Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

Abstract

The dual techniques have been widely used in many engineering papers, to deal with singularity and ill-conditioning of the boundary element method (BEM). In this paper, we consider Laplace's equation with circular domains with one circular hole. The explicit algebraic equations of the first and second kinds of the null field method (NFM) are provided for applications. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and the Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, however, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual NFM (DNFM) in this paper. The excellent stability and the optimal convergence rates are explored in this paper. By using the simple Gaussian elimination or the iteration methods, numerical solutions can be easily obtained. Recently, the study on degenerate scales is active, many removal techniques are proposed, where the advanced solution methods may be needed, such as the truncated singular value decomposition (TSVD) and the overdetermined systems. In contrast, the solution methods of the DNFM in this paper are much simpler, with a little risk of the algorithm singularity from degenerate scales.