An adaptive DPG method for high frequency time-harmonic wave propagation problems

The Discontinuous Petrov-Galerkin (DPG) method for high frequency wave propagation problems is discussed. The DPG method offers uniform pre-asymptotic stability for any wavenumber. This allows for a fully automatic adaptive hp algorithm, that can start from very coarse meshes. Moreover DPG always delivers a Hermitian positive definite system, suggesting the use of the Conjugate Gradient algorithm for its solution. We present a new iterative solution scheme which benefits from these attractive properties of DPG. This novel solver is integrated within the adaptive procedure by constructing a two-grid-like preconditioner for the Conjugate Gradient method, that exploits information from previous meshes. The construction of such a preconditioner is discussed and an example for the 2D acoustics problem is presented. Our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber.