as follows ((&II&Q) :
Tg’u = F (z, 8; u) - p’ g TP’u=F(2,y;u)-pa-gg where the operators notation
(2.3a)
+ J(1)(z, y; cp”‘),
a*u f .m (5, y; lp’) ,
(2.3b)
Th"),!fh")are defined by the formulae in (2.2) and we have used
the
Adding the two identities of (2.3) together for (Z,Y)E% and using the fact that n(s,EJ) is a solution of Es.(l), we obtain the following relationships for solutions of problem Cl)(3): TP’u+ TP’U = F (2, &!; u) + J(1) + J(r), (2, y) E Wh, uIq$=O. (2.4) As a difference scheme for (l)-(3) we take the following non-linear system of algebraic equations for the grid function v(s,y): Lhu = Tf’Uf Tf'U = F (x, I/: U), (x, g) E (oh, UIrh=O. where u is a 'solutionof problem (l)Lemma 3. The error Z~U(X, y)-U(s, g), (2, ~)Eo~, (3) and U a solution of (2.5), satisfies the estimate
ma= lz 6~ rdlb + k UEZ* Proof.
I(rmygmh I JC1) I + &xh
I m II*
By (2.4), (2.5), we conclude that ~(s,v), (3,Y)E?& Lhz- F,,% = - (J(1) + J(‘)), (2, y) E oh,
w-9
is a solution of the problem
zIrf$=O,
where we have used the mean-value theorem, and PyO=Fy(x,II; 6(z,y)),Clbeing an intermediate point of the interval whose endpoints are u and U. The truth of (2.6) now follows from (2) and the maximum principle for difference equations (see, e.g., /12, p.248, Theorem 3/. 2. Computation grid. We now introduce a special grid tjh'which takes the qualitative behaviour of the solution of problem (l)-(3) into consideration. It follows from (1.2) that in the neighbourhood of the sides rr,rr boundary layers form along the x-axis, and in the neighbourhood of r,,I's - along the y-axis, the dimensions of the boundary layer soy being O(&), &=t.l.llIl~]/x. 'l,'x'jjJh'l" The grid ZL'=G)* is defined as follows Z'E [O,h,]: zi=-pl[l 5Np=he,
x0=0, zt E(h,,
I, -h,)
-(I
-p)&j,
i=o,i,.
. .,Np,
&l/No”),
: m:x(x',,-s')=I#,
izS_NP'+l, . . ., N:‘,
=I, + -$ In(I- (1- p)6(N'l) - i)], 21FII,-'h,,z,]:z, i=Nf'+1
(2.7e)
(2.7b) (2.79
,...,N"',
x&i -1,-h,?
Z*W-=l,,
N"'=,'$'OW+N,'~'+l
The grid points in 8~~'~)' are defined by similar formulae in the intervals [O,h,], (&I,-&), N,@l,ho),iv'*).The total number of grid points Nlkh,, L I, using the notation N,'*)==N,"), N"'N'" is independent of p. The grid points in the boundary layer are so chosen that at two neighbouring grid points the variation of the solution is bounded uniformly in P by a quantity O(6). We stipulate furthermore that N"JI=N'~'. 3. Estimate of the convergence for scheme (2.5) on the grid Gi,,’ : Theorem 1.
of the difference
scheme.
The following theorem holds
The rate of convergence of a solution of the non-linear scheme (2.5) on the
12.3 grid
(2.71 to a solution
where
the constant
Proof. now estimate
K
of problem
(l)-(3) satisfies
the following
estimate
and the total number of grid points N are independent
We have already established an estimate (2.6) for the error ](I)J@' on the grid o,,'.Rewrite I"',I(') in (2.3) as
J(Z) (?p’)
=
J(2)
($P’) +
~ro,l;_pwu,
In /7/ we established
estimates
(
J(S)
-
pB
q+l)~F_pwU,
for terms of type
uniformly
in
I(:
of p. z-u-u.
We shall
g) , ’
li"(qc')), ~"'($"') on the grid
(2.7) :
(2.8) We claim that
(2.9) By (2.3), with
yj fixed, we have
(
J(1) xiv YJ;
-
1=
p’a @lJ(SlYJ) ayr
we have the estimate
(2.10) Indeed, differentiating
(1) with respect
to r, we obtain
a ac P’-ax ayl
ac
au
( >=--Ir’,,+F,,+F,.
