We study the numerical behaviours of the relaxed asynchronous multisplitting methods for the linear complementarity problems by solving some typical problems from practical applications on a real multiprocessor system. Numerical results show that the parallel multisplitting relaxation methods always perform much better than the corresponding sequential alternatives, and that the asynchronous multisplitting relaxation methods often outperform their corresponding synchronous counterparts. Moreover, the two-sweep relaxed multisplitting methods have better convergence properties than their corresponding one-sweep relaxed ones in the sense that they have larger convergence domains and faster convergence speeds. Hence, the asynchronous multisplitting unsymmetric relaxation iterations should be the methods of choice for solving the large sparse linear complementarity problems in the parallel computing environments.

The conjugate gradient method for unconstrained optimization problems varies with a scalar. In this note, a general condition concerning the scalar is given, which ensures the global convergence of the method in the case of strong Wolfe line searches. It is also discussed how to use the result to obtain the convergence of the famous Fletcher-Reeves, and Polak-Ribiére-Polyak conjugate gradient methods. That the condition cannot be relaxed in some sense is mentioned.  

This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonliner NDDEs, are given. In particular, it is shown thar the ($k,l$)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.

A finite element method for the solution of Oseen equation in exterior domain is proposed. In this method, a circular artificial boundary is introduced to make the computational domain finite. Then, the exact relation between the normal stress and the prescribed velocity field on the artificial boundary can be obtained analytically. This relation can serve as an boundary condition for the boundary value problem defined on the finite domain bounded by the artificial boundary. Numerical experiment is presented to demonstrate the performance of the method.

In this work, a singularly perturbed two-point boundary value problem of convection-diffusion type is considered. An $hp$ version finite element method on a strongly graded piecewise uniform mesh of Shishkin type is used to solve the model problem. With the analytic assumption of the input data, it is shown that the method converges exponentially and the convergence is uniformly valid with respect to the singular perturbation parameter.

Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDEs. In this treatment, the symplectic step is performed prior to the spatial step, as opposed to the spatial, as opposed to the standard approach of spatially discretising the PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied. In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatial step size.

We study structure-preserving algorithms to phase space volume for linear dynamical systems $\dot{y} = Ly$ for which arbitrarily high order explicit symmetric structure-preserving schemes, i.e. the numerical solutions generated by the schemes satisfy $\det(\frac{\partial y_1}{\partial y_0})=e^{htrL}$, where $trL$ is the trace of matrix $L$, can be constructed. For nonlinear dynamical systems $\dot{y}=f(y)$ Feng-Shang first-order volume-preserving scheme can be also constructed starting from modified $\theta-$methods and is shown that the scheme is structure-preserving to phase space volume.

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. The optimal and superconvergence error estimates for this new method are derived both in space and in time. Also, a class of new error estimates of convergence and superconvergence for the time-continuous finite element method is demonstrated in which there are no time derivatives of the exact solution involved, such that these estimates can be bounded by the norms of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.

A regular splitting and potential reduction method is presented for solving a quadratic programming problem with box constraints (QPB) in this paper. A general algorithm is designed to solve the QPB problem and generate a sequence of iterative points. We show that the number of iterations to generate an $\epsilon$-minimum solution or an $\epsilon$-KKT solution by the algorithm is bounded by $O(\frac{n^2}{\epsilon}\log{\frac{1}{\epsilon}}+n\log{(1+\sqrt{2n})})$, and the total running time is bounded by $O(n^2(n+\log n+\log \frac{1}{\epsilon})(\frac{n}{\epsilon}\log{\frac{1}{\epsilon}}+\log n))$ arithmetic operations.

In this paper, HFEM is proposed to investigate the circular arch problem. Optimal error estimates are derived, some superconvergence results are established, and an asymptotic exactness posteriori error estimator is presented. In contrast with the classical displacement variational method, the optimal convergence rate for displacement is uniform to the small parameter. In contrast with classical mixed finite element methods, our results are free of the strict restriction on h (the mesh size) which is preserved by all the previous papers. Furtheremore, we introduce an asymptotic exactness posteriori error estimator based on a global superconvergence result which is discovered in this kind of problem for the first time.