- Email: [email protected]
The piecewise-polynomial approximation in W21 of functions from Wp2 , 1<p ⩽ 2
114
ViN. Orlov
In view of the form of this system (see [S] ), we arrive at the familiar result (see [6,3] ) that, in the absence of integrals, Au is finite-dimensional. Translated by D. E. Brown REFERENCES 1.
PAVLOVSKII, Yu. N., Group properties of controlled dynamic systems, and phase-constraint structures, I., Group properties of controlled dynamic systems, Zh. vj@W. Mat. mat. Fiz., 14, No. 4, 862-872, 1914.
2.
PAVLOVSKII, Yu. N., Group properties of controlled dynamic system, and phase constraint structures, II, Phase constraint structures,Zh. vjkhisl. Mat. mat. Fiz., 14, No. 5, 1093-1103,1974.
3.
PAVLOVSKII, Yu. N., Group properties, aggregation, and constmint structures of controlled dynamic systems, Diss. dokt. fm.-matem. n., VTs Akad. Nauk SSSR. Moscow, 1975.
4.
OVSYANNIKOV, L. V., Lectures on the theory of group properties of differential equations (Lektsii PO teorii gruppovykh svoistv differentsial’nykh uravnenii), Izd-vo NGU, Novosibirsk, 1966.
5.
EISENHART, L. P., Continuous groups of transformations, Dover, 1933.
6.
YAKOVENKO, G. N., L-systemsand their investigation(on a generalization of the class of linear dynamic systems), Diss. kand. tIz.-matem. n., MFTI, Moscow, 1973.
U.S.S.R. Comput. MathsMath. Phys. Vol. 18, pp. 114-121 0 Pergamon Press Ltd. 1979. Printed in Great Britain.
0041-5553/78/0801-0114$07.50/O
THE PIECEWISE-POLYNOMIAL APPROXIMATION IN W21 OF FUNCTIONS FROM “5 , i c p 6 2 * V. N. ORLOV
Leningrad (Received 3 May 1977)
defined on the unit square, a piecewise-polynomial FOR A function u from w,,s, i-zp<~, function c from IV21 is constructed depending on O(N) parameters. The points at which the parameter values are defined may be locally condensed. As strategy of condensation the Birman-Solomyak algorithm [I] may be used. For this condensation the estimate (1) is obtained.
Introduction In the computer solution by the mesh method of elliptic type differential equations, it becomes necessary to condense the mesh at those places of the domain where the solution of the equation varies “sharply”. In the variational difference method this leads to a fine triangulation at places of “sharp” variation of the solution. The passage from the fme to the usual triangulation causes technical difficulties. In [2,3] two methods were presented for constructing an approximating function from IV21 based on a locally condensed mesh and the results of numerical experiments were given; however in these papers estimates of the approximation are not given. In [ 1] estimates were obtained for the approximation of the initial function by a piecewise-polynomial function constructed by partition of the domain into locally condensed squares, but the approximating function does not belong to IV21, which is important for the variational-difference method. *Zh. vjkhisl. Mat. mat. Fiz., 18,4, 935-942,1978.
Piecewise-polynomialapproximation
115
In this paper we construct a piecewise-polynomial function g from W-J1, depending on O(N) parameters, whose values are defined at the vertices of the squares into which the initial unit square is subdivided. By Theorem 2.1 of [ 1 ] we obtain estimates of the approximation error of the form Ifu-a IIrp,‘scN-“l/ ull wp’.
l-C&2.
