The natural boundary reduction,suggested by Feng and Yu^[2,3],is applied to deal with the exterior boundary value problem of 3-D Helmholtz equation.First,it explains how to derive the natural integral equation,namely the exact Dirichlet-to-Neumann condition,of Helmholtz problem in exterior spherical domain by expansion of spherical harmonics.Second,a numerical method for solving the hypersingular integral equation is developed.Third,some numerical examples are given to illustrate this method.

The adaptive finite element methods are very effective for solving partial differential equations in scientific researches and engineering designs.By using these methods the best possible results can be obtained at less computational costs.A posteriori error estimates serve as a key to realize the adaptive finite element computation.This paper surveys the progress in the adaptive finite element methods and a posteriori error estimates for solving elliptic equations,parabolic equations and hyperbolic equations.

The selection of relaxation factor plays a key role in D-N domain decomposition methods (DDM). Based on the natural boundary reduction, the D-N DDM for elliptic problems over unbounded domain is discussed. A simple and convenient approach to selection of relaxation factor is given, and an optimal convergence is obtained.