What is the complexity of related elliptic, parabolic, and hyperbolic problems?
Arthur G. Werschulz
Math. Comp. 47 (1986), 461-472
Primary 65P05; Secondary 68Q15
Full-text PDF Free Access
Similar Articles |
Abstract: Traub and Woźniakowski have dealt with the complexity of some simple partial differential equations. They chose three model problems and showed that the parabolic problem considered had significantly lower complexity than the elliptic problem, which in turn had significantly lower complexity than the hyperbolic problem considered. They asked whether this is true in general. We show that this is not the case by proving that if L is a reasonably well-behaved elliptic operator, then the steady-state heat equation , the heat equation , and the wave equation all have roughly the same worst-case complexity for f in the unit ball of a certain Sobolev space of smoothness r.
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.-Toronto-London, 1965. MR 0178246
Babuška and R.
B. Kellogg, Nonuniform error estimates for the finite element
method, SIAM J. Numer. Anal. 12 (1975), no. 6,
868–875. MR 0411201
M. Berezans′kiĭ, Expansions in eigenfunctions of
selfadjoint operators, Translated from the Russian by R. Bolstein, J.
M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical
Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.
0222718 (36 #5768)
Fairweather, Finite element Galerkin methods for differential
equations, Marcel Dekker, Inc., New York-Basel, 1978. Lecture Notes in
Pure and Applied Mathematics, Vol. 34. MR 0495013
B. Z. Kacewicz & G. Wasilkowski, "How powerful is continuous nonlinear information for linear problems?. (In preparation.)
G. Kreĭn and Ju.
I. Petunin, Scales of Banach spaces, Uspehi Mat. Nauk
21 (1966), no. 2 (128), 89–168 (Russian). MR 0193499
D. E. Knuth, "Big omicron and big omega and big theta," SIGACT News, April, 1976, pp. 18-24.
T. Oden and J.
N. Reddy, An introduction to the mathematical theory of finite
elements, Wiley-Interscience [John Wiley & Sons], New
York-London-Sydney, 1976. Pure and Applied Mathematics. MR 0461950
Fred Traub and H.
Woźniakowsi, A general theory of optimal algorithms,
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
York-London, 1980. ACM Monograph Series. MR 584446
G. Werschulz, Finite element methods are not always optimal,
Adv. in Appl. Math. 8 (1987), no. 3, 354–375.
Fanett Wheeler, A priori 𝐿₂ error estimates for
Galerkin approximations to parabolic partial differential equations,
SIAM J. Numer. Anal. 10 (1973), 723–759. MR 0351124
- S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, N.J., 1965. MR 0178246 (31:2504)
- I. Babuška & R. B. Kellogg, "Nonuniform error estimates for the finite element method," SIAM J. Numer. Anal., v. 12, 1975, pp. 868-875. MR 0411201 (53:14939)
- Ju. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Math. Monographs, vol. 17, Amer. Math. Soc., Providence, R.I., 1968. MR 0222718 (36:5768)
- G. Fairweather, Finite Element Galerkin Methods for Differential Equations, Lecture Notes in Pure and Appl. Math., vol. 34, Marcel Dekker, New York, 1978. MR 0495013 (58:13781)
- B. Z. Kacewicz & G. Wasilkowski, "How powerful is continuous nonlinear information for linear problems?. (In preparation.)
- S. G. Krein and Y. I. Petunin, "Scales of Banach spaces," Russian Math. Surveys, v. 21, 1966, pp. 85-160. MR 0193499 (33:1719)
- D. E. Knuth, "Big omicron and big omega and big theta," SIGACT News, April, 1976, pp. 18-24.
- J. T. Oden & J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, Wiley-Interscience, New York, 1976. MR 0461950 (57:1932)
- J. F. Traub & H. Woźniakowski, A General Theory of Optimal Algorithms, Academic Press, New York, 1980. MR 584446 (84m:68041)
- A. G. Werschulz, Finite Element Methods are Not Always Optimal, Mathematics Research Report 82-11, University of Maryland, Baltimore County, June, 1982. To appear in Adv. in Appl. Math. MR 898711 (89c:65127)
- M. F. Wheeler, "A priori -error estimates for Galerkin approximations to parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 723-759. MR 0351124 (50:3613)
Retrieve articles in Mathematics of Computation
Retrieve articles in all journals
Optimal solution of partial differential equations,
© Copyright 1986
American Mathematical Society