(1)
The method of constructing the function r differs from the methods presented in [2] and [3]. It is known that the error of the variation-difference method of solution in W2’ is estimated by the approximation error, consequently, the estimate given confirms the existence of a “best” mesh l
1. Generalizations, construction of the Binnan-Sofomyak subdivision, construction of the completions
Let p
be a unit cube in Rm with centre at the origin of coordinate
C is a constant whose value is a matter of indifference for the discussion. We consider the square Q2 on a plane. We subdivide it into 4 equal squares. Each of the squares obtained can again be subdivided into 4 equal squares. Any of the resulting squares can again be subdivided into 4 equal squares, and so on. We call the union of all the squares obtained at any stage of this algorit~ the 2R-subdivision of the original square. The size of the squares occurring in the ~-sub~~sion may differ, since at each stage it is not obligatory to subdivide ail the squares into 4 parts. We consider an arbitrary 2R-subdivision of the square Q2. We denote by 1S 1 the number of squares in the subdivision, by Ai an arbitrary square of the 2R-subdivision, and by I Ai I the measure of this square. The vertices of the squares belonging to the 2R-subdivision, we call real points. The real points situated on sides of the square L$ and not vertices of it, we call supplemental points for A, We define the vector of the values of the continuous function u at the real points as the mesh function and we denote it by U. Let the function FsL, (Q') , lCpS2 be defined on Q2. A fundamental lemma holds which is a particular case of Theorem 1 of [ 1].
V. N.
116
Orlov
Lemma 1
Let FEL, (Q”), pa 1,00. Then for any natural N a 2R-subdivision of Q2 into squares can be found such that 1B 1
max{lAil”liFll ~~(*iI}~C((I)N-(.+~)IIFII~~~~~~.
(2)
1
We call this subdivision the Birman-Solomyak subdivision. Following [ 11, we omit the algorithm for obtaining this special 2R-subdivision of the square. We subdivide Q2 into 4 equal squares. For each of them we find the numbers P
bt=IAtl”llFllt,~a,,. We subdivide into 4 equal squares those 4 for which bP2-2”max{b,).
(3)
i
We obtain a new X-partition of the square Q2. For each of the squares of this subdivision we find the numbers bj and repeat the process (3). If F # 0 almost everywhere, then in the new subdivision the number of squares is greater than in the old one. We stop the subdivision process when 1E 1 exceeds N. As shown in [l] , the estimate (2) holds for the subdivision obtained. We pass to the construction of the completion Z of the function UEW,~ (Q") , lCpG2. The proposed construction is constructed on any 2R-subdivision of the square. We will use it on the Birman-Solomyak subdivision. At each real point we defme a number which we call a real parameter. In our case we take u as the real parameters. We consider a typical case: the square 4 with side hi, on whose right side k supplementary points lie, but no supplementary points lie on the other sides. For convenience of notation we place its centre at the origin of cordinates of the system. We denote where 52, lies in the the squares obtained on subdividing it into 4 squares by f&, . . . , f&, Square I, 52, in the square II and so on. Some of the vertices Qi, , . . , Qc we mark specially: Ai(O, h/2),
A2(4*/2,
O),
As(0, -h/2),
A‘@,
0).
If k = 0, we construct the polylinear completion (see [4], p. 92) with respect to 4 real parameters at the vertices of the A,. If k = 1, there exists one supplementary point at the middle of the right side, at which a real parameter is defined. We introduce 4 fictitious parameters at the points A,,..;, A4. We set the values of the parameters at these points equal to the value of the polylinear function constructed for k = 0. On each of the squares Sz,, . . . , !& for the known parameters at the vertices we construct the polylinear completion. For k = 2 (depending on where the additional point lies), we again subdivide either 521 or a;24into 4 parts and repeat on it a construction similar to the case k = 1. Let the completion for the k-1-th additional point, k > 2, have been constructed. The completion for k additional points is obtained by reconstructing the function only on that square whose right side is bisected on the addition of the k-th point. The reconstruction itself is just the same as for the case k = 1. If there are additional points on other sides of the square Ar, the completion is constructed similarly: instead of the fictitious parameters at the points A 1, AZ, A3 the corresponding actual parameters must be used, and so on. Therefore, on each square 4 at the 2R-th subdivision a piecewise-polynomial function has been constructed, which is identical with the function u at the actual points and is continuous on Q2. Thereby the function CEW~’ (Q’) has been constructed on Q2.
