In this review we summarize the results on the cell boundary element methods (CBE methods) and the multiscale cell boundary element method (MsCBE method) based on papers by Jeon and his colleagues. In the CBE methods, flux is conserved on each cell and normal fluxes on intercell boundaries are continuous for unstructured triangulations. The CBE method can be understood as an finite element version of the finite volume method.

A class of Crouzeix-Raviart type nonconforming finite element methods are proposed for the parabolic variational inequality problem with moving grid on anisotropic meshes. By using some novel approaches and techniques, the same optimal error estimates are obtained as the traditional ones. It is shown that the classical regularity condition or quasi-uniform assumption on meshes is not necessary for the finite element analysis.

Our aim in this paper is to present two types of emi-implicit schemes based on multilevel wavelet-like incremental unknowns (WIU) for solving a one-dimensional reaction-diffusion equation with a polynomial growth nonlinearity. The stability of schemes is proved which also shows the advantage over explicit and implicit schemes in the same conceptual framework of multilevel WIU. Numerical examples are provided to test the efficiency of the new schemes.

The global convergence theory of quasi-Newton methods for optimization problems has well been established. Related work to the globalization of quasi-Newton methods for nonlinear equations is relatively less. The major difficulty in globalizing quasi-Newton methods for nonlinear equations lies in the lack of efficient line search technique. Recently, there have been proposed some derivative-free line searches. The study in the global convergence of some quasi-Newton methods has taken good progress. In this paper, we summarize some recent progress in the global convergence of quasi- Newton methods for solving nonlinear equations.

We consider a Rayleigh-Ritz type eigensolver for finding a limited set of eigenvalues and their corresponding eigenvectors in a certain region of generalized eigenvalue problems. When the matrices are very large, iterative methods are used to generate an invariant subspace that contains the desired eigenvectors. Approximations are extracted from the subspace through a Rayleigh-Ritz projection. In this paper, we present a Rayleigh-Ritz type method with a contour integral (CIRR method). In this method, numerical integration along a circle that contains relatively small number of eigenvalues is used to construct a subspace. Since the process to derive the subspace can be performed in parallel, the presented method is suitable for master-worker programming models. Numerical experiments illustrate the property of the proposed method.

The existence of center manifolds is closedly related to local behavior of dynamical systems. In this paper we consider center manifolds both of the discrete Toda equation and the discrete Lotka-Volterra system. Their solutions converge to eigenvalues and singular values of certain structured matrices. A free parameter plays a key role to show the existence of a center manifold of the discrete Lotka-Volterra system. A monotone convergence of the solution of the discrete Lotka-Volterra system is proved with the help of the existence of a center manifold. In contrast, a center manifold of the discrete Toda equation does not always exist.

We propose a method to enclose solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains. Our method is based on an infinite dimensional Newton-type formulation by using the finite element method with constructive error estimates and fixed point theorems. Numerical examples related to the step flow problems in $L$-shape domain are presented.

For large sparse system of linear equations with a non-Hermitian positive definite coefficient matrix, we review the recently developed Hermitian/skew-Hermitian splitting (HSS) iteration, normal/skew-Hermitian splitting (NSS) iteration, positive-definite/skew-Hermitian splitting (PSS) iteration, and block triangular/skew-Hermitian splitting (BTSS) iteration. These methods converge unconditionally to the exact solution of the linear system, with the upper bounds of their convergence factors being only dependent on the spectrum of the Hermitian (or normal, or positive-definite) splitting matrix, but independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved.

The solution of the interface problem is only in $H^{1+\alpha}(\Omega)$ with $\alpha>0$ possibly close to zero and, hence, it is difficult to be approximated accurately. This paper studies an accurate numerical method on quasi-uniform grids for two-dimensional interface problems. The method makes use of a singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors. Using continuous piecewise linear elements on quasi-uniform grids, our finite element approximation is shown to be optimal, $O(h)$, accurate in the $H^1$ norm. This is confirmed by numerical experiments for interface problems with $\alpha < 0.1$. An $O(h^{1+\alpha})$ error bound in the $L^2$ norm is also established by the standard duality argument. For small $\alpha$, this improvement over the $H^1$ error bound is negligible. However, numerical tests presented in this paper indicate that the $L^2$ norm accuracy is much better than the theoretical error bound.

The aim of this paper is to show numerical treatment of analytic continuation by high-accurate discretization with multiple-precision arithmetic. We deal with the Cauchy problem of the Laplace equation and an integral equation of the first kind with an analytic kernel. We propose high-accurate discretization based on the spectral method, and show some numerical examples with our proposed multiple-precision arithmetic.

In this paper we show that the monotone difference methods with smooth numerical fluxes possess superconvergence property when applied to strictly convex conservation laws with piecewise smooth solutions. More precisely, it is shown that not only the approximation solution converges to the entropy solution, its central difference also converges to the derivative of the entropy solution away from the shocks.

For two-fluid flow problems with surface tension we present finite element schemes based on energy-stable approximation. In the case of no surface tension, those schemes are unconditionally stable in the energy-sense. When there exists surface tension, they are proved to be stable if a quantity remains bounded in the computation. Some numerical results of rising bubble problems show the robustness and applicability of these schemes.

In this paper we analyze numerical dispersion relation of some conforming and nonconforming quadrilateral finite elements. The elements employed in this analysis are the standard $Q_1$ conforming finite element, the DSSY nonconforming element [5] and the $P_1$-nonconforming quadrilateral finite element [14]. Several aspects of comparative analyses of the above three elements for two or three dimensional problems are shown.