Estimating the right-hand side by means of (1.2) for points in (2.7) are so chosen that IarP(--r~)if and
erY(--~+l)l<6
n=l,3,
we verify
The
grid
m;r(mi+i- 4 <6/x,
+(,z,+~E[O,I~~]U[Z~-~~,I,]. Hence, by (2.10), we obtain (2.9). (2.9) and using Lemma 3 , we complete
(2.10).
Combining the proof of the theorem.
the estimates
(2.8)
3. An iterative algorithm for solving the non-linear difference scheme. 1. The iterative algorithm. We adopt the additional assumption that F(z, y; u) (see (1)) satisfies
a two-sided
estimate
m
M=const.
(3.1)
Construction of the iterative algorithm is based on a discrete analogue of the method upper and lower functions developed for one-dimensional schemes in /3, 7/. The iteration sequence {u(n)(r,Y), (5, Y)a,,) is defined by the recurrence formulae
,l’JC
fixed,
U(n+*),U(n)+Z(n+l)
cP)(Q=O'
n=O,l,...,
(3.2a)
,
Lh~(n+l)_M~("+I),-[Lbl(")-F(2, y;VW)], fj(') = b'"= M.
of
(I,Y)=a%
(3.2b) (3.2~)
124 Theorem 2.
If the initial approximation U to'in (3.2) satisfies the conditions L,.?Y'-F(t, y; u'O')Go (>O),
(3.3a)
(2,b()EOh,
(3.3b)
u@?lrh=0,
then,the functions U'*',n-O,&..., of (3.2) are upper (lower) functions of (2.5) and (3.11, where V(s, g) is a solution of problem (2.5). The sequence (U'"') i.e., u’GU(“’ (GP’) is monotone decreasidg (increasing)and converges to u' at the rate of a geometric progression with qd-mM-l. Proof. We prove the theorem for upper functions. By the maximum principle, if ~*l'+"-_Mz("+l',O,2("+"(,-0, then z'"+"
for (2,Y)=OA. Using the mean-value theorem, we conclude from (3.2) that ,*U'"+"-_F(r, y; U(n+r')=(M-F"O)z("+", (I,b()EOh. (3.4)
If U'"' in (3.2) is an upper function, then by the maximum principle Z'WO and by induction, using (3.1) and (3.4), we obtain z~"+~'GO, n=O, I,..., i.e., (Vn') is a monotone decreasing sequence. By (3.2) and (3.4),
The maximum principle yields the estimate (1 z(n+l'I( E max 1~(~+r'(z,y)l <
Hence the following estimate for the convergence of the iteration sequence: where
s("
z(n+"~<4n[jP'll, is determined by the system ' Lh2(l’ -
Jf$+” = -
[L’JP - F (U”)l,
(2, e) E %r
(3.5a) a(‘) ‘rh =o.
It follows from (3.4) that {U("'} converges to a solution of problem (2.5), (3.1). theorem is proved.
(3.5b) The
2. Choice of initial approximations. we describe one method of constructing U"' the basis of (3.3). Define V(z, u) on %I,+so that VIm=O. Solve the linear system RIfh=O. IL,,V-F(V)I, (%~)=@hl LhR-mR=Then
on (3.6)
U'O)=Vi-R is an upper function. Indeed,
Lh(V+R)-_((V+R)=mR-_Lhv-_((V)I+thV-_((V)F,,OR<(m- F,O)R
(GV)~a,,,
V+Rlr,=O,
where we have used the mean-value theorem, the non-negativity of R and the bounds (3.1). Solving the system &,S--8=(&V-F((V)Ir (r,#)Emh, S[r*=O, instead of (3.6), we obtain a lower function U@'-VSS. Note that if
GlO' and
0'0' are upper and lower functions, respectively, it follows
(ci'"') are upper and from Theorem 2 that all the terms of the respective sequences (8tn'), lower functions, with PO'<. . . < o(n’.$O(n+l’ <. , . < Uf < . , . < tP+l’<
i7’“‘g . . .
Parallel computation of the upper and lower functions enables one a posteriori to improve the zone in which problem (2.5) is solved. For example, if V==O,(z,y)=Z&, then the maximum principle yields the following estimate for W-R:
IIUco)II
Hence, using the maximum principle for (3.5b), we obtain
1’z”“‘< AI-1
max ImU IX, Y)EQl*
- IF@, y;O)I - F(x, y;U@))I< K,
125 i.e., the sequence (zcn)}of (3.5&, and hence also
(W)}, converges at a rate uniform inn.
