This paper discusses the optimal preconditioning in the domain decomposition method for Wilson element. The process of the preconditioning is composed of the resolution of a small scale global problem based on a coarser grid and a number of independent local subproblems, which can be chosen arbitrarily. The condition number of the preconditioned system is estimated by some characteristic numbers related to global and local subproblems. With a proper selection, the optimal preconditioner can be obtained, while the condition number is independent of the scale of the problem and the number of subproblems.

In this paper, a mean-field equation of motion which is derived by Penrose (1991) for the dynamic Ising model with Glauber dynamics is considered. Various finite difference schemes such as explicit Euler scheme, predictor-corrector scheme and some implicit schemes are given and their convergence, stabilities and dynamical properties are discussed. Moreover, a Lyapunov functional for the discrete semigroup $\{ S\}_{n>0}$ is constructed. Finally, numerical examples are computed and analyzed. it shows that the model is a better approximation to Cahn-Allen equation which is mentioned in Penrose (1991).

A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in $L_2$ norm, the convergence of the difference solution is proved in order $O(h^2+r^2)$. In the proof, a new skill is used to deal with the term of difference quotient $(e_{j,k}^n)t$. This is necessary, since there is no estimate of $E(x,y,t)$ in $L_\infty$ norm.

In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic integrators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no high order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are "essentially" maintained by the discrete map, at least for linear problems.

In this paper, the generalized complementarity problem is formulated as an unconstrained optimization problem. Our results generalize the results of [9]. The dimensionality of the unconstrained problem is the same as that of the original problem. If the mapping of generalized complementarity problem is differentiable, the objective function of the unconstrained problem is also differentiable. All the solutions of the original problem are global minimizers of the optimization problem. A generalized strict complementarity condition is given. Under certain assumptions, local properties of the correspondent unconstrained optimization problem are studied. Limited numerical tests are also reported.

We formulate a coupled vibration between plate and acoustic field in mathematically rigorous fashion. It leads to a non-standard eigenvalue problem. A finite element approximation is considered in an abstract way, and the approximate eigenvalue problem is written in an operator form by means of some Ritz projections. The order of convergence is proved based on the result of Babuška and Osborn. Some numerical examples are shown for the problem for which the exact analytical solutions are calculated. The results show that the convergence order is consistent with the one by the numerical analysis. 

We will study the convergence property of Schwarz alternating method for concave region where the concave region is decomposed into convex subdomains. Optimality of regular preconditioner deduced from Schwarz alternating is also proved. It is shown that the convergent rate and the condition number are independent of the mesh size but dependent on the relative geometric position of subdomains. Special care is devoted to non-uniform meshes, exclusively, local properties like the shape regularity of the finite elements are utilized.