A quadrilateral nonconforming finite element for linear elasticity problem

Abstract

In this paper, a four-parameter quadrilateral nonconforming finite element with DSP (double set parameters) is presented. Then we discuss the quadrilateral nonconforming finite element approximation to the linear elastic equations with pure displacement boundary. The optimal convergence rate of the method is established in the broken H 1 energy and L 2 -norms, and in particular, the convergence is uniform with respect to the Lamé parameter λ. Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.

On the error bounds of nonconforming finite elements

“…Nonconforming finite elements have been received much attention from scientists and engineers in last thirty years, since they have successfully provided stable numerical solutions to many practical fluid and solid mechanics problems: see, for instance, [3,6,7,9,[12][13][14]18,21,32] for linear or nonlinear Navier-Stokes problems and [2,5,10,11,19,22,25,27,28,30,31] for elasticity related problems, and the references therein.…”

“…where Π h is some conforming finite element interpolation operator, see [25] for details. It is easy to see that the first term on the right-hand side of (1) can be bounded by the approximation error after applying Schwarz inequality as long as u ∈ H 1+s (Ω) with any s > 0.…”

P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

“…Note that for b = 1, (P 1 h ) is solvable. We prove that (29) and u h ∈ N ξ hold for appropriately small h.…”

“…Recently, Mao et al give a more general framework to construct effective FEM by using the different ideas, and novel DSP element and QB element are proposed in [29,31,32]. In fact, QB element is equivalent to the nonparametic P 1 -nonconforming quadrilateral element [34] under rectangular meshes, but for general quadrilateral meshes, they are not equivalent.…”

“…the RQC4 element proposed in [44] (has been published 20 years ago). Furthermore, motivated by references [33,34,44], the parametric constrained nonconforming rotated Q 1 element are presented in [20] and the nonconforming double set parameter rotated Q 1 element are also presented in [28][29][30][31][32]. In 2006, Mao, Chen and Sun [31] presented a four-parameter quadrilateral nonconforming element with double set parameter by using different motivation and ideas (It is originated from the nonconforming rotated Q 1 element and bilinear element, they call it QB element) and proved that it is an anisotropic and superconvergent nonconforming quadrilateral element.…”