Remarks. 1. The exact solution of problem (l)-(3) on expressed as
[zs+(+I] with YJ fixed may
be
I) is the Green's function of the operator L,(ur)=~'m"-fi~')w where G,('l(r, (for the definition see /7/). Define the following function on the straight line yJ.j=l,Z,...,N(*)-i: up' (2,y,) = u (I*, y.) q?. (1)(4 + CJ (Q' 11
=[z,,W,!.
Uj) qp"'
(4.
(3.7)
i=o,1 ,...,NJ')-i,
where u(r,v),(I,JJ)EB~is a solution of the difference scheme (2.5). Then the following estimates hold throughout the interval .[O, Jr] : &zl I'(+* bJj)u~'(z*Yj)16 +?
j-1.2,...,NC*) -1.
Accordingly, on the straight line z,, I-1,2,...,N(Jl-i, the function u?'(IJ* U)= u (zi* Uj)Q:*(*J (V) + U (fJ* Uf+,)~~jf”‘“(U), Ye h
1J+Il, j=O,'t ,
. . , N(‘)-i,
satisfies the estimate
The proof of, for example, the estimate for (3.7) follows from Theorem 1, the fact that the functions O< +(')‘(I), &(l) (x)6; are bounded and the explicit form of the function G,(‘)(z, :) (see /7/). 2. If the parameters PC*) and b"' in the difference scheme (2.5) tend to zero, we obtain the following traditional scheme for a non-uniform grid LhU-CI*(U?;+tl~~)-F(z,y,U), (t, JJ)ee.% Ulm-0, for which Theorems 1 and 2 remain valid. 3. All our results (Lemmas l-3 and Theorems 1 and 2)‘ which were proved for a two-dimensional singularly perturbed problem, may be generalized to equations of dimension 1>2, considered on parallelepipeds. REFERENCES lISHIK M.I. and LYUSTERNIK L.A., Regular degeneration and the boundary layer for linear differential equations with a small parameter. Uspekhi Mat. Nauk, 12, 5, 3-122, 1957. 2. 3UTUZOV V.F., Asymptotic behaviour of the solution of the equation p*Adu--Kp(z, r)u-fft, y) in a rectangular domain. Differents. Uravn., 9, 9, 1654-1660, 1973. 3. 30GLAYEV I.P., On numerical analysis of the main equations simulating the operation of semiconductor structures. Preprint, ChernogolovkaInst. Prikl. Teoret. Mat. Akad. Nauk SSSR, 1986. 4. I 3AKHVALOVN.S., On the optimization of methods for solving boundary-value problems in the presence of a boundary layer. Zh. vychisl. Mat. mat. Fiz., 9, 4, 841-859, 1969. 5. ;HISHKIN G.I., A difference scheme for elliptic equations with a small parameter as the coefficient of the highest derivatives. Dokl. Akad. Nauk SSSR, 286, 1, 57-61, 1986. 6. I .lAKARQV V.L. and SAMARSKII A.A., Application of exact difference schemes to estimating the rate of convergence of the method of straight lines. 2h. vychisl. Mat. mat. Fix., 20, 2, 371-387, 1980. 7. I 3OGLAYEV I.P., Approximate solution of a non-linear boundary-value problem with a small parameter as the coefficient of the highest derivative. 2h. vychisl. Mat. mat. Fix., 24, 11, 1649-1656, 1984. 8. JOLKOV B.A., Differential properties of solutions of boundary-valueproblems for the Laplace and Poisson equations on a rectangle. Trudy MIAN SSSR, 77, Moscow, 89-112, 1965. 9. 1LADYEHENSKAYAO.A. and URAL'TSEVA N.N., Quasilinear elliptic equations and variational problems with several independent variables. Uspekhi Mat. Nauk, 16, 1, 19-90, 1961. Izd. Akad. Nauk SSSR, 1960. 10. BERNSHTEIN S.N., Collected Papers, III, Moscow, 11. OLEINIK O.A. and RADKEVICH E.V., Second-order equations with non-negative characteristic form. Itogi Nauki. MatematicheskiiAnaliz, Moscow, VINITI, 7-252, 1969. 12. SAMARSKII A.A., Theory of Difference Schemes, Moscow, Nauka, 1977. 1.
Translated by D.L.