Discontinuous galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs

We analyze the discontinuous Galerkin method in time combined with a finite element method with symmetric stabilization in space to approximate evolution problems with a linear, first-order differential operator. A unified analysis is presented for space discretization, including the discontinuous Galerkin method and H¹-conforming finite elements with interior penalty on gradient jumps. Our main results are error estimates in various norms for smooth solutions. Two key ingredients are the post-processing of the fully discrete solution by lifting its jumps in time and a new time-interpolate of the exact solution. We first analyze the L^∞(L²) (at discrete time nodes) and L²(L²) errors and derive a superconvergent bound of order (τ^k+2 + h^r+1/2) for static meshes for k ≥ 1. Here, τ is the time step, k the polynomial order in time, h the size of the space mesh, and r the polynomial order in space. For the case of dynamically changing meshes, we derive a novel bound on the resulting projection error. Finally, we prove new optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm.