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2014-02

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

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