This paper is concerned with the initial value problem for non-stationary Stokes flows, under a certain non-linear boundary condition which can be called the leak boundary condition of friction type. Theoretically, our main purpose is to show the strong solvability (i.e., the unique existence of the $L^2$-strong solution) of this initial problem by means of the non-linear semi-group theory originated with Y. Kōmura. The method of analysis can be applied to other boundary or interface conditions of friction type. It should be noted that the result yields a sound basis of simulation methods for evolution problems involving these conditions.

In this contribution we analyse some fundamental features of an iterative method to solve systems of linear equations, following the approach introduced in a previous work [1]. Such questions range from optimal parameters and initial conditions to comparison with other methods. An interesting result is that a priori we can give an estimation of the number of iterations to get a given accuracy.

The subject matter of this paper is the representation of the solution of the linear differential equation $Y'= AY - YB, Y(0) = Y_0,$ in the form $Y(t) = e^{Ω(t)}Y_0$ and the representation of the function n as a generalization of the classical Magnus expansion. An immediate application is a new recursive algorithm for the derivation of the Baker-Campbell-Hausdorff formula and its symmetric generalisation.

We examine a nonlinear partial differential equation of elliptic type with the homogeneous Dirichlet boundary conditions. We prove comparison and maximum principles. For associated finite element approximations we introduce a discrete analogue of the maximum principle for linear elements, which is based on nonobtuse tetrahedral partitions.

Interior point methods are very efficient methods for solving large scale linear programming problems. The central path plays a very important role in interior point methods. In this paper we propose a new central path, which scales the variables. Thus it has the advantage of forcing the path to have roughly the same distance from each active constraint boundary near the solution.

In this study we will consider moving mesh methods for solving one-dimensional time dependent PDEs. The solution and mesh are obtained simultaneously by solving a system of differential-algebraic equations. The differential equations involve the solution and the mesh, while the algebraic equations involve several geometrical variables such as $\theta$ (the tangent angle), $U$ (the normal velocity of the solution curve) and $T$ (tangent velocity). The equal-arclength principle is employed to give a close form for $T$. For viscous conservation laws, we prove rigorously that the proposed system of moving mesh equations is well-posed, in the sense that first order perturbations for the solution and mesh can be controlled by the initial perturbation. Several test problems are considered and numerical experiments for the moving mesh equations are performed. The numerical results suggest that the proposed system of moving mesh equations is appropriate for solving (stiff) time dependent PDEs.  

This paper present a way to construct orthogonal piece-wise polynomials on an arbitrary triangular domain via barycentric coordinates. A boundary value problem for Laplace equation and its eigenvalue problem can be solved as two applications of this approach.

Based upon a new mixed variational formulation for the three-field Stokes equations and linearized Non-Newtonian flow, an $h-p$ finite element method is presented with or without a stabilization. As to the variational formulation without stabilization, optimal error bounds in $h$ as well as in $p$ are obtained. As with stabilization, optimal error bounds are obtained which is optimal in $h$ and one order deterioration in $p$ for the pressure, that is consistent with numerical results in [9,12] and therefore solved the problem therein. Moreover, we proposed a stabilized formulation which is optimal in both $h$ and $p$.

In this paper, a nonoverlapping domain decomposition method, which is based on the natural boundary reduction (cf. [4,13,15]), is developed to solve the boundary value problem in exterior three-dimensional domain of general shape. Convergence analyses both for the exterior spherical domain and the general exterior domain are made. Some numerical examples are also provided to illustrate the method.  

An explicit upwind finite element method is given for the numerical computation to multi-dimensional scalar conservation laws. It is proved that this scheme is consistent to the equation and monotone, and the approximate solution satisfies discrete entropy inequality. To guarantee the limit of approximate solutions to be a measure valued solution, we prove an energy estimate. Then the $L^p$ strong convergence of this scheme is proved.

Composite Legendre-Laguerre approximation in unbounded domains is developed. Some approximation results are obtained. As an example, a composite spectral scheme is provided for the Burgers equation on the half line. The stability and convergence of proposed scheme are proved strictly. Two-dimensional exterior problems are discussed.