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The nonconforming virtual element method for elasticity problems. (English) Zbl 1416.74092
Summary: We present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation. This method is uniformly convergent for the nearly incompressible case. The optimal convergence in the \(H^1\) norm is proved under some regularity assumptions. We mention that, for the lowest-order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by S. C. Brenner and L.-Y. Sung [Math. Comput. 59, No. 200, 321–338 (1992; Zbl 0766.73060)] up to an approximation of the right-hand side. Besides, we also present the nonconforming virtual element method for the pure traction problem, which is proved to be uniformly convergent with respect to the Lamé constant. Finally, we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
PolyMesher
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References:
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