With A Long Strip Of The Inner Boundary Of The Two-dimensional Numerical Method, Based On Natural Boundary Reduction Of Harmonic Exterior Problems

The natural boundary reduction, first suggested and developed by Prof.K.Feng, is the third academic contribution of Prof.Feng besides the finite element method and the sympletic algorithms. Later Prof.Yu has done a lot of outstanding work in the field. Now not only can the NBEM(natural boundary element method) be used to deal with some boundary value problems over several special domains directly, but also the DDM(domain decomposition method) and the coupled algorithm, based on the natural boundary reduction, are two kinds of efficient methods in solving problems over unbounded or cracked domains. Up to now, the above work has achieved many important results for 2-D problems, 3-D problems and time-dependent problems successfully. However all of them are done based on an circular or a spherical artificial boundary. As for some boundary value problems over an unbounded domain outside an elongated domain, computing number will be reduced largely and storage space and computing time will be saved with an elliptic or an ellipsoidal artificial boundary.In this paper, through the idea of domain decomposition method and the theory of natural boundary reduction, the coupling of natural boundary element and finite element methods and a non-overlapping domain decomposition method based on natural boundary reduction, are investigated for 2-D exterior harmonic problem outside an elongated domain. In the first chapter, we propose the coupling of NDEM and FEM, discuss well-posedness of the coupled variational problem , and give error estimates of approximate solutions and some numerical examples. In the second chapter, a non-overlapping domain decomposition method (i.e. D-N alternating algorithm) based on natural boundary reduction on an elliptic artificial boundary is suggested for 2-D exterior harmonic problem outside an elongated domain. The convergence of discrete D-N alternating algorithm for a general exterior domain is investigated, the convergence rate independent of the mesh size is proved, the convergence of the D-N alternating algorithm and the choice of the contraction factor for an exterior elliptic domain are analyzed, and some numerical results are given.