Choosing the penalty for Discontinuous Galerkin

Question

So I am attempting to solve a 3D poisson problem with Discontinuous Galerkin (interior penalty method). The weak form (written in FEniCS) is as following:

a = dot(grad(v), grad(u))*dx \
  - dot(avg(grad(v)), jump(u, n))*dS \
  - dot(jump(v, n), avg(grad(u)))*dS \
   + alpha/h_avg*dot(jump(v, n), jump(u, n))*dS \
   - dot(grad(v), u*n)*ds(1) \
   - dot(v*n, grad(u))*ds(1) \
   + (gamma/h)*v*u*ds(1)
L = v*f*dx - u0*dot(grad(v), n)*ds(1) + (gamma/h)*u0*v*ds(1) + g*v*ds(2)

where u/v are the trial/test functions, u0 is Dirichlet BC, g is Neumann BC, h is cell size, n is facet normal, and alpha and gamma are the penalties.

For many 2D unit square problems such as one of the undocumented FEniCS/DOLFIN examples, I have seen alpha and gamma set to 4 and 8 respectively, but when I do a 3D problem, I need much higher values like 40 and 80. How do I systematically determine what the values for alpha and gamma need to be?

Answer 1

The penalty usually scales as $O(p^2/h)$ for polynomial discontinuous Galerkin methods regardless of dimension, but for a more precise constant, it's usually enough to take the penalty large enough to guarantee coercivity with respect to the DG norm. Shahbazi gives an expression for the penalty over a given face which just depends on $p$ and geometric mapping factors for volume/faces of simplicial elements, which is then refined to include mesh angles by Epshteyn and Riviere.

For succinctness, Shahbazi's penalty constant $\gamma_f$ over a given face $f$ is roughly

$$\gamma_f = \max(c_{K^+},c_{K^-})$$

where $K^+, K^-$ are the elements which share that common face, and $c_K$ is an order-dependent constant for each element

$$c_K = \frac{(p+1)(p+d)}{d}\frac{\text{Area of face}}{\text{Volume of element}}.$$

The ratio of face area to element volume is $\approx h$, so this has the $O(p^2/h)$ scaling expected (you can also sharpen this slightly by scaling the contributions of non-boundary faces by 1/2). You can also take the global penalty as the max over the individual penalties, but this may worsen conditioning.

I'm less sure of what the optimal $\alpha$ constant is, since it's not strictly necessary for well-posedness of SIPDG formulations.