Matched convergence-rate estimates of the mesh method for the axisymmetric Poisson equation in spherical coordinates

U.S.S.R. Comput.Maths.Math.Phys.,Vol.27,No.4,pp.195-197,1987

OO41-5553/87 $10.OO+O.OO ~ 1 9 8 8 P e r g a m o n Press plc

P r i n t e d in G r e a t B r i t a i n

MATCHED CONVERGENCE-RATEESTIMATES OF THE MESHMETHODFOR THE AXISYMMETRIC POISSON EQUATION IN SPHERICALCOORDINATES* V.L. M A K A R O V and A.I. R Y Z H E N K O

S o l v a b i l i t y c o n d i t i o n s in w e i g h t e d Sobolev spaces W=li+,(~) are o b t a i n e d for the a x i s y m m e t r i c D i r i c h l e t p r o b l e m for Poisson's e q u a t i o n in spherical coordinates. A d i f f e r e n c e scheme is constructed, w h i c h in the m e s h L=,,+.(~) norm has first order of accuracy, m a t c h e d w i t h the smoothness of the s o l u t i o n of the d i f f e r e n t i a l problem. C o n s i d e r a b l e a t t e n t i o n has been p a i d to the c o n s t r u c t i o n of d i f f e r e n c e schemes w h i c h have c o n v e r g e n c e - r a t e e s t i m a t e s that are m a t c h e d w i t h the smoothness of the solution of the r e l e v a n t d i f f e r e n t i a l problem. This is p a r t i c u l a r l y true of the main p r o b l e m s of m a t h e m a t i c a l p h y s i c s in C a r t e s i a n c o o r d i n a t e s but is m u c h less true of p r o b l e m s in c u r v i l i n e a r coordinates. In /3-5/ d i f f e r e n c e schemes for linear and q u a s i l i n e a r second-order equations were studied on the a s s u m p t i o n that the s o l u t i o n of the d i f f e r e n t i a l p r o b l e m is s u f f i c i e n t l y smooth. For the D i r i c h l e t p r o b l e m for P o i s s o n ' s e q u a t i o n in c y l i n d r i c a l coordinates, a d i f f e r e n c e scheme w i t h a m a t c h e d c o n v e r g e n c e - r a t e e s t i m a t e was d e v i s e d in /6/. Similar estimates for schemes of the f i n i t e - e l e m e n t m e t h o d were o b t a i n e d in /7, 8/. i. We c o n s i d e r the a x i s y m m e t r i c D i r i c h l e t p r o b l e m for Poisson's e q u a t i o n in spherical coordinates : =-r:

z Ou r ~

ulr=O,

,=o=0,

O~O<~u,

sin O/(r, 0),

(r, O) ~ fl,

a= I0=0= =0, sin0 a0

(ha)

0
(ib)

( ~ { ( r , O): O<:r
where

We denote b y ~)'~.I+~(L ~) the class of g e n e r a l i z e d functions w h i c h are o b t a i n e d by the c l o s u r e of the set of functions of class C2(~)~CI(~) and w h i c h satisfy condition (ib) in the norm

where e is a real p a r a m e t e r and ~=~ua~. We assume that the f u n c t i o n /(r,0) in Eq. (la) has the form 0 l(r, O) -- (,-3+, sin O)'.'21o(r, O) - r - ' - ' sin -V~ O - - (r('+~i/21, (r, 0)) Or 0 r -(3+~,'a sin -1 0 - - (sin 'i' O/2(r, 0)). 00

Definition. We call it satisfies the i d e n t i t y

u(r,0) a g e n e r a l i z e d solution of p r o b l e m (1) in space a(u, ~ ) = l(~)

WI2.1+,(Q) if

v ~ ~ W2i,~+~(~),

(2)

where a(u,~)= /(~)= n and

we a s s u m e

r ~ + ~ - - - -Or + r * - ~ - - - -O0 + ( e -O0 l)r'-'Or

sin 'h O r(~-l)12fogt+r(~+~)/2/~ - - + r('-il/2]z Or

l~(r,O)EL',.(-°-),i=O,l,2 .

that

Theorem i.

sinO

If

0
-~-r Iz dfl, d~,

We h a v e :

(2) has a unique g e n e r a l i z e d solution in the space

The p r o o f is o b t a i n e d by c h e c k i n g the c o n d i t i o n s of the L a x - M i l l g r a m lemma for

W~.,+

(2).

