Hybrid Schwarz preconditioners for linear systems arising from hp-discontinuous Galerkin method

We deal with the numerical solution of elliptic problems by the $hp$-discontinuous Galerkin method. We develop a two-level hybrid Schwarz preconditioner for the arising linear algebraic systems. The preconditioner is additive with respect to the local components and multiplicative with respect to the mesh levels. We derive the $hp$ spectral bound of the preconditioned operator in the form $O((H/h)(p^2/q))$, where $H$ and $h$ are the element sizes of the coarse and fine meshes, respectively, and $p$ and $q$ are the polynomial approximation degrees on the fine and coarse meshes. Further, we present a numerical study showing that the hybrid Schwarz preconditioner dominates the additive one from the point of view of the speed of convergence and also computational costs. Finally, the combination with a $hp$-mesh adaptation for the solution of nonlinear problem demonstrates the potential of this approach.