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NIA Seminar by Bram van Leer |
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Date: November 3, 2005
Time: 10:30am
Location: NIA, Rm 137
Discontinuous Galerkin for Diffusion
Bram van Leer, University of Michigan
Discontinuous Galerkin methods are the Finite Element analyst's answer to Finite Voluime methods. Originally inspired by upwind (Godunov-type) methods for the advection equation and hyperbolic systems, the DG community soon turned to the diffusion equation, with much less success.
It seems that the DG approach is fundamentally unsuited for second-order operators. The most successful method of today, the Local Discontinuous Galerkin method of Shu and Cockburn requires that the diffusion equation be rewwritten as a system of first-order equations.
In this lecture I will first show that there is absolutely no conflict between the DG approach and the diffusion equation. In order to make it work two insights are needed:
- the realization that there are multiple representations of the
numerical solution which all are equivalent in the weak sense,
and that one may have to switch between these for the sake of
getting useful schemes;
- for a second-order PDE integration by parts must be done TWICE in order
to obtain the DG equations - which is not standard DG practice.
Next, I will present the Reovery method, developed from the above starting points. In particular, a smooth locally recovered solution is used that in the weak sense is indistinguishable from the discontinuous discrete solution. The recovery principle creates schemes that are not included in the family of traditional DG diffusion schemes, and are potentially more accurate. A way is presented to extend the family so that the recovery-based schemes are included.
An eigenvalue/eigenvector analysis suggests that the order of accuracy of the recovery schemes is at least 2p + 2, where p is the highest order of the basis functions used, and maybe as high as 2**(p+2}.
This conclusion is supported by 1-D numerical tests, in which the new and old schemes are pitted against each other.
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