[petsc-users] Implementing FEM with PLEX/DS/SNES

[Matthew Knepley]

On Mon, Oct 8, 2018 at 5:50 PM Ellen M. Price <ellen.price at cfa.harvard.edu>
wrote:

> Can anyone shed some light on SNES example 12?
>
>
> https://www.mcs.anl.gov/petsc/petsc-current/src/snes/examples/tutorials/ex12.c.html
>
> I am solving a Poisson problem, but with a source term defined on mesh
> points, so I don't have an analytic form.


Can you explain more about this source term? It sounds like a bunch of
delta functions. That
would still work in this framework, but the convergence rate for a rhs with
these singularities
is reduced (this is a generic feature of FEM).


> The part I am confused about
> is the role of the functions f0, f1, and g3 that go into the DS.
>
> Suppose I have the Poisson problem nabla^2 u = f. Then the weak form is,
> for test function phi,
>
> \int_{\partial \Omega} \phi \nabla u - \int_\Omega \nabla u \cdot \nabla
> \phi = \int_\Omega \phi f
>

I think you need \int_{\partial \Omega} \phi \nabla u \cdot n instead.


> Based on the documentation of PetscDSSetResidual(), I'm a little
> confused about f0 and f1:
>
> \int_\Omega \phi f_0(u, u_t, \nabla u, x, t) + \nabla\phi \cdot {\vec
> f}_1(u, u_t, \nabla u, x, t)
>
> So f0 is, for the Poisson problem, zero, and f1 is just -grad u?


No. f1 is indeed -grad u, but f0 is -f.

Note that ex12 solves -nabla^2 u = f since that operator is positive
definite.


> Do we
> need to do anything with the surface term, or is that handled
> internally?


The surface term can be included using


https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/DT/PetscDSSetBdResidual.html

and we use that in ex12 for inhomogeneous Neumann conditions.


> Because neither of these terms are a surface integral term,
> both are volume integrals.
>
> I can't figure out the purpose of g0, g1, g2, g3 in
> PetscDSSetJacobian(), either. Is g supposed to be the source term?


No, it is the coefficient of the different combinations of basis functions
and their gradients.
For instance, the g3 term (grad phi and grad psi), g3 plays the same role
as the elasticity
tensor C in mechanics.


> Is
> psi a second basis function?


Yes. If the method is Galerkin, it comes from the same space as phi.


> What even *is* the Jacobian in a FEM
> problem?


The Frechet derivative of the residual.


> I've only seen them formulated as linear problems in tutorials,
> so I'm not even clear why we need a nonlinear solver or Jacobian...
>

The Frechet derivative of a linear operator is itself. ex12 also
incorporates
nonlinear variants of Poisson.


> Any help is appreciated. I like the idea of a general framework for FEM,
> as I think it will reduce human error on my part, but I also want to
> understand what I'm doing.
>

If you have any more questions, feel free to keep mailing.

  Thanks,

     Matt


> Ellen Price
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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