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.

Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

The paper analyzes the Morley element method for the Cahn–Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on 1/ε polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on 1/ε polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825–847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L∞(H2) error order, or we can merely obtain the error bounds which are exponentially dependent on 1/ε. Numerical results are presented to validate the optimal L∞(H2) error order and the asymptotic behavior of the solutions of the Cahn–Hilliard equation.

A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids

This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.