In this paper we consider the nonoverlapping domain decomposition method based on mixed element approximation for elliptic problems in two dimensional space. We give a kind of discrete domain decomposition iterative algorithm using mixed finite element, the subdomain problems of which can be implemented parallelly. We also give the existence, uniqueness and convergence of the approximate solution.

A new column recurrence algorithm based on the classical Greville method and modified Huang update is proposed for computing generalized inverse matrix and least squares solution. The numerical results have shown the high efficiency and stability of the algorithm.

A simple one-dimensional $2\times 2$ hyperbolic system is considered in the paper. The model contains a linear hyperbolic equation, as well as a hyperbolic equation of which the coefficients are about the solution of the linear one. The exact solution is presented and discussed, then numerical experiments are given by TVD (or MmB) type schemes for Riemann problems. From the results, we know that the solutions do have $\delta-$waves for some suitable initial data.

The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estimates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems is based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.

We examine a simple averaging formula for the gradient of linear finite elements in $R^d$ whose interpolation order in the $L^q$-norm is $O(h^2)$ for $d<2q$ and nonuniform triangulations. For elliptic problems in $R^2$ we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. Local error estimates up to a regular part of the boundary and the effect of numerical integration are also investigated.

The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate $O(N_kN)$ where $N_k$ is the dimension of the finite element space and $N$ is the number of time steps.

In this paper, we consider solving dense linear equations on Dawning1000 by using matrix partitioning technique. Based on this partitioning of matrix, we give a parallel block LU decomposition method. The efficiency of solving linear equations by different ways is analysed. The numerical results are given on Dawning1000. By running our parallel program, the best speed up on 32 processors is over 25.