DISCONTINUOUS GALERKIN FOR DIFFUSION

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. One of the more successful methods of today, the Local Discontinuous Galerkin method of Shu and Cockburn, requires that the diffusion equation be rewritten as a system of first-order equations. While working with first-order systems is computationally advantageous, and a general trend in CFD, it evades the question how to directly discretize a second-order operator.

In this lecture I will first show there are essential differences between discretizing advection and diffusion equations: what works for one does not work for the other, and vice versa. This means that, when formulating a DG method for diffusion, one can not blindly copy what's done for the advection equation.

I will show, however, there is absolutely no conflict between the DG approach and the diffusion equation. In order to make it work two insights are needed:

(1) 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;

(2) 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 Recovery method, developed from the above starting points. Specifically, 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 some recovery-based schemes are included.

An eigenvalue/eigenvector analysis suggests that the order of accuracy of the recovery schemes may be as high as 2**(p+2}, i.e., exponential. This conclusion is supported by numerical tests.