Developments in overlapping Schwarz preconditioning of high-order nodal discontinuous Galerkin discretizations

A preconditioned two-level overlapping Schwarz method for solving unstructured nodal discontinuous Galerkin discretizations of the indefinite Helmholtz problem is studied. We employ triangles in two dimensions and in a local discontinuous Galerkin (LDG) variational setting. We highlight the necessary components of the algorithm needed to achieve efficient results in the context of high-order elements and indefinite algebraic systems. More specifically, we demonstrate the importance of not only coarse-grid solution sweeps, but also for increased overlap in the subdomain solves as the order of the elements increases. In this paper, we detail the discretization strategy and offer an effective approach to solving the resulting system of equations, with numerical evidence in support.