An Optimal Multilevel Method with One Smoothing Step for the Morley Element

In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem. Based on an appropriate space splitting of the finite element spaces associated with the refinements and the abstract Schwarz framework, we prove that the proposed multilevel methods with one smoothing step are optimal, i.e., the convergence rate is independent of the mesh sizes and mesh levels. Moreover, the computational complexity is also optimal since the smoothers are performed only once on each level in the algorithm. Numerical experiments are provided to confirm the optimality of the suggested methods.

New High Accuracy Analysis of a Double Set Parameter Nonconforming Element for the Clamped Kirchhoff Plate Unilaterally Constrained by an Elastic Obstacle

In this paper, a non-C0 double set parameter finite element method is presented for the clamped Kirchhoff plate with an elastic unilateral obstacle. A new high accuracy error estimate with order O(h2) in the broken energy norm is derived by use of a series of novel approaches, including some special features of the element and an incomplete biquadratic interpolation operator. At the same time, some experimental results are provided to verify the theoretical analysis.

Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem

We establish a general framework to study the conforming and nonconforming virtual element methods (VEMs) for solving a Kirchhoff plate contact problem with friction, which is a fourth-order elliptic variational inequality (VI) of the second kind. This VI contains a non-differentiable term due to the frictional contact. This theoretical framework applies to the existing virtual elements such as the conforming element, the $C^0$-continuous nonconforming element and the fully nonconforming Morley-type element. In the unified framework we derive a priori error estimates for these virtual elements and show that they achieve optimal convergence order for the lowest-order case. For demonstrating the performance of the VEMs we present some numerical results that confirm the theoretical prediction of the convergence order.

Study of Quadri Qunitic Element Applicable to Deep Beams

A lot of studies have been performed to explore the use and application of conforming elements in meshing of two and three-dimensional structures. Here, a combination of 5 nodes on X axis and 3 nodes on Y axis has been considered. Hence, the element used here is a 12 nodded rectangular nonconforming element, which can be termed as Quadri-Quintic nonconforming element. The development of a computer program to study the behavior of Quadri-Quintic nonconforming element and its application in meshing of deep beams has been targeted. The study may able to show that the proposed element gives results with higher accuracy and with faster convergence.

A Two-Grid Method of Nonconforming Element Based on the Shifted-Inverse Power Method for the Steklov Eigenvalue Problem

This paper establishes a new kind of two-grid discretization scheme of nonconforming Crouzeix-Raviart element based on the shifted-inverse power method for the Steklov eigenvalue problem. The error estimates are provided from the work of Yang and Bi (SIAM J. Numer. Anal., 49, pp.1602-1624, 2011). Finally, numerical experiments are reported to illustrate the high efficiency of the two-grid discretization scheme proposed in this paper.