Conservative schemes and degenerate scale problems in the null-field method for Dirichlet problems of Laplaceʹs equation in circular domains with circular holes

Recently, the null-field method (NFM) has been proposed by Chen and his co-researchers for solving boundary value problems involving circular domains with circular holes. The explicit algebraic equations of the NFM are derived in our recent paper [31]. However, even for the Dirichlet problem of Laplaceʹs equation, when the logarithmic capacity (transfinite diameter) C Γ = 1 is given, the solutions may not exist, or not unique if existing, to cause a singularity of the discrete algebraic equations. The non-uniqueness of the solutions of Dirichlet problems by the boundary integral equations is first reported in Christiansen [20] due to some special geometry, and then in [14,15] called the degenerate scale problems. In this paper, the new conservative schemes of NFM are proposed. The conservative schemes can always bypass the degenerate scale problems; though numerically it causes a severe instability. A new pseudo-singularity property is discovered that only the minimal singular value σ min of the discrete matrices is infinitesimal to cause the instability. To restore good stability of the conservative schemes, the over-determined systems and the truncated singular value decomposition (TSVD) are proposed. The over-determined systems are more favorable than TSVD due to simpler algorithms and slightly better performances in error and stability. More importantly, such numerical techniques can also be used to deal with all the degenerate scale problems of the original NFM in [11–13] as well as the boundary element method (BEM).