In this paper, the numerical solutions of heat equation on 3-D unbounded spatial domain are considered. An artificial boundary $\Gamma$ is introduced to finite the computational domain. On the artificial boundary $\Gamma$, the exact boundary condition and a series of approximating boundary conditions are derived, which are called artificial boundary conditions. By the exact or approximating boundary condition on the artificial boundary, the original problem is reduced to an initial-boundary value problem on the bounded computational domain, which is equivalent or approximating to the original problem. The finite difference method and finite element method are used to solve the reduced problems on the finite computational domain. The numerical results demonstrate that the method given in this paper is effective and feasible.

In this paper, we present an algorithm for the CDT subproblem. This problem stems from computing a trust region step of an algorithm, which was first proposed by Celis, Dennis and Tapia for equality constrained optimization. Our algorithm considers general case of the CDT subproblem, and convergence of the algorithm is proved. Numerical examples are also provided.

In this paper, we propose an Expanded Characteristic-mixed Finite Element Method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection part in time and an expanded mixed finite element spatial approximation to deal with the diffusion part. The scheme is stable since fluid is transported along the approximate characteristics on the discrete level. At the same time it expands the standard mixed finite element method in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux. Our analysis shows the method approximates the scalar unknown, its gradient, and its flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. A numerical example is presented to show that the scheme is of high performance.  

In this paper, we describe a method to solve large-scale structural optimization problems by sequential convex programming (SCP). A predictor-corrector interior point method is applied to solve the strictly convex subproblems. The SCP algorithm and the topology optimization approach are introduced. Especially, different strategies to solve certain linear systems of equations are analyzed. Numerical results are presented to show the efficiency of the proposed method for solving topology optimization problems and to compare different variants. 

Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of $R^n$ satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of $C^{\mu}$ piecewise semialgebraic sets are also discussed.  

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$, where $Q$ is a square Hermitian positive definite matrix and $A^*$ is the conjugate transpose of the matrix $A$. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation $X+A^*X^{-2}A=Q$. At last, we further generalize these results to the nonlinear matrix equation $X+A^*X^{-n}A=Q$, where $n \ge 2$ is a given positive integer.

In this paper we study perturbations of the stiffly weighted pseudoinverse $ (W^{1\over 2}A)^†W^{1\over 2}$ and the related stiffly weighted least squares problem, where both the matrices $A$ and $W$ are given with $W$ positive diagonal and severely stiff. We show that the perturbations to the stiffly weighted pseudoinverse and the related stiffly weighted least squares problem are stable, if and only if the perturbed matrices $\widehat A=A+\delta A$ satisfy several row rank preserving conditions.  

The mortar element method is a new domain decomposition method(DDM) with nonoverlapping subdomains. It can handle the situation where the mesh on different subdomains need not align across interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. But until now there has been very little work for nonlinear PDEs. In this paper, we will present a mortar-type Morley element method for a nonlinear biharmonic equation which is related to the well-known Navier-Stokes equation. Optimal energy and $H^1$-norm estimates are obtained under a reasonable elliptic regularity assumption.