Alexandre Ern
CERMICS, Ecole nationale des ponts et chaussees, Champs sur Marne, France

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. We first analyze the L∞(L2) and L2(L2) errors and derive a super-convergent bound of order (τ^{k+2}+h^{r+1/2}) in the case of 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 new bound on the resulting projection error. Finally, we prove optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm. This is joint work with F. Schieweck (University of Magdeburg).