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, with its attractive uniform (mesh and wavenumber independent) pre-asymptotic stability property, allows for a fully automatic adaptive hp-algorithm that can be initiated 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 capitalizes on 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 our preconditioner is discussed, and its efficacy is illustrated with an example of a 2D acoustics problem. Our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber.