A class of domain decomposition preconditioners for \(hp\)-discontinuous Galerkin finite element methods (Q639391)

The discontinuous Galerkin finite element method (DGFEM) using \(h\) and \(p\) refinement is applied to symmetric elliptic boundary value problems. As example the Poisson equation in a polygonal domain is considered either in two or three space dimensions. The authors focus on the \(p\)-refinement and investigate the question whether the class of non-overlapping Schwarz preconditioners introduced for the case of \(h\)-refinement can be extended to \(p\)- and \(hp\)-refinement versions of the DGFEM, while preserving their efficiency. Spectral bounds for the preconditioned stiffness matrix are shown to be of order \(p^2\) for \(p\)-DGFEM. Combining these results with those known for \(h\)-DGFEM, spectral bounds of order \(p^2H/h\) are proven for \(hp\)-DGFEM. Finally, numerical results for the Poisson equation on the unit square illustrate the theoretical estimates.