In the paper, we analyze the $L^2$ norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the $L^2$ norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including: the Powell-Sabin $C^1-P_2$ macro element, the nonconforming Morley element, the $C^1-Q_2$ macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang-Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature: can the $L^2$ norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm?

This paper deals with preconditioners for the iterative solution of the discrete Oseen problem with variable viscosity. The motivation of this work originates from numerical simulations of multiphase flow, governed by the coupled Cahn-Hilliard and incompressible Navier-Stokes equations. The impact of variable viscosity on some known preconditioning technique is analyzed. Theoretical considerations and numerical experiments show that some broadly used preconditioning techniques for the Oseen problem with constant viscosity are also efficient when the viscosity is varying.

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a challenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface problems with multi-domains and triple junctions. The resulting linear system of equations is positive definite if the matrix coefficients for the elliptic equations in the domains are positive definite. Numerical experiments show that this method is about second order accurate in the $L^∞$ norm for piecewise smooth solutions. Corner singularity can be handled in a way such that the accuracy does not degenerate. The triple junction is carefully resolved and it does not need to be placed on the grid, giving our method the potential to treat moving interface problems without regenerating mesh.

In a previous paper, some particular multistep cosine methods were constructed which proved to be very efficient because of being able to integrate in a stable and explicit way linearly stiff problems of second-order in time. In the present paper, the conditions which guarantee stability for general methods of this type are given, as well as a thorough study of resonances and filtering for symmetric ones (which, in another paper, have been proved to behave very advantageously with respect to conservation of invariants in Hamiltonian wave equations). What is given here is a systematic way to analyse and treat any of the methods of this type in the mentioned aspects.

We give general expressions, analyze algebraic properties and derive eigenvalue bounds for a sequence of Toeplitz matrices associated with the sinc discretizations of various orders of differential operators. We demonstrate that these Toeplitz matrices can be satisfactorily preconditioned by certain banded Toeplitz matrices through showing that the spectra of the preconditioned matrices are uniformly bounded. In particular, we also derive eigenvalue bounds for the banded Toeplitz preconditioners. These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential equations, and are useful in analyzing algebraic properties and deriving eigenvalue bounds for the corresponding preconditioned matrices. Numerical examples are given to show effectiveness of the banded Toeplitz preconditioners.

A novel two level spline method is proposed for semi-linear elliptic equations, where the two level iteration is implemented between a pair of hierarchical spline spaces with different orders. The new two level method is implemented in a manner of p-adaptivity. A coarse solution is obtained from solving the model problem in the low order spline space, and the solution with higher accuracy is generated subsequently, via one step Newton or modified Newton iteration in the high order spline space. We also derive the optimal error estimations for the proposed two level schemes. At last, the illustrated numerical results confirm our error estimations and further research topics are commented.

The classical eigenvalue problem of the second-order elliptic operator is approximated with bi-quadratic finite element in this paper. We construct a new superconvergent function recovery operator, from which the $O(h^8|\ln h|^2)$ ultraconvergence of eigenvalue approximation is obtained. Numerical experiments verify the theoretical results.