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 <daniel.wheel...@gmail.com> wrote: > Hi Jamie, > > Sorry for the delayed reply. > > On Thu, Jul 21, 2016 at 11:31 AM, James Pringle <jprin...@unh.edu> 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 > _______________________________________________ > fipy mailing list > fipy@nist.gov > > https://urldefense.proofpoint.com/v2/url?u=http-3A__www.ctcms.nist.gov_fipy&d=DQICAg&c=c6MrceVCY5m5A_KAUkrdoA&r=7HJI3EH6RqKWf16fbYIYKw&m=F8ThJ6NxKtDlYIWkDUqGq2ql_J0shUHQMI1lQCB1Wvo&s=A7D0acjE011uAk-Vl-5OhrTNVdH9kgmq0rshSN6tbsg&e= > [ NIST internal ONLY: > https://urldefense.proofpoint.com/v2/url?u=https-3A__email.nist.gov_mailman_listinfo_fipy&d=DQICAg&c=c6MrceVCY5m5A_KAUkrdoA&r=7HJI3EH6RqKWf16fbYIYKw&m=F8ThJ6NxKtDlYIWkDUqGq2ql_J0shUHQMI1lQCB1Wvo&s=V8y_Fr-VcNq0fTtFnfiyabj8MXL8mmGa5LFCMv0Sx_Q&e= > ] >
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