Investigation on near-boundary solutions by singular boundary method

This study is to solve the dramatic drop of numerical solution accuracy of the singular boundary method (SBM) in near-boundary regions, also known as the boundary layer effect in the literature of boundary element method (BEM), where we encounter ‘nearly singular’ interpolation functions when the field point is close to the boundary source points. It is noted that the SBM uses the singular fundamental solution as its interpolation basis function. Different from singularity at origin, the fundamental solution interpolation at near-boundary regions remains finite. However, instead of being a flat function, the interpolation function develops a sharp peak as the field point approaches the boundary, namely, nearly singular behaviors. Consequently, the evaluation of the physical quantities at this point is much less accurate than at central region points. To remove or damp out the rapid variations of this nearly singular interpolation, this paper introduces a nonlinear transformation, based on the sinh function, to remedy this boundary layer effect. Our numerical experiments verify that the proposed approach can improve the SBM near-boundary solution accuracy by several orders of magnitude in terms of relative errors. The SBM solution appears accurate at a point as close to the boundary as 1.0E–10 scale.