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 [18, 5], or Lagrange-multiplier approaches [10]. The advantage of Trefftz methods is that they often require fewer degrees of freedom than standard polynomial finite element methods since the basis functions already oscillate with the correct wavenumber. The disadvantage is that the resulting linear systems are often significantly ill-conditioned, making direct solvers or efficient preconditioning for iterative solvers necessary. For very large problems, especially in three dimensions, direct solvers become prohibitively expensive, and preconditioning iterative solvers is a difficult problem for the Helmholtz equation as demonstrated in [9]. Domain decomposition methods, in particular optimized Schwarz methods, have proven to still be effective iterative solvers for finite elements and discontinuous Galerkin methods with polynomial basis functions; for the Helmholtz equation, see [11, 12], and for Maxwell’s equation, see [1, 7]. In this paper we consider block Jacobi relaxation methods for the PWDG method. In the classical finite element case a block Jacobi relaxation is equivalent to a classical Schwarz method with Dirichlet transmission conditions, see for example [13]. This is however not necessarily the case for discontinuous Galerkin methods, see [14]. We investigate in this short paper what kind of domain decomposition methods one obtains when simply performing a block Jacobi relaxation in a PWDG discretization of the Helmholtz equation, and also show how one can obtain optimized Schwarz methods for such discretizations. Motivated by the block Jacobi relaxation we present a simple algebraic decomposition approach of the system matrix in PWDG methods and demonstrate for an example problem with plane wave basis functions its performance for iterative solvers.