### 2014-02

by Ern, A.; Schieweck, F..

**Series:** 2014-02, Preprints

- MSC:
- 65M12 Stability and convergence of numerical methods
- 65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 65J10 Equations with linear operators (do not use [[65Fxx|65Fxx]])

**Abstract:**

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 $H^1$-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 $L^infty(L^2)$ and $L^2(L^2)$ errors and

derive a super-convergent bound of order $(tau^{k+2}+h^{r+1/2})$

in the case of static meshes for $kge 1$.

Here, $tau$ 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.

**Keywords:**

discontinuous Galerkin in time, stabilized FEM, first-order PDEs, graph norm error estimates, superconvergence, dynamic meshes