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**A uniformly accurate finite element method for the Reissner-Mindlin plate.**
*(English)*
Zbl 0696.73040

A simple finite element method for the Reissner-Mindlin plate model is presented in the paper. The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging. It is proved that the approximate values of the displacement and rotation, together with their first derivatives, all converge at an optimal order uniformly with respect to the thickness.

The Reissner-Mindlin model is simple and its discretization is not straight-forwarded. One of the problems to be solved is to avoid the locking problem, it is the spurious effect of the shear forces. It can not be overcome as simply as in the case of beams, for example by the reduced integration or modification of the variational formulation. Another possibility proposed in the paper is to project the discrete transverse shear strain into a lower-order finite element space.

A discrete analogue of the Helmholtz theorem, giving the decomposition of a vector field into an irrotational and solenoidal field has been proved. It appears to be a new result which can be useful in other analysis of finite element methods.

The paper is strictly mathematical. It can be read by those working in the field of development of mathematical base of discrete solution methods, error estimation, convergence etc.

The Reissner-Mindlin model is simple and its discretization is not straight-forwarded. One of the problems to be solved is to avoid the locking problem, it is the spurious effect of the shear forces. It can not be overcome as simply as in the case of beams, for example by the reduced integration or modification of the variational formulation. Another possibility proposed in the paper is to project the discrete transverse shear strain into a lower-order finite element space.

A discrete analogue of the Helmholtz theorem, giving the decomposition of a vector field into an irrotational and solenoidal field has been proved. It appears to be a new result which can be useful in other analysis of finite element methods.

The paper is strictly mathematical. It can be read by those working in the field of development of mathematical base of discrete solution methods, error estimation, convergence etc.

Reviewer: Cz.I.Bajer

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74K20 | Plates |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |