On rank-deficiency in direct Trefftz method for 2D Laplace problems

Abstract

The application of the direct Trefftz method to the solution of Laplace equation defined on a 2D domain is frequently hindered by the numerical instability of the solving system, which may become ill-conditioned or even rank-deficient. Ill-conditioning is typically caused by a lack of domain scaling or by the oscillatory nature of the functions included in the weighting basis. Conversely, rank-deficiency may occur even for scaled domains and for low-order weighting bases. Its causes are related to the regularity properties of the weighting functions and to a lack of completeness of the weighting basis. The objective of this paper is to contribute to a better understanding of the mathematical grounds of rank-deficiency, and of its sensitivity to the definition of the referential and the (over-)determination of the basis. It shows that, while frequent, rank-deficiency can be avoided by slightly skewing the referential and by meshing the boundary such as to ensure that the basis is complete.