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

Lee, Ming‐Gong; Li, Zi‐Cai; Zhang, Li Ping; Huang, Hung‐Tsai; Chiang, John Y.

doi:10.1016/j.enganabound.2014.01.013

Cited by 9 publications

(43 citation statements)

References 33 publications

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“…Moreover, since the Neumann boundary conditions (i.e., Neumann conditions) are given, the leading terms involving lnr are no longer troublesome, and no more degenerate scale problems exist. 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.…”

Section: Introductionmentioning

confidence: 89%

“…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.…”

Section: Numerical Experiments By Two Kinds Of Mfesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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. Hence, we may employ two techniques, such as the overdetermined system and the truncated singular value decomposition (TSVD) in [19,21], to restore good stability.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Lee¹,

Li²,

Huang³

et al. 2015

Engineering Analysis with Boundary Elements

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Section: Introductionmentioning

confidence: 89%

Section: Numerical Experiments By Two Kinds Of Mfesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Lee¹,

Li²,

Huang³

et al. 2015

Engineering Analysis with Boundary Elements

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Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

Zhang,

Li,

Lee

et al. 2024

Numerical Methods Partial

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Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.

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Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Zhang

Wei

et al. 2015

Engineering Analysis with Boundary Elements

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

Cited by 9 publications

References 33 publications

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

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

Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

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