# [petsc-users] 3D Model Coupled to 0D Model

Jed Brown jed at jedbrown.org
Wed Aug 10 07:17:22 CDT 2022

```Have you considered PCFieldSplit? The SCHUR variant gives you all sorts of approximate factorization schemes, though you might find that an additive or multiplicative split is almost as good for this problem.

"Karabelas, Elias (elias.karabelas at uni-graz.at)"	<elias.karabelas at uni-graz.at> writes:

> Hey all,
>
> so I have a nonlinear problem that can be abstractly written as
>
> | F(u,P) |           | 0 |
>
> |        |    =      |   |
>
> | G(u,P) |           | 0 |
>
>
> Here u is a variable that comes from a discretization of a PDE and P are four scalars that come from some coupled attached ODEs (P = (P1,P2,P3,P4) )
>
> I know abstract how to apply Newton-Raphson in this context as the Jacobian is simply
>
> d_u F   d_P F
>
> d_u G   d_P G
>
> where d_P F and d_u G are formed from 4 Vecs resp, and d_P G is a 4x4 matrix.
>
> So what I have troubles with is how I could squeeze something like this into an SNES context, at the moment I'm doing a Schur-Complement for Solving this problem for each Newton solve. This however entails, that I'm, solving (d_u F)^-1 to a very low tolerance inside the SC.
> In the End I want to have something that can work with an inexact Newton method, but I don't know which would be the correct tool (MATSHELL for the jacobian maybe?) to squeeze this into an SNES.
>
> Any ideas?
>
> Best regards
> Elias
>
>
> --
> Dr. Elias Karabelas
> Research Associate
> University of Graz
> Institute of Mathematics and Scientific Computing
> Heinrichstraße 36
> A-8010 Graz
> Austria
>
> Phone: +43 316 380 8546
> Email: elias.karabelas at uni-graz.at<mailto:elias.karabelas at uni-graz.at>
> Web:  https://ccl.medunigraz.at/
```