[petsc-users] Discontinuous Galerkin and BDDC

[Stefano Zampini]

Il giorno mar 2 feb 2021 alle ore 14:06 Carsten Uphoff <
uphoff at geophysik.uni-muenchen.de> ha scritto:

> 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?
>

I don't think so. I know Lawrence gave it some thoughts but never heard
about a final solution about how to represent subdomain DG matrices via a
MATIS.

>
> - Does it make sense to apply PCBDDC on A_{h,C}? Could I combine an
> additive correction with PCBDDC using PCCOMPOSITE, e.g.?
>

You either use PCComposite or write a small PCSHELL that implements PCApply
as additive combination


> - Does anyone already test PCBDDC for DG?
>

Not that I know


> 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
>
>

-- 
Stefano
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