2. We will n o w c o n s t r u c t a n d study the d i f f e r e n c e scheme for p r o b l e m (2). We cover the d o m a i n -~ by a m e s h ~=~,×~2, where ~,;~={r~=i/h: i=0,|.....N,, I,~--I/N,},~2={0j=~h~: i=0, I....,N2, h2=~/,\'2}, N2=2h, I. is an integer, k>~2 We put ~=~nF, ~=~\7, Hh is the Hilbert space of functions d e f i n e d on the mesh ~ and v a n i s h i n g on the b o u n d a r y y. The other n o t a t i o n is the same as in /9/. We a p p r o x i m a t e p r o b l e m (2) by the d i f f e r e n c e scheme

*Zh.vychisl.Mat.mat.Fiz. ,27,8,1252-1255,1987 195

196

Au~(At+Az)y=-%

(r,O)~w,

v(r,O)=O,

(r, 0)~'f,

(3)

where A, (V)=TO (sin 0)

(ay;)~,

Azy=T ~(1) (bye) 0,

i=t, 2. . . . . N,-I,

i=l,2,...,.Yz-l, --

_

-

_

h,(r,-t~ r, ) ,

a,

i=2,3 .... ,N,,

tg (Q~/2) ~ , tg(Q~_/2)

b~=h21n -

at=0,

j=2,3,...,Nz-l,

rt÷t

Tr,(w(.))= j ~,-,~,,(ri)w(DaL

,=1,2..... N,-i,

(4a)

rt.t ot,t

r o t ( w ( . ) ) = ~ p.¢2C~)w([)d[,

]=l,2,...,Nz-t,

(4b)

B/.l -t

-t

r~_t~r~ri,

ht-trl-'(r~_ - r - )/(r _ -r~- ), ht-tr -,(r-'-r~+t)/(ri-i-ri+t),

bt~l(r) =

ri<~r<~ri+l, r e[0, i]\[r~_l, ri+t],

0, i=2, 3. . . . , N t - t ,

i ptt (r)=

hi-ir ~-', hi-trl-~(r-~-rz-~)/(r~-t-rz-t),

O~r~r,,

rt<~r<~r2,

O, j

,

r ~[0, t]\[0, rz],

tg(0/2) • -- m tg (0~_,/2)

tg(0fi2)

hz- i n - -

tg(0j-i/2)

tt? (0) = / h~-~ In tg(O~+t/2)- .in- , tg(0~+,/2) tg(0j/2) (012)

[

0j~<0<0j+t,

"-tg

0 ~[0, .~]\[0j-,, 0,+,1,

0,

j=2, 3. . . . . N2-2, h2-~, tg (02/9--

~J(0)=

0~<0~01, tg (02/2)

hz- l l n - - l n

'~

tg(0/2)

tg(0,/2)

,

01~<0~0z,

0,

0 ~[0, :d\[0, 0z],

tg (0/2)

h~-' lnt-g(0~.~_2/2)~

2

ix.,~,-, (0) =

(0,.¢~-,/2)

tg(0N~-2/2)

0:~'-~<~0~<0N'-t'

h2-', O,

• For the error of the m e t h o d

0av,-l~0~, 0 *[0, ~]\[0~,,-:, :*1.

z=y-[Tr(1)]-'Tr(u)

Az=Al'q, where

In- tg

(r, 0)co),

we can write

z(r, 0)=0,

(5)

rl=[T o (sinO)]-~{T ° (a sin 0)-[Tr(i)]-~T° (sinO)T~(u)}. The p r o p e r t i e s

Lemma i.

of the o p e r a t o r s

For the o p e r a t o r s

T",

Tel

will be u s e d below.

T rt, T °t, given by

(4) we have

c, sin 0j
t
the estimates:

hz-~/4,

(6a)

i=2, 3. . . . . N , - i ,

Tr'(t)r~-l-t-e(e+t)T~'(~-t-e)In2>O, where

the p r o b l e m

(r, 0)~7,

(6b)

i=l,2,...,N,-t,

e e(0, ~.],

(6c)

cl=7/,:, cz=2, e,=0.700050473. The d i f f e r e n c e

Lemma 2.

analogue

For a n y m e s h

of the i m b e d d i n g function

Nt--1

theorem

w(0~), d e f i n e d

is:

on the m e s h ~J2. we have

N,--I

Z h'bj wOt
T h e o r e m 2.

For any f u n c t i o n

u~IL~ w i t h

0
we have

(7)

([r'+'rr(l)T ° (sin 0) ]-tt,,v, A,v)<([r'+'T~(t)T o (sin 0)]-'Av, Av). The p r o o f is o b t a i n e d by f i n d i n g a lower bound of the r i g h t - h a n d side of aid of Lermma 2 and r e l a t i o n (6a) of Lemma i. We will n o w o b t a i n an a p r i o r i e s t i m a t e for the error z. We have:

Lemma 3.

