Convergence of Lowest Order Semi-Lagrangian Schemes |
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Authors: | Holger Heumann Ralf Hiptmair |
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Institution: | 1. LJAD, Université Nice-Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France 2. SAM, ETH Zürich, R?mistrasse 101, 8092, Zürich, Switzerland
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Abstract: | We consider generalized linear transient advection-diffusion problems for differential forms on a bounded domain in ? d . We provide comprehensive a priori convergence estimates for their spatiotemporal discretization by means of a first-order in time 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 of order $O(\tau+h^{r}+h^{r+1}\tau^{-\frac{1}{2}}+\tau^{\frac{1}{2}})$ , where h is the spatial meshwidth, τ denotes the time step, and r is the polynomial degree of the forms used as trial functions. This estimate can be improved considerably in a variety of special settings. |
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