  Trefftz methods are numerical methods for the approximation of solutions to
boundary and/or initial value problems. They are Galerkin methods with
particular test and trial functions, which solve locally the governing partial
differential equation (PDE). This property is called the Trefftz property.
Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz
methods for problems governed by variable coefficient PDEs, by relaxing the
Trefftz property into a so-called quasi-Trefftz property: test and trial
functions are not exact solutions but rather local approximate solutions to the
governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics
problems governed by the convected Helmholtz equation, the present work tackles
the question of the definition, construction and approximation properties of
three families of quasi-Trefftz functions: two based on generalizations on
plane wave solutions, and one polynomial. The polynomial basis shows
significant promise as it does not suffer from the ill-conditioning issue
inherent to wave-like bases.
