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Convergence of lowest order semi-Lagrangian schemes

by H. Heumann and R. Hiptmair

(Report number 2011-47)

Abstract
We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in $R^n$. We provide comprehensive a priori convergence estimates for their spatio-temporal discretization by means of a semi-Lagrangian approach combined with a discontinuous Galerkin method. Under rather weak assumptions on the velocity underlying the advection we establish an asymptotic $L^2$-estimate $O(\tau + h^r + h^{r+1} \tau^{-1/2} + \tau^{1/2})$, where $h$ is the spatial meshwidth, $\tau$ denotes the timestep, and $r$ the polynomial degree of the forms used as trial functions. This estimate can even be improved considerably in a variety of special settings.

Keywords:

@Techreport{HH11_82,
  author = {H. Heumann and R. Hiptmair},
  title = {Convergence of lowest order semi-Lagrangian schemes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-47.pdf },
  year = {2011}
}

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