[petsc-users] Discontinuous Galerkin and BDDC

[Carsten Uphoff]

Hi everyone,

I'm interested in testing a BDDC preconditioner for Poisson and 
Elasticity equations using the symmetric interior penalty Galerkin 
method. However, I wonder how one would apply PCBDDC for discontinuous 
Galerkin.

The major problem is that for DG you cannot write the bilinear form as a 
sum of local bilinear forms which only involve degrees of freedom of the 
respective local subdomain. In particular, coupling terms at the 
interface of two subdomains, such as [[u]], require DOFs from two 
subdomains. Therefore, it is not straightforward to write the operator A 
as sum of local operators in the form
A = sum_{i=1}^N R_i A_i R_i^T
where A_i are local operators and R_i is the local-to-global map.

In the literature, I found two possible solutions:
- Double degrees of freedom at the subdomain interface [1]
- Split the bilinear form a_h in the two parts a_{h,D} and a_{h,C}, 
where the first leads to an easy-to-invert operator that is 
discontinuous across the subdomain interface, and the second is 
continuous across the subdomain interface. As a_{h,C} is continuous, one 
may write the bilinear form as sum of local bilinear forms only 
involving the local degrees of freedom [2]

The first approach [1] seems unattractive as you double the DOFs in the 
Schur complement. For [2] I think one might be able to apply PCBDDC on 
A_{h,C} and apply A_{h,D}^{-1} as an additive correction, cf. (2.23) in [2].

Questions:
- Is there any straightforward way to apply PCBDDC for DG which I am 
missing?
- Does it make sense to apply PCBDDC on A_{h,C}? Could I combine an 
additive correction with PCBDDC using PCCOMPOSITE, e.g.?
- Does anyone already test PCBDDC for DG?

I appreciate your help and I'm looking forward for your comments!

Best regards,
Carsten

[1] Dryja and Galvis and Sarkis, Numer. Math. 131:737-770, 2015, 
doi:10.1007/s00211-015-0705-x
[2] Brenner and Park and Sung, ETNA 46:190-214, 2017, 
http://etna.mcs.kent.edu/vol.46.2017/pp190-214.dir/pp190-214.pdf