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

We analyze and optimize two-level methods applied to a symmetric interior penalty discontinuous Galerkin finite element discretization of a singularly perturbed reaction-diffusion equation. Previous analyses of such methods have been performed numerically by Hemker et. al. for the Poisson problem. Our main innovation is that we obtain explicit formulas for the optimal relaxation parameter of the two-level method for the Poisson problem in 1D, and very accurate closed form approximation formulas for the optimal choice in the reaction-diffusion case in all regimes. Our Local Fourier Analysis, which we perform at the matrix level to make it more accessible to the linear algebra community, shows 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. Our 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. We illustrate our analysis with numerical experiments and comparisons in higher dimensions and different geometries.