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Estimate of the Constant in Two Strengthened C.B.S. Inequalities for F.E.M. Systems of 2D Elasticity: Application to Multilevel Methods and a Posteriori Error Estimators
Author(s) -
Achchab B.,
Maître J. F.
Publication year - 1996
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199603/04)3:2<147::aid-nla75>3.0.co;2-s
Subject(s) - mathematics , estimator , rate of convergence , poisson distribution , finite element method , a priori and a posteriori , polygon mesh , quadratic equation , constant (computer programming) , linear elasticity , elasticity (physics) , convergence (economics) , mathematical analysis , geometry , statistics , channel (broadcasting) , physics , philosophy , electrical engineering , epistemology , computer science , thermodynamics , programming language , economics , economic growth , materials science , engineering , composite material
The constant γ in the strengthened Cauchy‐Buniakowski‐Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems. We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ 2 ≤ 3/4$ uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2