• Abstract

Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution $u$ but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0‹\alpha‹1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.  

• Abstract

A new method for nonlinearly constrained optimization problems is proposed. The method consists of two steps. In the first step, we get a search direction by the linearly constrained subproblems based on conic functions. In the second step, we use a differentiable penalty function, and regard it as the metric function of the problem. From this, a new approximate solution is obtained. The global convergence of the given method is also proved.

• Abstract

• Abstract

In this paper the results in [5] and [6] related to two-parameter nonlinear problems and computing the folds of degree 3 are generalized to any n-parameter nonlinear problems. Constructing a repeatedly extended system for an n-parameter nonlinear problem we prove that a fold of degree n+1 corresponds to a regular solution of its n-th extended system. Also, the equivalence between the n-th extended system and its reduced system is proved. Finally, some examples are computed.

• Abstract

This paper presents some results on finite dimensional approximation of branches of solutions of nonlinear problems near a cusp point. These result can be applied to numerical methods of solving nonlinear differential equations.

• Abstract

The spectral function $\hatμ(t)=\sum\limits_{j=1}^\infty e^{-itλ^{\frac{1}{2}}_j}$ where $\{λ_j\}^\infty_{j=1}$ are the eigenvalues of the three-dimensional Laplacian is studied for a variety of domains, where $- \infty＜t＜\infty$ and $i=\sqrt{-1}$. The dependence of $\hat{\mu}(t)$ on the connectivity of a domain and the impedance boundary condition (Robbin conditions) are analyzed. Particular attention is given to the spherical shell together with Robbin boundary conditions on its surface.

• Abstract

In this paper, a new algorithm for the eigenvalue problem of matrices is given. Numerical examples show that it could be a remarkable approach for practical purposes. Some open problems are listed.