The interior field method for Laplace׳s equation in circular domains with circular holes

Abstract

The null field method (NFM) was proposed by Chen and his co-researchers, and has been discussed in numerous papers, see Chen et al. (2007 [11]), Chen et al. (2002 [12]), Chen et al. (2001 [13]), and Chen and Shen (2009 [14]). The further developments of the NFM have been made in our recent papers (Huang et al., 2013 [21]; Lee et al., 2013 [23]; Lee et al., 2014 [25]; Li et al., 2012 [29]). In this paper, the interior field method (IFM) is proposed, which offers the best performance of the NFM when the field nodes are located exactly on the domain boundary. The algorithms of the IFM are much simpler than those of the NFM, because only one formula of the interior solutions is needed, compared with multiple formulas in the NFM. Since the IFM can be classified into the family of the Trefftz method (Li et al., 2008 [31]), a new error analysis of the IFM and the collocation IFM (CIFM) can be explored, to achieve the optimal convergence rates. Moreover, new proof techniques for aliasing errors in this paper are straightforward, heuristic and much easier to follow, because of direct derivations from trigonometric functions, which are distinct from Canuto and Quarteroni (1982 [8]), Canuto et al. (2006 [9]), and Kreiss and Oliger (1979 [22]). Based on this paper, the IFM and the CIFM may be recommended for those problems solvable by the NFM before.