| Title:
|
Two simple derivations of universal bounds for the C.B.S. inequality constant (English) |
| Author:
|
Axelsson, Owe |
| Author:
|
Blaheta, Radim |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
57-72 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations. (English) |
| Keyword:
|
finite element method |
| Keyword:
|
$h$- and $p$-refinement |
| Keyword:
|
strengthened Cauchy-Bunyakowski-Schwarz inequality |
| MSC:
|
65F10 |
| MSC:
|
65N22 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1099.65103 |
| idMR:
|
MR2032148 |
| DOI:
|
10.1023/B:APOM.0000024520.06175.8b |
| . |
| Date available:
|
2009-09-22T18:16:45Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134558 |
| . |
| Reference:
|
[1] B. Achchab, O. Axelsson, L. Laayouni and A. Souissi: Strengthened Cauchy-Bunyakowski-Schwarz inequality for a three dimensional elasticity system.Numer. Linear Algebra Appl. 8 (2001), 191–205. MR 1817796, 10.1002/1099-1506(200104/05)8:3<191::AID-NLA229>3.0.CO;2-7 |
| Reference:
|
[2] B. Achchab, J. Maitre: Estimate of the constant in two strengthened C.B.S. inequalities for F.E.M. systems of 2D elasticity. Applications to multilevel methods and a posteriori error estimators.Numer. Linear Algebra Appl. 3 (1996), 147–159. MR 1379558, 10.1002/(SICI)1099-1506(199603/04)3:2<147::AID-NLA75>3.0.CO;2-S |
| Reference:
|
[3] O. Axelsson: On multigrid methods of the two-level type.In: Multigrid Methods, Lecture Notes in Math. 960, W. Hackbusch, U. Trottenberg (eds.), Springer-Verlag, Berlin, 1982, pp. 352–367. Zbl 0505.65040, MR 0685778 |
| Reference:
|
[4] O. Axelsson: Stabilization of algebraic multilevel iteration methods; additive methods.Numer. Algorithms 21 (1999), 23–47. Zbl 0937.65135, MR 1725714, 10.1023/A:1019136808500 |
| Reference:
|
[5] O. Axelsson, V. A. Barker: Finite Element Solution of Boundary Value Problems. Theory and Computation.Academic Press, Orlando, 1984, reprinted as SIAM Classics in Applied Mathematics 35, SIAM, Philadelphia, 2001. MR 0758437 |
| Reference:
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[6] O. Axelsson, I. Gustafsson: Preconditioning and two-level multigrid methods of arbitrary degree of approximation [Report 8120 (July 1981), Department of Mathematics, University of Nijmegen, The Netherlands].Math. Comp. 40 (1983), 219–242. MR 0679442 |
| Reference:
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[7] O. Axelsson, A. Padiy: On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems.SIAM J. Sci. Comput. 20 (1999), 1807–1830. MR 1694685, 10.1137/S1064827597320058 |
| Reference:
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[8] R. Blaheta: Adaptive composite grid methods for problems of plasticity.Math. Comput. Simulation 50 (1999), 123–134. MR 1717646, 10.1016/S0378-4754(99)00064-6 |
| Reference:
|
[9] R. Blaheta: Nested tetrahedral grids and strengthened C.B.S. inequality.Numer. Linear Algebra Appl. 10 (2003), 619–637. Zbl 1071.65164, MR 2030627, 10.1002/nla.340 |
| Reference:
|
[10] M. Jung, J. F. Maitre: Some remarks on the constant in the strengthened C.B.S. inequality: estimate for hierarchic finite element discretization of elasticity problems.Numer. Methods Partial Differential Equations 15 (1999), 469–488. MR 1695748, 10.1002/(SICI)1098-2426(199907)15:4<469::AID-NUM4>3.0.CO;2-B |
| Reference:
|
[11] J. F. Maitre, F. Musy: The contraction number of a class of two-level methods, an exact evaluation for some finite element subspaces and model problems.In: Multigrid Methods, Lecture Notes in Math. 960, W. Hackbusch, U. Trottenberg (eds.), Springer-Verlag, Berlin, 1982, pp. 535–544. MR 0685787 |
| Reference:
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[12] J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction.Elsevier, Amsterdam, 1981. MR 0600655 |
| Reference:
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[13] S. D. Margenov: Upper bound on the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations.Numer. Linear Algebra Appl. 1 (1994), 65–74. MR 1269944, 10.1002/nla.1680010107 |
| . |
