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Refined convergence theory for semi-Lagrangian schemes for pure advection

by H. Heumann and R. Hiptmair

(Report number 2011-60)

Abstract
We consider generalized linear transient advection 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. We establish a new asymptotic estimate O(h^(r+1)/tau^(I I -1/2)) for the L^2-norm of the error, where h is the spatial meshwidth I I denotes the timestep, and r is the polynomial degree of the piecewise polynomial discrete differential forms used as trial functions. Numerical experiments hint that the estimate is sharp for certain trial spaces and may be sub-optimal for others.

Keywords:

@Techreport{HH11_112,
  author = {H. Heumann and R. Hiptmair},
  title = {Refined convergence theory for semi-Lagrangian schemes for pure advection},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-60},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-60.pdf },
  year = {2011}
}

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