Piecewise-polynomi
117
approximation
The function i? constructed by the method described above on Q2 for the actual parameters, is called the R-polylinear completion of the mesh function U. We note that when additional points are present u is a piecewise-polylinear (and not a polylinear) function on the square.
2. Approximation theorems In estimating the rate of convergence of u” to uin the Birman-Solomyak subdivision an important part is played by the following lemma. Lemma 2
the square Q2 has been subdivided into squares by the Let UEW,~(Q*), l
IU41,,Ai aypllnllL;(ai
(4)
),
Proof: We again consider the typical case which we considered when constructing the completion in section 1. Let uk be the completion on Ar for k additional points on the right side. For k = 0 estimates hold which are proved by the standard method (see, for example, [l]):
(6)
The difficulty is to obtain constants independent of k in inequalities (4) and (5). For uk the inequality
holds, where ck denotes a constant which is constructed’below by induction. For k = 0 we have co from (7). Let ck satisfying the condition ck > c, for k > s have already been constructed. We obtain c~+~ by a recurrence formula. We have
IU-&+il,,a, < b-&+,I I,Aj +I%+I-&+III,Ai
*
(8)
As vk+l we take a completion which differs from LQ+~only by the values of the parameters at the pointsAl,. . . , A4. As the parameters for uk+l at these points we take the values of the function u. Then
In the squares C21and Q4 on the right sides there cannot be more than k additional points. Let there be s points in al and t points in fl4. Then s + t = k and
V. N. Orlov
118
IU-~~+,I:,,~(C,k2;2’p22’p-2JIUIILI Writing q~2z’p--2, ‘/6q
we
(*
P f
))“.
(10)
obtain
(11) Adding (10) and (11) and extracting the root, we obtain
(12) Here we have used the monotonic increase of ck and the inequality ll~llrY,2~n,~+ll~ll~~~~,~~ll~ll~~~~i~. For 522 and 04 we have the estimate (13) If it can be shown that the inequality (14) holds, then, substituting (9), (12), (13) and (14) in (8), we have 121--u” kfll i,A iG (qc~+c)~2~2'p11UllL~(Ai).
(15)
The estimate (15) is independent of the position of the additions points on the right side. In (15) From this it is obvious that we can obviously take c > cu. Therefore we can put ck+,=qci-kc. k
%+-I=:
(c,--c,) y.
q’-tc,
G
const..
We will prove the estimate (14). Let I be the least distance between the additional points. We subdi~de the square 4 into identical squares with side equal to I. Then the side Ar is subdi~ded into g = hi/Z parts. The function Fk+ 1 can be represented on Ar as a linear combination of basis functions (pS.Then s runs through all the vertices of the squares with side 1 into which the square 4 was subdivided:
where pS is the value of the function Fk+l at the points. In our case these values are linear combinations of the actual parameters belonging to Ai:
pa=
c
t+1.2,3.4
uth: +
c
I) u&‘,
t-1,2,3,4
c
G=l,
t
tit is the value of the function u at the point t. The contribution of the points A,, . . . , A4 is extracted specially.
Piecewise-polynomialapproximation
For ;k+l Al,...,Aqby
a similar representation
119
is obtained by replacing the parameters at the points
the fictitious:
i&+i =
c
v.cp.9
vs =
where utl are the values of the function
c
Ulh,’ +
c
Ut’ht’,
1=,,2,3,4
i#t,2,3,4
I
G’ + 1at the points A 1, . . . , A4. Therefore,
We write
Then
Using (6) we obtain
h+i-&+, I,,, ,Qch2,-2" I
II”IIL:,‘i,~
II~,I,,A~. I-1,2,3,4
t=l, 2, 3, 4. We consider +I, since the discussion is We prove that M~,AI- (21’2, similar for the other rjr. In view of the method of constructing the completion, the following estimates hold on any of the squares with side 1:
since the sum contains g2 equal terms. The estimate (4) is proved. We obtain the estimate (5) similarly by considering
that
Lemma 2 is proved.
