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I adapted this FEniCS code for my problem and I'm wondering if there is any good resource about how to choose the penalty parameter $\alpha$? Best case would be, if I can define it through some relation of the diffusion coefficient $D$ and element height $h$.

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    $\begingroup$ I think it would really help if you describe the problem you are trying to solve. $\endgroup$ – Anton Menshov Oct 8 '18 at 2:41
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TL;DR: Let $k$ be the diffusion coefficient, $\theta$ the minimum angle between any two edges of the mesh, $d$ the space dimension, and $p$ the polynomial degree of the finite element basis you're using. Then taking

$$\alpha \ge \frac{2 \cdot d \cdot (p + d - 1)}{\cos\theta\cdot \tan\theta/2} \cdot \frac{\max k}{\min k}$$

should guarantee that the discretized diffusion operator will be positive-definite. You'll still have to be careful about which numerical flux you use for the convective part in order to guarantee stability.

DG in part evolved from Nitsche's method for weakly imposing Dirichlet boundary conditions, so a lot of the relevant information for this is contained in the literature on Nitsche's method. When using either the symmetric interior penalty DG method or Nitsche's method, the penalty parameter $\alpha$ has to be greater than a certain lower bound in order to guarantee positivity of the resulting linear operator. It would be nice if there were a formula in terms of the diffusion coefficient and the element shape so that you could select a good value $\alpha$ without any guesswork.

Unfortunately a lot of papers on DG or Nitsche's method never address this point. Hansbo 2005 has an analytical lower-bound in terms of the element shape, but only for linear finite elements. It has some discussion of Nitsche's method which you can apply equally well to DG, so although more limited in scope it's worth the read. Warburton and Hesthaven 2003 has the full analysis for higher-order finite elements. I also have some example code demonstrating Nitsche's method for the Poisson equation using firedrake which, if you're not familiar with it, shares a lot of features with fenics.

Another paper that I think is worth looking at is Freund and Stenberg 1995, in particular for the discussion of convection-diffusion problems like the one you're solving. In that case, the operator is non-symmetric, so it might arguably make more sense to use the asymmetric interior penalty method. This approach has no lower bound on the size of the penalty parameter, it just has to be positive.

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  • $\begingroup$ This is actually the most accurate answer I've ever get in any internet forum. Thanks a lot, Daniel! $\endgroup$ – Maxi Köhler Oct 9 '18 at 21:34

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