Direction-Adaptive Plane-Wave Discontinuous Galerkin Methods for the Helmholtz Equation

Abstract

We consider plane-wave discontinuous Galerkin (PWDG) approximations of the Helmholtz equation with adaptive local propagation directions. The directions are chosen by minimizing a weighted residual on the mesh skeleton. We study two formulations: in Part A the PWDG system is solved for the coefficients at fixed directions, while in Part B the same residual is minimized jointly over coefficients and directions, with the coefficients eliminated by variable projection. Complex angles unify propagating and evanescent Trefftz waves. The discrete system is normalized in the Trefftz-DG norm, and a local Cauchy-trace Gramian is used to remove numerically dependent directions. On straight edges the residual integrals are exact. For all-Dirichlet problems, the residual equals the squared DG error and provides local adaptive indicators. We prove local quadratic growth of the reduced direction functional near an identifiable zero-residual solution of fixed rank. Numerically, an exact circular DtN test shows that the dominant phase direction of a Hankel wave can be recovered accurately even when a small plane-wave fan gives a larger field error. We also separate the effects of the coefficient basis, trace cutoff, and trace-spectrum arithmetic on the high-$p$ DtN floor. Finally, for finite sums of plane waves, residual-based ENRICH--MOVE continuation recovers all directions to roundoff for $M=1,\ldots,19$. At $M=20$ the automatic birth step enters a false basin, whereas a nearby birth again reaches roundoff. The results indicate that direction adaptation is most effective for solutions of low directional complexity.