772eorem1 Let UEww,’ (Q”), lc~G2, FE& (Q’), I V,U IGF almost everywhere. The square ~2 is subdivided into I % 1G/V squares of the Birman-Solomyak algorithms for F and ‘a= (I- l/p) p; * G is the R-polylinear completion of the mesh function u on this subdivision. Then the following estimates hold:
1u-fi I,, cp~CN-"'IIFlltp~Q~~,
(16)
v.Iv.
120
orlov
(17) Roof: We have
=CN’-(2’p)(Q+‘) IIFIl;p(0’). Here we have used the estimates (2) and (4). Calculating the exponent of N and extracting the root, we obtain (16). The estimate (17) is obtained similarly by using the estimates (2) and (5). Theorem 1 is proved. Remarks. 1. In the partition of Q2 into identical squares the rate in IV21 would equal h2--2/P,l\r--1*-UP).
2. The result mentioned in the introduction is obtained in Theorem 1 for F= I V2uI.
In order that the length of the side of the greatest square of the 2R-subdivision 4 should be of order I@ (this may be necessary in the automatic construction of a locally condensed mesh), we can take for example F= IV24+1. We formulate a similar theorem for subdivisions of the cube em in Rm . The changes of all the definitions given in section 1 are obvious. In the Bi~an-Solomy~ algorithm condition (3) is replaced by bi>2-“” max{ bJ and the cubes for which it is satisfied are subdivided into 2” equal cubes. This theorem is more general even for m = 2, since the errors of approximation are measured in L,’ (Q”) and L,(Q”) respectively. i%eorem 2
I_et UEE’~~(Q~), m>2, 2p>m, rapI q&p, i-m/p + m/r>O, F=L,(Q”), cubes by the almost everywhere. Let the cube p be subdivided into 1%:1
IIu-~ll~~~~~~~~~N-*‘mllFII~~~~~, lbEII
~p~p~~~~~~2’mll~ll~p~4~~t
lb-u” II~~~q~~~~~-“m-“~II~lI~p~~~~,
r-Cm, ram.
l/m-l/r+l/q>O,
The scheme of reasoning is quite similar for the subdivision of the domain into triangles also.
Piecewise-polynomialapproximation
121
We call the subdivision of a triangle into 4 equal triangles a 2R-subdivision. The union of triangles obtained after a series of 2R-subdivisions is also called a 2R-subdivision. Lemma 1 will hold for 2Rsubdiviions of a triangle also. This is verified by repetition of the proof of [ 1] . On a triangle 4 with additional points on the sides it is easy, as above, to construct a completion based on linear functions. We call it the R-linear completion of the mesh function U. The following theorem holds. Theorem 3 Let 52be an arbitrary 2R-subdivision consisting of a finite number of triangles, UEW,~ (Q) , l
The authors sincerely thanks L. A. Oganesyan for help in formulating the problem and for his continued interest. Translated by J. Berry.
REFERENCES 1.
BIRMAN, M. Sh. and SOLOMYAK, M. Z. Piecewise polynomial approximations of functions of the classes Wp%Matem sb. 73 (115), 3, 331-355, 1967.
2.
CAREY, G. F. A mesh refinement scheme for finite element computations. Comput. Meth. Appl. Mech. Engng. 7,1,93-105, 1976.
3.
CAVENDISH, J. C. Local mesh refinement using rectangular blended finite elements. J. Comput. Phys., 19,2,211-228,197s.
4.
OCANESYAN, L. A., RIVKIND, V. Ya. and RUKHOVETS, L. A. Differential equations and their application. Proceedings of a seminar. No. 5. (Dlfferentsial’nye uravneniya i ikh primenenie. Tr. seminara. Vyp. 5.) In-t fll. i matem. Akad. Nauk Lit SSR, Viinyus, 1973.
