Approximation Properties of Vectorial Exponential Functions

This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference “Essentially Hyperbolic Problems: Unconventional Numerics, and Applications”. With respect to classical Finite Elements Methods, Trefftz methods are unconventional methods because of the way the basis functions are generated. Trefftz discontinuous Galerkin (TDG) methods have recently shown potential for numerical approximation of transport equations [ 6 , 26 ] with vectorial exponential modes. This paper focuses on a proof of the approximation properties of these exponential solutions. We show that vectorial exponential functions can achieve high order convergence. The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.