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?