ViN. Orlov
In view of the form of this system (see [S] ), we arrive at the familiar result (see [6,3] ) that, in the absence of integrals, Au is finite-dimensional. Translated by D. E. Brown REFERENCES 1.
PAVLOVSKII, Yu. N., Group properties of controlled dynamic systems, and phase-constraint structures, I., Group properties of controlled dynamic systems, Zh. vj@W. Mat. mat. Fiz., 14, No. 4, 862-872, 1914.
2.
PAVLOVSKII, Yu. N., Group properties of controlled dynamic system, and phase constraint structures, II, Phase constraint structures,Zh. vjkhisl. Mat. mat. Fiz., 14, No. 5, 1093-1103,1974.
3.
PAVLOVSKII, Yu. N., Group properties, aggregation, and constmint structures of controlled dynamic systems, Diss. dokt. fm.-matem. n., VTs Akad. Nauk SSSR. Moscow, 1975.
4.
OVSYANNIKOV, L. V., Lectures on the theory of group properties of differential equations (Lektsii PO teorii gruppovykh svoistv differentsial’nykh uravnenii), Izd-vo NGU, Novosibirsk, 1966.
5.
EISENHART, L. P., Continuous groups of transformations, Dover, 1933.
6.
YAKOVENKO, G. N., L-systemsand their investigation(on a generalization of the class of linear dynamic systems), Diss. kand. tIz.-matem. n., MFTI, Moscow, 1973.
U.S.S.R. Comput. MathsMath. Phys. Vol. 18, pp. 114-121 0 Pergamon Press Ltd. 1979. Printed in Great Britain.
0041-5553/78/0801-0114$07.50/O
THE PIECEWISE-POLYNOMIAL APPROXIMATION IN W21 OF FUNCTIONS FROM “5 , i c p 6 2 * V. N. ORLOV
Leningrad (Received 3 May 1977)
defined on the unit square, a piecewise-polynomial FOR A function u from w,,s, i-zp<~, function c from IV21 is constructed depending on O(N) parameters. The points at which the parameter values are defined may be locally condensed. As strategy of condensation the Birman-Solomyak algorithm [I] may be used. For this condensation the estimate (1) is obtained.
Introduction In the computer solution by the mesh method of elliptic type differential equations, it becomes necessary to condense the mesh at those places of the domain where the solution of the equation varies “sharply”. In the variational difference method this leads to a fine triangulation at places of “sharp” variation of the solution. The passage from the fme to the usual triangulation causes technical difficulties. In [2,3] two methods were presented for constructing an approximating function from IV21 based on a locally condensed mesh and the results of numerical experiments were given; however in these papers estimates of the approximation are not given. In [ 1] estimates were obtained for the approximation of the initial function by a piecewise-polynomial function constructed by partition of the domain into locally condensed squares, but the approximating function does not belong to IV21, which is important for the variational-difference method. *Zh. vjkhisl. Mat. mat. Fiz., 18,4, 935-942,1978.
Piecewise-polynomialapproximation
115
In this paper we construct a piecewise-polynomial function g from W-J1, depending on O(N) parameters, whose values are defined at the vertices of the squares into which the initial unit square is subdivided. By Theorem 2.1 of [ 1 ] we obtain estimates of the approximation error of the form Ifu-a IIrp,‘scN-“l/ ull wp’.
l-C&2.
