14. The Coupling of Natural BEM and Composite Grid FEM

The coupling of boundary elements and finite elements is of great importance for the numerical treatment of boundary value problems posed on unbounded domains. It permits us to combine the advantages of boundary elements for treating domains extended to infinity with those of finite elements in treating the complicated bounded domains. The standard procedure of coupling the boundary element and finite element methods is described as follows. First, the (unbounded) domain is divided into two subregions, a bounded inner region and an unbounded outer one, by introducing an auxiliary common boundary. Next, the problem is reduced to an equivalent one in the bounded region. There are many ways to accomplish this reduction (refer to [Cos87], [FY83], [GHW94], [HZ94], [JN80], [Med98] and [ESH79]). The FEM-BEM coupled method can be viewed as a domain decomposition method to solve unbounded domain problems. The natural boundary reduction method proposed by [FY83] has obvious advantages over the usual boundary reduction methods: the coupled bilinear form preserve automatically the symmetry and coerciveness of the original bilinear form,so not only the analysis of the discrete problem is simplified, but also the optimal error estimates and the numerical stability are restored (see [FY83] and [Yu93]). It is well known that the analytic solution of the Dirichlet exterior problem is in general singular at the corner points. The fast adaptive composite grid (iteration) method advanced by McCormick (refer to [BPWX91], [MT86] and [McC89]) is very effective in dealing with this kind of local singularity. However, it can not be applied directly to the case of unbounded domain. In the present paper we combine the composite grid method with the coupling method of natural boundary element and finite element to handle the corner singularity of the Dirichlet exterior problems. Under suitable assumptions we obtain the optimal error estimates of the corresponding approximate solutions. The underlying linear system is expensive to solve directly due to the complicated structure (which is neither sparse nor band). Instead, we introduce two iterative methods to solve this coupled system: (1) a combination algorithm between the inexact two-level multiplicative Schwarz method and the steepest descent method; (2) the preconditioning