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

Consider Dirichlet problems of Laplace's equation in a bounded simply-connected domain S$$ S $$, and use the null field equation (NFE) of Green's representation formulation, where the source nodes Q$$ Q $$ are located on a pseudo-boundary Gamma R$$ {\Gamma}_R $$ outside S$$ S $$ but not close to its boundary Gamma(= partial differential S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$. Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives u nu$$ {u}_{\nu } $$ of the solutions on the boundary Gamma(= partial differential S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$ 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 q$$ q $$-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 O(10-4)$$ O\left(1{0}<^>{-4}\right) $$ 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.