Some remarks on Trefftz type approximations

For the Trefftz method sequences of linearly independent functions satisfying the governing differential equations are needed. For two- and three-dimensional elasticity problems some useful options for obtaining Trefftz trial functions are discussed. The discussion also includes some very useful particular solutions. For three-dimensional elasticity problems four methods are considered: the displacement representation of Papkovich/Neuber, Piltner's complex representation, a hypercomplex displacement representation of Bock, Gürlebeck, Weisz-Patrault, Legatiuk, and H.M. Nguyen, as well as Slobodyanskii's representation written in real and complex form. For two-dimensional problems the option of using discretized Cauchy integrals is illustrated. Very briefly Trefftz type Radial Basis Functions are mentioned satisfying a homogeneous or inhomogeneous differential equation. This paper is not meant as a complete survey or review on Trefftz methods. The paper presents a collection of personal choices to help future developers of numerical methods based on Trefftz trial functions.