Algorithm singularity of the null-field method for Dirichlet problems of Laplace׳s equation in annular and circular domains

For circular domains with circular holes, the null field method (NFM) is proposed by Chen and his co-researchers when solving boundary integral equation (BIE). The explicit algebraic equations of the NFM are recently derived in Li et al. (2012) [33], and their conservative schemes are proposed in Lee et al. (2013) [28]. However, even for the Dirichlet problem of Laplace׳s equation, there may exist a singularity of the original boundary integral equation (BIE) and/or its numerical algorithms such as the NFM. Such a singularity is called the degenerate scale problem due to special domain scales, and was studied in Christiansen (1975) [22]. Since to bypass the singularity is imperative for both theory and computation, the degenerate scale problem has been extensively discussed in the literature. An algorithm singularity means the singularity of the coefficient matrix of collocation methods, but we confine ourselves to the singularity caused by the degenerate scale problem. So far, for the algorithm singularity of the NFM of degenerate scales, no advanced analysis exists, although a preliminary discussion was given in Chen and Shen (2007) and Lee et al. (2013) [15] and [28]. In this paper, all kinds of field nodes of degenerate scales leading to algorithm singularity are revealed in detail. To remove singularity of discrete matrices and to restore good stability, several effective techniques are proposed. Numerical experiments are carried out to verify the theoretical analysis made. Based on the analysis and computation in this paper, not only can the algorithm singularity of the NFM be bypassed, but also the highly accurate solutions with good stability may be achieved.