Equivalence of finite element methods for problems in elasticity. (English) Zbl 0722.73068

The authors first show that there exists equivalence of various boundary value problems for the equations of elasticity, the stationary Stokes equation and the biharmonic equation. Then the question pointed in the paper is whether any finite element methods based on these formulations are also equivalent. It is shown how a modification of the Morley method for the biharmonic equation can be obtained from the standard continuous piecewise linear approximation of the elasticity equations in the case when the force \(f=\text{grad} \phi\) for some potential \(\phi\). The elimination procedures analogous to those used in the continuous case was used. The key idea was a discrete version of the orthogonal decomposition of some symmetric tensors. Later it is shown how another modified Morley method for biharmonic equation arises from the nonconforming piecewise linear approximation for Stokes equation. The error estimates derived for all discussed versions of the Morley method were compared. Finally a mixed formulation is given, which is equivalent in the incompressible limit to the nonconforming approximation of the Stokes problem.
The refered Morley method can be found neither in the text of the paper nor in references. That is why a proper background for considerations can be hardly found. The paper can be read by people working in the field of technical physics, in the range of theory rather than of practical use.


74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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