Neumann problems of Laplace׳s equation in circular domains with circular holes by methods of field equations

Abstract

For Laplace׳s equation in circular domains with circular holes, the null field method (NFM) is proposed by Chen with his groups. In NFM, the fundamental solutions (FS) with the exterior field nodes to the solution domain are used in the Green formulas, where the FS are replaced by the infinite expansion series. The explicit algebraic equations are derived and reported in Li et al. (2012) [20]. The explicit algebraic equations are essential not only to practical computation, but also to the algorithm analysis, such as algorithm singularity, error and stability analysis. So far, the study of the NFM is confined to the Dirichlet problems (i.e., the Dirichlet boundary value problems) by the first kind NFM. This paper is devoted mainly to the Neumann problems (i.e., the Neumann boundary value problems) of Laplace׳s equation by the second kind NFM. When the field nodes are pulled to the domain boundary, this special (i.e., the optimal) NFM is equivalent to the interior field method (IFM) (Huang et al., 2013) [16]. In fact, the IFM results from the Trefftz method, where the interior field solutions are chosen to satisfy the Neumann boundary conditions. For simplicity, we call the IFM and the specific NFM as the method of field equations (MFEs). For the Neumann problems, there do not exist the degenerate scale problems, but the pseudo-singularity may be encountered if the numbers of the unknown coefficients and the collocation equations are exactly the same. To bypass this pseudo-singularity, the overdetermined system and the truncated singular value decomposition (TSVD) are solicited, to restore good stability. Interestingly, the first kind MFE can also be used for the Neumann problems. Numerical experiments with comparisons are carried out by two kinds of MFEs. The stability is made for two kinds of MFEs, and a theoretical argument is provided to verify the effectiveness of the overdetermined system. In summary, two kinds of MFEs are effective for solving the Neumann problems, and their numerical performances are excellent.