(1)
The method of constructing the function r differs from the methods presented in [2] and [3]. It is known that the error of the variation-difference method of solution in W2’ is estimated by the approximation error, consequently, the estimate given confirms the existence of a “best” mesh l
1. Generalizations, construction of the Binnan-Sofomyak subdivision, construction of the completions
Let p
be a unit cube in Rm with centre at the origin of coordinate
C is a constant whose value is a matter of indifference for the discussion. We consider the square Q2 on a plane. We subdivide it into 4 equal squares. Each of the squares obtained can again be subdivided into 4 equal squares. Any of the resulting squares can again be subdivided into 4 equal squares, and so on. We call the union of all the squares obtained at any stage of this algorit~ the 2R-subdivision of the original square. The size of the squares occurring in the ~-sub~~sion may differ, since at each stage it is not obligatory to subdivide ail the squares into 4 parts. We consider an arbitrary 2R-subdivision of the square Q2. We denote by 1S 1 the number of squares in the subdivision, by Ai an arbitrary square of the 2R-subdivision, and by I Ai I the measure of this square. The vertices of the squares belonging to the 2R-subdivision, we call real points. The real points situated on sides of the square L$ and not vertices of it, we call supplemental points for A, We define the vector of the values of the continuous function u at the real points as the mesh function and we denote it by U. Let the function FsL, (Q') , lCpS2 be defined on Q2. A fundamental lemma holds which is a particular case of Theorem 1 of [ 1].
V. N.
116
Orlov
Lemma 1
Let FEL, (Q”), pa 1,00. Then for any natural N a 2R-subdivision of Q2 into squares can be found such that 1B 1
max{lAil”liFll ~~(*iI}~C((I)N-(.+~)IIFII~~~~~~.
(2)
1
We call this subdivision the Birman-Solomyak subdivision. Following [ 11, we omit the algorithm for obtaining this special 2R-subdivision of the square. We subdivide Q2 into 4 equal squares. For each of them we find the numbers P
bt=IAtl”llFllt,~a,,. We subdivide into 4 equal squares those 4 for which bP2-2”max{b,).
(3)
i
We obtain a new X-partition of the square Q2. For each of the squares of this subdivision we find the numbers bj and repeat the process (3). If F # 0 almost everywhere, then in the new subdivision the number of squares is greater than in the old one. We stop the subdivision process when 1E 1 exceeds N. As shown in [l] , the estimate (2) holds for the subdivision obtained. We pass to the construction of the completion Z of the function UEW,~ (Q") , lCpG2. The proposed construction is constructed on any 2R-subdivision of the square. We will use it on the Birman-Solomyak subdivision. At each real point we defme a number which we call a real parameter. In our case we take u as the real parameters. We consider a typical case: the square 4 with side hi, on whose right side k supplementary points lie, but no supplementary points lie on the other sides. For convenience of notation we place its centre at the origin of cordinates of the system. We denote where 52, lies in the the squares obtained on subdividing it into 4 squares by f&, . . . , f&, Square I, 52, in the square II and so on. Some of the vertices Qi, , . . , Qc we mark specially: Ai(O, h/2),
A2(4*/2,
O),
As(0, -h/2),
A‘@,
0).
If k = 0, we construct the polylinear completion (see [4], p. 92) with respect to 4 real parameters at the vertices of the A,. If k = 1, there exists one supplementary point at the middle of the right side, at which a real parameter is defined. We introduce 4 fictitious parameters at the points A,,..;, A4. We set the values of the parameters at these points equal to the value of the polylinear function constructed for k = 0. On each of the squares Sz,, . . . , !& for the known parameters at the vertices we construct the polylinear completion. For k = 2 (depending on where the additional point lies), we again subdivide either 521 or a;24into 4 parts and repeat on it a construction similar to the case k = 1. Let the completion for the k-1-th additional point, k > 2, have been constructed. The completion for k additional points is obtained by reconstructing the function only on that square whose right side is bisected on the addition of the k-th point. The reconstruction itself is just the same as for the case k = 1. If there are additional points on other sides of the square Ar, the completion is constructed similarly: instead of the fictitious parameters at the points A 1, AZ, A3 the corresponding actual parameters must be used, and so on. Therefore, on each square 4 at the 2R-th subdivision a piecewise-polynomial function has been constructed, which is identical with the function u at the actual points and is continuous on Q2. Thereby the function CEW~’ (Q’) has been constructed on Q2.
