Spectral two-level deflation for DG: A preconditioner for CG that does not need symmetry

Despite their geometric flexibility and high accuracy, an important drawback of Discontinuous Galerkin (DG) methods is that the resulting linear systems cannot be solved efficiently by standard iterative solution methods. In search of an effective preconditioning strategy for these systems, we cast a uniform spectral two-level preconditioner proposed by Dobrev et al. [6] into the deflation framework [17]. This report discusses how the resulting spectral twolevel deflation technique can be implemented in a Conjugate Gradient (CG) algorithm, and studies its performance for Symmetric Interior Penalty Galerkin (SIPG) discretizations for diffusion problems with extreme contrasts in the coefficients. Furthermore, it investigates the influence of the SIPG penalty parameter, which is not well-understood for such applications. Interestingly, we find that the proposed deflation technique essentially follows from the original preconditioning variant by skipping one of the two smoothing steps involved, and dropping the inconvenient restriction on the smoother which the original preconditioner required to be SPD. Despite these simplifications, the spectral two-level deflation method yields uniform convergence (independent of the mesh element diameter), and it is faster and more robust than the original preconditioner for large problems. Regarding the penalty parameter, we find that it can best be based on local values of the diffusion coefficient, instead of the usual strategy to use one global constant for the entire domain. The former approach allows for local minimization of the penalty parameter, resulting in smaller condition numbers and faster convergence of both the SIPG and the CG method.Electrical Engineering, Mathematics and Computer Scienc