A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.