Piecewise-polynomi
117
approximation
The function i? constructed by the method described above on Q2 for the actual parameters, is called the R-polylinear completion of the mesh function U. We note that when additional points are present u is a piecewise-polylinear (and not a polylinear) function on the square.
2. Approximation theorems In estimating the rate of convergence of u” to uin the Birman-Solomyak subdivision an important part is played by the following lemma. Lemma 2
the square Q2 has been subdivided into squares by the Let UEW,~(Q*), l
IU41,,Ai aypllnllL;(ai
(4)
),
Proof: We again consider the typical case which we considered when constructing the completion in section 1. Let uk be the completion on Ar for k additional points on the right side. For k = 0 estimates hold which are proved by the standard method (see, for example, [l]):
(6)
The difficulty is to obtain constants independent of k in inequalities (4) and (5). For uk the inequality
holds, where ck denotes a constant which is constructed’below by induction. For k = 0 we have co from (7). Let ck satisfying the condition ck > c, for k > s have already been constructed. We obtain c~+~ by a recurrence formula. We have
IU-&+il,,a, < b-&+,I I,Aj +I%+I-&+III,Ai
*
(8)
As vk+l we take a completion which differs from LQ+~only by the values of the parameters at the pointsAl,. . . , A4. As the parameters for uk+l at these points we take the values of the function u. Then
In the squares C21and Q4 on the right sides there cannot be more than k additional points. Let there be s points in al and t points in fl4. Then s + t = k and
V. N. Orlov
118
IU-~~+,I:,,~(C,k2;2’p22’p-2JIUIILI Writing q~2z’p--2, ‘/6q
we
(*
P f
))“.
(10)
obtain
(11) Adding (10) and (11) and extracting the root, we obtain
(12) Here we have used the monotonic increase of ck and the inequality ll~llrY,2~n,~+ll~ll~~~~,~~ll~ll~~~~i~. For 522 and 04 we have the estimate (13) If it can be shown that the inequality (14) holds, then, substituting (9), (12), (13) and (14) in (8), we have 121--u” kfll i,A iG (qc~+c)~2~2'p11UllL~(Ai).
(15)
The estimate (15) is independent of the position of the additions points on the right side. In (15) From this it is obvious that we can obviously take c > cu. Therefore we can put ck+,=qci-kc. k
%+-I=:
(c,--c,) y.
q’-tc,
G
const..
We will prove the estimate (14). Let I be the least distance between the additional points. We subdi~de the square 4 into identical squares with side equal to I. Then the side Ar is subdi~ded into g = hi/Z parts. The function Fk+ 1 can be represented on Ar as a linear combination of basis functions (pS.Then s runs through all the vertices of the squares with side 1 into which the square 4 was subdivided:
where pS is the value of the function Fk+l at the points. In our case these values are linear combinations of the actual parameters belonging to Ai:
pa=
c
t+1.2,3.4
uth: +
c
I) u&‘,
t-1,2,3,4
c
G=l,
t
tit is the value of the function u at the point t. The contribution of the points A,, . . . , A4 is extracted specially.
Piecewise-polynomialapproximation
For ;k+l Al,...,Aqby
a similar representation
119
is obtained by replacing the parameters at the points
the fictitious:
i&+i =
c
v.cp.9
vs =
where utl are the values of the function
c
Ulh,’ +
c
Ut’ht’,
1=,,2,3,4
i#t,2,3,4
I
G’ + 1at the points A 1, . . . , A4. Therefore,
We write
Then
Using (6) we obtain
h+i-&+, I,,, ,Qch2,-2" I
II”IIL:,‘i,~
II~,I,,A~. I-1,2,3,4
t=l, 2, 3, 4. We consider +I, since the discussion is We prove that M~,AI- (21’2, similar for the other rjr. In view of the method of constructing the completion, the following estimates hold on any of the squares with side 1:
since the sum contains g2 equal terms. The estimate (4) is proved. We obtain the estimate (5) similarly by considering
that
Lemma 2 is proved.
