Block Jacobi Relaxation for Plane Wave Discontinuous Galerkin Methods

Nonpolynomial finite element methods for Helmholtz problems have seen much attention in recent years in the engineering and mathematics community. The idea is to use instead of standard polynomials Trefftz-type basis functions that already satisfy the Helmholtz equation, such as plane waves [17], Fourier–Bessel functions [8] or fundamental solutions [4]. To approximate the inter-element interface conditions between elements several possibilities exist, such as the ultra-weak variational formulation (UWVF [6]), plane wave discontinuous Galerkin methods (PWDG [15]), partition of unity finite elements (PUFEM [3]), least-squares methods [5, 18], or Lagrange-multiplier approaches [10].