Complexity of indefinite elliptic problems

Author:
Arthur G. Werschulz

Journal:
Math. Comp. **46** (1986), 457-477

MSC:
Primary 65N30; Secondary 35J40, 65N15, 68Q25

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829619-1

MathSciNet review:
829619

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the approximate solution of a linear regularly-elliptic 2*m*th-order boundary-value problem , with for . Suppose that the problem is indefinite, i.e., the variational form of the problem involves a weaklycoercive bilinear form. Of particular interest is the quality of the finite element method (FEM) of degree *k* using *n* inner products of *f*. The error of the approximation is measured in the Sobolev *l*-norm ; we assume that . We assume that an *a priori* bound is known for either the Sobolev *r*-norm or for the Sobolev *r*-seminorm of *f*. We first consider the normed case. We find that the FEM has minimal error if and only if . Regardless of the values of *k* and *r*, there exists a linear combination (called the spline algorithm) of the inner products used by the FEM which *does* have minimal error. For the seminormed case, we give a very restrictive condition which is necessary and sufficient for the error of the FEM to have a bound which is independent of *f*. When this condition holds, we find that the FEM has minimal error if and only if . However, we once again find that the spline algorithm (using the same inner products as does the FEM) has minimal error, no matter what values *k* and *r* have and regardless of whether the FEM has uniformly bounded error. We also show that the inner products used by the FEM is the best set of linear functionals to use.

**[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.-Toronto-London, 1965. MR**0178246****[2]**P. M. Anselone and P. J. Laurent,*A general method for the construction of interpolating or smoothing spline-functions*, Numer. Math.**12**(1968), 66–82. MR**0249904**, https://doi.org/10.1007/BF02170998**[3]**A. K. Aziz (ed.),*The mathematical foundations of the finite element method with applications to partial differential equations*, Academic Press, New York-London, 1972. MR**0347104****[4]**Paul L. Butzer and Hubert Berens,*Semi-groups of operators and approximation*, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR**0230022****[5]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[6]**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**, https://doi.org/10.1016/0045-7825(72)90006-0**[7]**Joseph W. Jerome,*Asymptotic estimates of the \cal𝐿₂𝑛-width*, J. Math. Anal. Appl.**22**(1968), 449–464. MR**0228905**, https://doi.org/10.1016/0022-247X(68)90188-1**[8]**J. 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****[9]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832****[10]**G. Strang & G. Fix, "A Fourier analysis of the finite element variational method,"*Constructive Aspects of Functional Analysis, Part II*, C.I.M.E., Rome, 1973.**[11]**Joe 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****[12]**J. M. Trojan, "Asymptotic model for linear problems," (In preparation).**[13]**L. B. Wahlbin, "Quasi-optimality of the projection into finite element spaces,"*Lectures on the Numerical Solution of Partial Differential Equations*:*Proceedings of the Special Year in Numerical Analysis*(I. Babuška, T.-P. Liu, and J. Osborn, eds.), Dept. of Math., Univ. of Maryland, College Park, MD, Lecture Notes #20, 1981.**[14]**Arthur G. Werschulz,*Does increased regularity lower complexity?*, Math. Comp.**42**(1984), no. 165, 69–93. MR**725985**, https://doi.org/10.1090/S0025-5718-1984-0725985-4**[15]**A. G. Werschulz, "Finite element methods are not always optimal," (Submitted for publication).**[16]**Alexander Ženíšek,*Hermite interpolation on simplexes in the finite element method*, Proceedings of Equadiff III (Third Czechoslovak Conf. Differential Equations and their Appl., Brno, 1972) Purkyně Univ., Brno, 1973, pp. 271–277. Folia Fac. Sci. Natur. Univ. Purkynianae Brunensis, Ser. Monograph., Tomus 1. MR**0373224**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
35J40,
65N15,
68Q25

Retrieve articles in all journals with MSC: 65N30, 35J40, 65N15, 68Q25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829619-1

Keywords:
Indefinite elliptic problems,
variational methods,
finite element methods,
optimal algorithms,
computational complexity

Article copyright:
© Copyright 1986
American Mathematical Society