Re: [Libmesh-users] High order geometry and finite volume methods

[John P. <jwp...@gm...>]

On Fri, Sep 7, 2018 at 12:16 PM Miguel Angel Salazar de Troya <
sal...@gm...> wrote:

> Hi,
>
> I want to solve a simple Poisson equation with a diffusion coefficient, a
> simple reaction term and apply Adaptive Mesh Ref (AMR). The problem is that
> if the diffusion coefficient is  much smaller than the mesh resolution (as
> it can happen before applying AMR), I obtain over and undershoots of the
> solution. I need this solution to be stable and not be over 1 or under 0
> (bounds from the source term).
>
> One way I was able to achieve this was by implementing a Finite Volume
> solver within libMesh. Very easy for simple cartesian geometries. However,
> now I need to extend it to non-cartesian geometries that are second order.
> I am wondering if there are alternatives in libMesh to solve this
> over/undershoot problem for the poisson equation.
>

If you have a reaction-dominated reaction/diffusion equation, you may be
able to stabilize it using a Galerkin least-squares (GLS) formulation.

There is a good discussion in the 1995 paper by Thompson:
https://doi.org/10.1002/nme.1620380303

-- 
John