Optimal error properties of finite element methods for second order elliptic Dirichlet problems
Author:
Arthur G. Werschulz
Journal:
Math. Comp. 38 (1982), 401413
MSC:
Primary 65N30
MathSciNet review:
645658
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We use the informational approach of Traub and Woźniakowski [9] to study the variational form of the second order elliptic Dirichlet problem on . For , where , a quasiuniform finite element method using n linear functionals has norm error . We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if , where , then there is a finite element method whose norm error is for , and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals are approximated by using n evaluations of f, then there is a finite element method with quadrature with norm error where . We show that when , there is no method using n function evaluations whose error is better than ; thus for , the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f.
 [1]
Shmuel
Agmon, Lectures on elliptic boundary value problems, Prepared
for publication by B. Frank Jones, Jr. with the assistance of George W.
Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co.,
Inc., Princeton, N.J.TorontoLondon, 1965. MR 0178246
(31 #2504)
 [2]
Ivo
Babuška and A.
K. Aziz, Survey lectures on the mathematical foundations of the
finite element method, The mathematical foundations of the finite
element method with applications to partial differential equations (Proc.
Sympos., Univ. Maryland, Baltimore, Md., 1972), Academic Press, New York,
1972, pp. 1–359. With the collaboration of G. Fix and R. B.
Kellogg. MR
0421106 (54 #9111)
 [3]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., Amsterdam, 1978. Studies in Mathematics and
its Applications, Vol. 4. MR 0520174
(58 #25001)
 [4]
P.
G. Ciarlet and P.A.
Raviart, Interpolation theory over curved elements, with
applications to finite element methods, Comput. Methods Appl. Mech.
Engrg. 1 (1972), 217–249. MR 0375801
(51 #11991)
 [5]
Joseph
W. Jerome, Asymptotic estimates of the
\𝑐𝑎𝑙𝐿₂𝑛width, J. Math.
Anal. Appl. 22 (1968), 449–464. MR 0228905
(37 #4484)
 [6]
Joseph
W. Jerome, On 𝑛widths in Sobolev spaces and applications
to elliptic boundary value problems, J. Math. Anal. Appl.
29 (1970), 201–215. MR 0254484
(40 #7692)
 [7]
Martin
H. Schultz, Multivariate spline functions and elliptic
problems, Approximations with Special Emphasis on Spline Functions
(Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969), Academic Press,
New York, 1969, pp. 279–347. MR 0257560
(41 #2210)
 [8]
S.
L. Sobolev, On the order of convergence of cubature formulae,
Dokl. Akad. Nauk SSSR 162 (1965), 1005–1008
(Russian). MR
0179528 (31 #3776)
 [9]
Joe
Fred Traub and H.
Woźniakowsi, A general theory of optimal algorithms,
Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.
ACM Monograph Series. MR 584446
(84m:68041)
 [1]
 S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, N. J., 1965. MR 0178246 (31:2504)
 [2]
 I. Babuška & A. K. Aziz, "Survey lectures on the mathematical foundations of the finite element method," in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972. MR 0421106 (54:9111)
 [3]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [4]
 P. G. Ciarlet & P. A. Raviart, "Interpolation theory over curved elements," Comput. Methods Appl. Mech. Engrg., v. 1, 1972, pp. 217249. MR 0375801 (51:11991)
 [5]
 J. W. Jerome, "Asymptotic estimates of the nwidth," J. Math. Anal. Appl., v. 22, 1968, pp. 449464. MR 0228905 (37:4484)
 [6]
 J. W. Jerome, "On nwidths in Sobolev spaces and elliptic boundary value problems," J. Math. Anal. Appl., v. 29, 1970, pp. 201215. MR 0254484 (40:7692)
 [7]
 M. H. Schultz, "Multivariate spline functions and elliptic problems," in Approximation with Special Emphasis on Spline Functions (J. J. Schoenberg, ed.), Academic Press, New York, 1969. MR 0257560 (41:2210)
 [8]
 S. L. Sobolev, "On the order of convergence of cubature formulas," Dokl. Akad. Nauk SSSR, v. 162, 1965, pp. 10051008; English transl. in Soviet Math. Dokl., v. 6, 1965, pp. 808812. MR 0179528 (31:3776)
 [9]
 J. F. Traub & H. Woźniakowski, A General Theory of Optimal Algorithms, Academic Press, New York, 1980. MR 584446 (84m:68041)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N30
Retrieve articles in all journals
with MSC:
65N30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206456584
PII:
S 00255718(1982)06456584
Article copyright:
© Copyright 1982 American Mathematical Society
