Interior field methods for Neumann problems of Laplace's equation in elliptic domains, comparisons with degenerate scales

The interior field method (IFM) is applied to Neumann problems for Laplace's equation in elliptic domains. The polynomial convergence rates are derived, and small condition number as O(N) can be obtained, where N   is the number of particular solutions used. Moreover, the effective condition number as O(1)O(1) is explored, to display excellent stability (Li et al., 2015 [21]). Numerical experiments are carried out, to support the analysis made. The error analysis of the IFM for Dirichlet problems in circular domains is reported in Li et al. (2016) [19]. The error and stability analysis of the IFM for Neumann problems in elliptic domains is more advanced and challenging; this is the first goal of this paper. The second goal is to compare the Neumann problems with the degenerate scales of Dirichlet problems; some useful guidances are found for application. From the comparisons, the conservation law is essential to guarantee the unique solutions. The adaptive processes are also proposed to deal with the algorithm singularity of Dirichlet problems; they may be applied to the boundary element method (BEM), the original NFM, and the indirect BIEM for the arbitrary smooth boundary or the convex polygonal boundary.