For the m e s h

function

z, i.e.,

the s o l u t i o n

of p r o b l e m

(7) w i t h

(5), we have

the

197

II(rl+'sinO)'azil
(8)

where c is a positive constant, independent of u(~ 0) and of the mesh steps h, and h~. The proof uses the difference analogue of Nietche's method. In short, to obtain the convergence-rate estimate, we have to estimate the norm in the right-hand side of (8). It can be shown that, for ~=[Tr(1)r'+'Te (sin0)]~J~, we have the estimate

I~1 ~ e,¢=(r~-~, r~+,)x(Sj_,,

where

and

8~+~), i = t

2. . . . .

c[ (hl/h2)'t,+ (hdh,)'"l Jlu]h.t+,., O,

N l - t , j = t , 2. . . . .

N2-1.

In

this

(9) connection,

from

relations

(8)

(9) we have.

Theorem 3. Let the conditions of Theorem 1 hold. Then the difference scheme (3) with 0
II(rt+'sinO)'J'zU~clhl[lulh.,+~.a, where

riO)

[hl=(h,'+h2=)'I'

Note. In accordance with the definition /3/ of a matched convergence-rate estimate for a difference scheme for an equation of elliptic type, our estimate (iO) belongs to the class of matched estimates. REFERENCES i. MAKAROV V.L. and SAMARSKII A.A., Application of exact difference schemes for estimating the rate of convergence of the method of straight lines, Zh. vych. Mat. i mat. Fiz., 20, 2, 371-387, 1986. 2. LAZAROV R.D., MAKAROV V.L. and SAMARSKII A.A., Application of exact difference schemes for constructing and studying difference schemes on generalized solutions, Matem. sb., 117, 4, 469-480, 1982. 3. FRYAZINOV I.V., On difference schemes for Poisson's equation in polar, cylindrical, and spherical coordinates, zh. vych. Mat. i mat. Fiz., ll, 5, 1219-1228, 1971. 4. GLUSHENKOV V.D. and LYASHKO A.D., Difference schemes for quasilinear elliptic equations in polar coordinates, Differents, ur-niya, 12, 6, 1052-1060, 1976. 5. KARCHEVSKII M.M. and LYASHKO A.D., Difference schemes for quasilinear elliptic equations on a polar mesh. Numerical methods of the mechanics of a continuous medium, VTs and ITPM SO AN SSSR, Novosibirsk, 3, 4, 77-86, 1972. 6. LAZAROV R.D. and MAKAROV V.L., A difference scheme of second order of accuracy for the axisymmetric Poisson's equation in generalized solutions, Zh. vych. Mat. i mat. Fiz., 21, 5, 1168-1179, 1981. 7. BENDALI A., Approximation of a degenerate elliptic boundary value problem by a finite element method, R.A.U.R.O., Analyse Numer., 15, 2, 87-99, 1981. 8. ZHOU S.Z., The linear finite element method for a two-dimensional singular boundary value problem, SIAM J. Numer. Analys., 20, 5, 976-984, 1983. 9. SAMARSKII A.A., Theory of difference schemes (Teoriya raznostnykh skhem), Nauka, Moscow, 1983.

Translated by D.E.B.

U.S.S.R.. Comput.Maths.Math.Ph~s.,Vol.27,No.4,pp.197-201,1987 Printed in Great Britain

0041-5553/87 $10.00+O.OO 0 1 9 8 8 Pergamon Press plc

CALCULATION OF TRANSIENTS IN A MICROWAVE AMPLIFIER WITH A DIELECTRIC FILLING"

A.R. MAIKOV, A.D. P O Y E Z D a n d S.A. YAKUNNIN

Using as an example the design of a microwave amplifier with a dielectric filling, a numerical algorithm for solving the non-stationary system of Maxwell's equations is described. Integral laws of conservation and variation of momentum and energy are satisfied at the discrete level for the method. The results of a non-linear approach and of the linear theory are compared. The strict statement of non-stationary boundary conditions simulating radiation from a bounded domain is discussed. The development of modern electronics demands the development of new effective algorithms for the numerical analysis of the physical processes in microwave devices. The traditional method in the electrodynamics of slowly varying amplitudes /1/ often does not give sufficient information about the evolution of the fields and beam instabilities in the device. The most

*Zh.vychisl.Mat.mat.Fiz.,27,8,1256-1261,1987