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

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 $k\ge 1$. 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.
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Alexandre Ern, Friedhelm Schieweck. Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs. Mathematics of Computation, American Mathematical Society, 2016, 85 (301), pp.2099-2129. ⟨10.1090/mcom/3073 ⟩. ⟨hal-00947695v2⟩

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