Norm Preconditioners for Discontinuous Galerkin $hp$-Finite Element Methods

We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order partial differential equations with a non-negative characteristic form. Our solution method is a norm-preconditioned three-term GMRES routine. We find that for symmetric positive-definite diffusivity tensors the convergence of our solver is independent of discretization, while for the semidefinite case both theory and experiment indicate dependence on both h and p. Numerical results are included to illustrate performance on several test cases.