Trefftz methods with cracklets and their relation to BEM and MFS

Engineering Analysis and Boundary Elements, 95 (2018), 93-104
https://www.sciencedirect.com/science/article/pii/S0955799718302650

In this paper we consider Trefftz methods which are based on functions defined by the single layer or double layer potentials, integrals of the fundamental solutions, or their normal derivative, on cracks. These functions are called cracklets, and verify the partial differential equation, as long as the crack support is not placed inside the domain. A boundary element method (BEM) interpretation is to consider these cracks as elements of the original boundary, in a direct BEM approach, or elements of an artificial boundary, in an indirect BEM approach. In this paper we consider the cracklets just as basis functions in Trefftz methods, as the method of fundamental solutions (MFS). We focus on the 2D Laplace equation, and establish some comparisons and connections between these cracklet methods and standard approaches for the BEM, the indirect BEM, and the MFS. Namely, we propose the enrichment of the MFS basis with these cracklets. Several numerical simulations are presented to test the performance of the methods, in particular comparing the results with the MFS and the BEM.