Open AccessNumerical Experiments of the Dual Null Field Method for Dirichlet Problems of Laplace’s Equation in Elliptic Domains with Elliptic Holes
Author(s) -
ZiCai Li,
HungTsai Huang,
Liping Zhang,
A. A. Lempert,
MingGong Lee
Publication year - 2022
Publication title -
izvestiâ irkutskogo gosudarstvennogo universiteta. seriâ "matematika"/izvestiâ irkutskogo gosudarstvennogo universiteta. seria matematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.411
H-Index - 3
eISSN - 2541-8785
pISSN - 1997-7670
DOI - 10.26516/1997-7670.2022.39.80
Subject(s) - mathematics , laplace's equation , collocation (remote sensing) , mathematical analysis , degenerate energy levels , dirichlet boundary condition , dirichlet problem , singularity , elliptic curve
Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). In the first part of the two-part article, our efforts were focused on studying the theoretical aspects of this problem, including the analysis of errors and the study of stability. We provided the theoretical analysis for Laplace equation in elliptic domains with elliptic holes. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the null field methods (NFM) are used for the exterior and the interior boundaries, simultaneously. This approach is called the dual null field method (DNFM). This paper is the second part of the study. Numerical results for degenerate models of an elliptic domain with one elliptic hole at + = 2 are carried out to verify the theoretical analysis obtained. The collocation Trefftz method (CTM) is also designed for comparisons. Both the DNFM and the CTM can provide excellent numerical performances. The convergence rates are the same but the stability of CTM is excellent and can achieve the constant condition numbers, Cond = (1).