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

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

“…This is distinct to the Dirichlet problems by the NFM, as discussed in [19]. However, the pseudosingularity may occur, as discovered in the conservative schemes in [18], to cause an extreme instability. The pseudo-singularity means that the discrete matrix is nonsingular; but the minimal singular value σ min is infinitesimal, and much less than the others.…”

“…In computation, choose ϵ ¼ ϵ ¼ 0 in the NFM, which are equivalent to the IFM. Note that the pseudo-singularity as in [18] is again discovered in numerical solutions, when the number of the collocation equations is just equal to that of unknowns. To bypass the pseudo-singularity, the overdetermined system of linear algebraic equations and the truncated singular value decompositions (TSVD) in [19,21] are solicited to achieve good and even excellent stability.…”

“…For the Neumann problem, the conservative law (i.e., consistent condition), R Γ ∂u=∂ν ¼ 0, must be obeyed when the solutions exist. Then two given constants p 0 and p 0 must satisfy [18]:…”

“…The null field method of Chen and the study of Chen in [13][14][15] seem to be more focused on the degenerate scale problems. For the circular domains with circular holes by the MFEs, our efforts are paid not only for the degenerate scales in [18,19], but also for the numerical algorithms themselves in [20] and this paper. The more specific the problem is, the more efficient the numerical algorithms are.…”

“…Recently, the NFM has been further studied in [16][17][18][19][20]. The new interior field method (IFM) is proposed in [16], which is equivalent to the specific NFM when the field nodes are located on the domain boundary.…”