772eorem1 Let UEww,’ (Q”), lc~G2, FE& (Q’), I V,U IGF almost everywhere. The square ~2 is subdivided into I % 1G/V squares of the Birman-Solomyak algorithms for F and ‘a= (I- l/p) p; * G is the R-polylinear completion of the mesh function u on this subdivision. Then the following estimates hold:
1u-fi I,, cp~CN-"'IIFlltp~Q~~,
(16)
v.Iv.
120
orlov
(17) Roof: We have
=CN’-(2’p)(Q+‘) IIFIl;p(0’). Here we have used the estimates (2) and (4). Calculating the exponent of N and extracting the root, we obtain (16). The estimate (17) is obtained similarly by using the estimates (2) and (5). Theorem 1 is proved. Remarks. 1. In the partition of Q2 into identical squares the rate in IV21 would equal h2--2/P,l\r--1*-UP).
2. The result mentioned in the introduction is obtained in Theorem 1 for F= I V2uI.
In order that the length of the side of the greatest square of the 2R-subdivision 4 should be of order I@ (this may be necessary in the automatic construction of a locally condensed mesh), we can take for example F= IV24+1. We formulate a similar theorem for subdivisions of the cube em in Rm . The changes of all the definitions given in section 1 are obvious. In the Bi~an-Solomy~ algorithm condition (3) is replaced by bi>2-“” max{ bJ and the cubes for which it is satisfied are subdivided into 2” equal cubes. This theorem is more general even for m = 2, since the errors of approximation are measured in L,’ (Q”) and L,(Q”) respectively. i%eorem 2
I_et UEE’~~(Q~), m>2, 2p>m, rapI q&p, i-m/p + m/r>O, F=L,(Q”), cubes by the almost everywhere. Let the cube p be subdivided into 1%:1
IIu-~ll~~~~~~~~~N-*‘mllFII~~~~~, lbEII
~p~p~~~~~~2’mll~ll~p~4~~t
lb-u” II~~~q~~~~~-“m-“~II~lI~p~~~~,
r-Cm, ram.
l/m-l/r+l/q>O,
The scheme of reasoning is quite similar for the subdivision of the domain into triangles also.
Piecewise-polynomialapproximation
121
We call the subdivision of a triangle into 4 equal triangles a 2R-subdivision. The union of triangles obtained after a series of 2R-subdivisions is also called a 2R-subdivision. Lemma 1 will hold for 2Rsubdiviions of a triangle also. This is verified by repetition of the proof of [ 1] . On a triangle 4 with additional points on the sides it is easy, as above, to construct a completion based on linear functions. We call it the R-linear completion of the mesh function U. The following theorem holds. Theorem 3 Let 52be an arbitrary 2R-subdivision consisting of a finite number of triangles, UEW,~ (Q) , l
The authors sincerely thanks L. A. Oganesyan for help in formulating the problem and for his continued interest. Translated by J. Berry.
REFERENCES 1.
BIRMAN, M. Sh. and SOLOMYAK, M. Z. Piecewise polynomial approximations of functions of the classes Wp%Matem sb. 73 (115), 3, 331-355, 1967.
2.
CAREY, G. F. A mesh refinement scheme for finite element computations. Comput. Meth. Appl. Mech. Engng. 7,1,93-105, 1976.
3.
CAVENDISH, J. C. Local mesh refinement using rectangular blended finite elements. J. Comput. Phys., 19,2,211-228,197s.
4.
OCANESYAN, L. A., RIVKIND, V. Ya. and RUKHOVETS, L. A. Differential equations and their application. Proceedings of a seminar. No. 5. (Dlfferentsial’nye uravneniya i ikh primenenie. Tr. seminara. Vyp. 5.) In-t fll. i matem. Akad. Nauk Lit SSR, Viinyus, 1973.