Optimization of two-level methods for DG discretizations of reaction-diffusion equations

In this manuscript, two-level methods applied to a symmetric interior penalty discontinuous Galerkin finite element discretization of a singularly perturbed reaction-diffusion equation are analyzed. Previous analyses of such methods have been performed numerically by Hemker et al. for the Poisson problem. The main innovation in this work is that explicit formulas for the optimal relaxation parameter of the two-level method for the Poisson problem in 1D are obtained, as well as very accurate closed form approximation formulas for the optimal choice in the reaction-diffusion case in all regimes. Using Local Fourier Analysis, performed at the matrix level to make it more accessible to the linear algebra community, it is shown that for DG penalization parameter values used in practice, it is better to use cell block-Jacobi smoothers of Schwarz type, in contrast to earlier results suggesting that point block-Jacobi smoothers are preferable, based on a smoothing analysis alone. The analysis also reveals how the performance of the iterative solver depends on the DG penalization parameter, and what value should be chosen to get the fastest iterative solver, providing a new, direct link between DG discretization and iterative solver performance. Numerical experiments and comparisons show the applicability of the expressions obtained in higher dimensions and more general geometries.