In this paper we consider two biharmonic problems [13] which will be conventionally indicated as "simply supported" and "clamped plate" problem.
We construct a decomposition method [16], [19] related to the partition of the plate in two, or more, subdomains. We carry on the numerical treatment of the method first decoupling these fourth order problems into two second order problems, then discretizing these problems by mixed linear finite element and obtaining an algebraic system. Moreover,  we present an iterative block algorithm for solving the foregoing system, which can be efficiently developed on parallel computers.
In the end, we extend the method to the respective biharmonic variational inequalities [10].  

A necessary and sufficient condition for the convexity of adjoint patches for a Bezier triangular surface is presented. Furthermore, it is proved that this condition is equivalent to the fact that the adjoint patches form a decreasing sequence as the corresponding degree decreases. The condition can be easily computationally verified.

In this paper we present a method for solving the matrix differential equation $X^{(2)}(t)-AX(t)=F(t)$, without increasing the dimension of the problem. By introducing the concept of co-square root of a matrix, existence and uniqueness conditions for solutions of boundary value problems related to the equation as well as explicit solutions of these solutions are given, even for the case where the matrix $A$ has no square roots.

In this paper we construct a new type of symmetrical dissipative difference scheme. Except discontinuity these schemes have uniformly second-order accuracy. For calculation using these, the simple-wave is very exact, the shock has high resolution, the programing is simple and the CPU time is economical.
Since the paper [1] introduced that in some conditions Lax-Wendroff scheme would converge to nonphysical solution, many researchers have discussed this problem. According to preserving the monotonicity of the solution preserving monotonial schemes and TVD schemes have been introduced by Harten, et. According to property of hyperbolic wave propagation the schemes of split-coefficient matrix (SCM) and split-flux have been formed. We emphasize the dissipative difference scheme, and these schemes are dissipative on arbitrary conditions.  

We consider the singular perturbation problem $$-\varepsilon^2u"+\mu b(x,u)u'+c(x,u)=0,u(0),u(1)$$ given with two small parameters $\varepsilon$ and $\mu$ , $\mu =\varepsilon^{1+p},p>0$. The problem is solved numerically by using finite difference schemes on the mesh which is dense in the boundary layers. The convergence uniform in $\varepsilon$ is proved in the discrete $L^1$ norm. Some convergence results are given in the maximum norm as well.

A spectral method is proposed, the existence and uniqueness of the global and smooth solution are proved for the periodic initial value problem of the generalized K-S equation. The error estimates are established and the convergence is proved for the approximate solution of the spectral method.

In this paper a modified MacCormack's scheme is presented. The scheme is based on flux vector splitting. The test computations show that the proposed modified scheme produces much better numerical results than original MacCormack's scheme.

In this paper, we present a dual algorithm for minimizing a convex quadratic function with two quadratic constraints. Such a minimization problem is a subproblem that appears in some trust region algorithms for general nonlinear programming. Some theoretical properties of the dual problem are given. Global convergence of the algorithm is proved and a local superlinear convergence result is presented. Numerical examples are also provided.

This paper proves a Kantorovich-type inequality on the matrix of the type $\frac{1}{2}(Q^H_1 AQ_1 Q^H_1A^{-1} Q_1+Q^H_1A^{-1}Q_1Q^H_1AQ_1)$, where $A$ is an $n\times n$ positive definite Hermitian matrix and $Q_1$ is an $n\times m$ matrix with rank $(Q_1)=m$. The result is applied to get an extension of the Bauer-Fike inequality on condition numbers of similarities that block diagonalized matrices.

In this paper a general numerical method for solving Riemann problems is discussed. It can be used to solve the Riemann problems of various hyperbolic systems of differential equations with two independent variables. The problem of reflection of discontinuities from external boundaries can also be solved by this general numerical method.

In this paper we study the polynomial acceleration methods for solving singular linear systems. We establish iterative schemes, show their convergence and find iteration error bounds.