The whys and hows of conditioning of DG plane wave Trefftz methods: a single element

Abstract

Plane-wave Trefftz methods (PWB) for the Helmholtz equation offer significant advantages over standard discretization approaches whose implementation employs more general polynomial basis functions. A disadvantage of these methods is the poor conditioning of the system matrices. In the present paper, we carefully examine the conditioning of the plane-wave discontinuous Galerkin method with reference to a single element. The properties of the mass and stiffness matrices depend on the size and geometry of the element. We study the mass and system matrices arising from a PWB on a single disk-shaped element. We then examine some preconditioning strategies, and present results showing their behaviour with three different criteria: conditioning, the behaviour of GMRES residuals, and impact on the $L^2$-error.