Thank you -- since my problem solves for a stream function, this was a very
helpful tip, and has made my solutions appear much better.
Jamie
On Mon, Aug 1, 2016 at 4:49 PM, Daniel Wheeler
wrote:
> Hi Jamie,
>
> Sorry for the delayed reply.
>
> On Thu, Jul 21, 2016 at 11:31 AM, James Pringle wrote:
> >
> > However, I am still getting large errors in estimates of the gradient
> even
> > when the mesh is nearly perfectly orthogonal. These errors DO NOT become
> > smaller as resolution is increased. I have two question:
> >
> > Are these errors to be expected in unstructured meshes?
>
> Cell centered FV has two issues with unstructured meshes. You know
> about the orthogonality issue, but there is a second issue. The
> docstring from the following code might help explain,
>
>
> https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_usnistgov_fipy_blob_develop_fipy_variables_cellVariable.py-23L257&d=DQICAg&c=c6MrceVCY5m5A_KAUkrdoA&r=7HJI3EH6RqKWf16fbYIYKw&m=F8ThJ6NxKtDlYIWkDUqGq2ql_J0shUHQMI1lQCB1Wvo&s=RM15FSI_eegFR5VfTb9il1PZ2B1_yaQ6eKmZHq-8Hx0&e=
>
> See, `\frac{1}{V_P} \sum_f \vec{n} \phi_f A_f`. This is the
> discretized gradient calculation. In this formulation using the
> triangular mesh, the calculated \phi_f is not the correct average for
> the face as the line between the cell centers doesn't cross the face
> in the middle of the face. I think this is called non-conjunctionality
> in finite volume speak. See equation 9.8 in the following link for
> further explanation (we should probably implement this approach).
>
>
> https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2aqRiHt&d=DQICAg&c=c6MrceVCY5m5A_KAUkrdoA&r=7HJI3EH6RqKWf16fbYIYKw&m=F8ThJ6NxKtDlYIWkDUqGq2ql_J0shUHQMI1lQCB1Wvo&s=EJ34X0r43RF50EgJ2p_clCS-H5yY8nSTitKGqtbINts&e=
>
> I believe that this is the source of the error in the gradient
> calculation. It's a consistent error independent of the mesh
> refinement I think (or maybe first order). An alternative gradient
> calculation is the least squares gradient, see
>
>
> https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_usnistgov_fipy_blob_develop_fipy_variables_cellVariable.py-23L270&d=DQICAg&c=c6MrceVCY5m5A_KAUkrdoA&r=7HJI3EH6RqKWf16fbYIYKw&m=F8ThJ6NxKtDlYIWkDUqGq2ql_J0shUHQMI1lQCB1Wvo&s=l6F4maMJnB3bfM6--FXXLzrUKyaH1i7KHuo1atLYJqM&e=
>
> Using that
>
>
> import fipy as fp
> nx = 25
> dx = 1. / nx
> mesh = fp.Tri2D(nx=nx, ny=nx, dx=dx, dy=dx)
> psi = fp.CellVariable(mesh=mesh)
> psi.constrain(1.0, where=mesh.facesLeft)
> psi.faceGrad.constrain(1.0, where=mesh.facesRight)
> eq = fp.DiffusionTerm().solve(psi)
> print psi.leastSquaresGrad[:, :100]
>
>
> seems to give lots of 1s and 0s which seems correct. The cells next to
> the boundary are not correct as they aren't using the boundary
> conditions in the calculation. I'm not sure about the order of
> accuracy of the least squares or why it happens to work for this mesh.
>
> > Is it acceptable to average the gradient over space to reduce the error?
>
> I don't think so unless it can be shown to improve the accuracy with
> analysis.
>
> --
> Daniel Wheeler
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