Let us have simplicial mesh and continuous function $u$ which is piece-wise linear and non-constant on every cell. Then normal vector to level-sets of $u$ is given $$\mathbf{n}=\frac{\nabla u}{|\nabla u|}$$ and is piece-wise constant function. What is then possible definition of curvature $\kappa$ which would be for $u\in{\cal C}^2$ $$\kappa = \mathrm{div}\,\mathbf{n}.$$ This definition is not suitable as it yields $\kappa$ being zero in every cell and a multiple of Dirac $\delta$ on every facet.

In conclusion: what is a suitable finite-element space for $\kappa$ and weak formulation for $\kappa = \mathrm{div}\,\mathbf{n}$ with given piece-wise constant $\mathbf{n}$?

EDIT: I got another idea: first project $\mathbf{n}$ on some $H(\mathrm{div})$-conforming space; then it is no problem calculating $\mathrm{div}\,\mathbf{n}$ directly. I did some numerical experiments with $u$ being projection of $$ u^\mathrm{exact} = (1+x)(1+y) $$ to CG1 (continuous, piece-wise linear) on domain $\Omega=(0,1)\times(0,1)$. Exact curvature for $u^\mathrm{exact}$ is $$ \kappa^\mathrm{exact} = \frac{-2(1+x)(1+y)}{[(1+x)^2+(1+y)^2]^{3/2}} .$$ Taking Raviart-Thomas degree 1 space (piece-wise linears, continuous on facet midpoints) or CG1 space as space for $\mathbf{n}$ method seems to converge in norms but with annoying wiggles on $\partial\Omega$. Here is calculated $\kappa$ $\kappa$

and error $\kappa-\kappa^\mathrm{exact}$ $\kappa-\kappa^\mathrm{exact}$

Can you see the problem?

  • $\begingroup$ I typically try to project derivatives onto one order lower spaces to get some averaging and avoid the wiggles. Depending on your purposes, you might try using piecewise constants for $\kappa$. $\endgroup$ – Bill Barth Jul 16 '13 at 12:51
  • $\begingroup$ Well, that's what I did - $\mathbf{n}\in$CG1/RT1 implies $\kappa\in$DG0. I think the problem may be that during projection: DG0$\rightarrow$RT1/CG1: $\frac{\nabla u}{|\nabla u|}\mapsto\mathbf{n}$ boundary values are subject to sorcery. I'm gonna try ignore them now. $\endgroup$ – Jan Blechta Jul 16 '13 at 13:16
  • $\begingroup$ OK, I didn't want to try to read anything into what you did. You could easily try to project back into the same space (which could cause a lot more trouble). Also, you plotted something that looks CG1, so I wasn't sure. $\endgroup$ – Bill Barth Jul 16 '13 at 14:35
  • $\begingroup$ Yes, this is boring post:) And yes, plots are misleading. This is FEniCS deficiency, it plots only CG1 interpolations:( $\endgroup$ – Jan Blechta Jul 16 '13 at 14:49
  • $\begingroup$ I don't think it's boring! Just trying to be perfectly clear about things. I think you're right that there's some issues near the boundaries. What happens as you refine the mesh? Do the errors grow or shrink? $\endgroup$ – Bill Barth Jul 17 '13 at 19:10

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