In this paper, the method of non-conforming mixed finite element for second order elliptic problems is discussed and a format of real optimal order for the lowest order error estimate.

In this paper, a Kantorovitch-Ostrowski type convergence theorem and an error estimate of $\frac{\|f'(z_0)^{-1}f(x_{n+1})\|}{\|f'(z_0)^{-1}f(x_n)\|}$ using the information of higher derivatives at the center between initial points for King-Werner iteration method in Banach space are established.  

This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we extend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the computational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.

In this paper, we choose the trigonometric rational functions as wavelet filters and use them to derive various wavelets. Especially for a certain family of wavelets generated by the rational filters, the better smoothness results than Daubechies' are obtained.

In solving integral equations with logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional Helmholtz equation, we combine the Galerkin method with Beylkin's ([2]) approach, series of dense and nonsymmetric matrices may appear if we use traditional method. By appealing the so-called periodic quasi-wavelet (PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we can find an algorithm with only $O(K(m)^2)$ arithmetic operations where $m$ is the highest level. The Galerkin approximation has a polynomial rate of convergence.  

This paper refers to Clarke generalized gradient for a smooth composition of max-type functions of the form: $f(x)=g(x,\ {\rm max}_{j \in J_1} \ f_{1j}(x),\cdots, \ {\rm max}_{j \in J_m} \ f_{mj}(x))$, where $x\in R^n, \ J_i, \ i=1, ..., \ m$ are finite index sets, $g$ and $f_{ij}, \ j \in J_i, \ i=1,...,m$, are continuously differentiable on $R^{m+n}$ and $R^n$ respectively. In a previous paper, we proposed an algorithm of finding an element of Clarke generalized gradient for $f$, at a point. In that paper, finding an element of Clarke generalized gradient for $f$ , at a point, is implemented by determining the compatibilities of systems of linear inequalities many times. So its computational amount is very expensive. In this paper, we will modify the algorithm to reduce the times that the compatibilities of systems of linear inequalities have to be determined.  

In this paper, the Crank-Nicholson+component-consistent pressure correction method for the numerical solution of the unsteady incompressible Navier-Stokes equation of [1] on the rectangular half-staggered mesh has been extended to the curvilinear half-staggered mesh. The discrete projection, both for the projection step in the solution procedure and for the related differential-algebraic equations, has been carefully studied and verified. It is proved that the proposed method is also unconditionally (in $Δt$) nonlinearly stable on the curvilinear mesh, provided the mesh is not too skewed. It is seen that for problems with an out flow boundary, the half-staggered mesh is especially advantageous. Results of preliminary numerical experiments support these claims.

In this paper, we construct a projection interpolation for cubic triangular element by using orthogonal expansion triangular method. We show two fundamental formulas of estimation on a special partion and obtain a superconvergence result of 1-ε order higher for the placement function and its tangential derivative on the third order Lobatto points and Gauss points on each edge of triangular element.

In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.