[Gmsh] Are curved and high order gmsh meshes really high order ?

[Christophe Geuzaine]

Hello Pierre,

Indeed. We are currently working on improving the algorithm to correct this.

Thanks for the feedback!

Christophe

On 12 Jan 2012, at 22:52, Pierre.Saramito at imag.fr wrote:

> Dear gmsh developers and users,
> 
> It seems that there is a major problem with the computation of internal node for curved 2d and 3d elements : an isoparametric Pk finite element computation of a basic PDE problem does not converge as expected with such meshes.
> 
> I have tested with the following problem:
>    - Laplace u = f in the domain
>    u           = 0 on the boundary
> The domain is the unit circle. The right-hand side f is computed such that the exact solution is u(x) = cos(pi*r) with r=sqrt(x1^2 + x2^2).
> 
> I put a small .pdf in attachment: Fig. 2, left column shows the convergence curves |uh - pi_h(u)| for various norms (L2, L^infty, H1), where pi_h(u) denotes the Lagrange interpolation of the exact solution u, and uh is the isoparametric finite element solution on the mesh, as generated by gmsh.
> 
> I do not known how internal nodes are computed in gmsh.
> Nevertheless, by using a blending formulae, such as eqn (27) in:
>  S. Dey, M. S. Shephard and J. E. Flaherty.
>  Geometry representation issues associated with p-version finite
>  element computations.
>  Comput. Meth. Appl. Mech. Engrg., 150:39-55, 1997.
> for the computation of boundary and internal nodes, then the isoparametric finite element method converges as expected. See Fig. 2, right column. Fig. 1 represents the difference between gmsh nodes and those obtained by this procedure: this difference is subtle ; nevertheless it has a dramatic effect on convergence properties. All finite element computations has been performed with the development version of Rheolef :
>  http://www-ljk.imag.fr/membres/Pierre.Saramito/rheolef
> 
> Please, could you check the convergence ?
> If it is confirmed, could you consider in the future a boundary and internal node placement procedure in such way that isoparametric FEM converges as expected ?
> 
> I take this opportunity to congratulate all the developers for the wonderful features in gmsh : the only one to my knowledge that consider high-order and curved elements.
> 
> Pierre
> -- 
> Pierre.Saramito at imag.fr
> Directeur de Recherche CNRS
> Laboratoire Jean Kuntzmann, Grenoble, France
> http://www-ljk.imag.fr/membres/Pierre.Saramito
> <curved-convergence.pdf>_______________________________________________
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-- 
Prof. Christophe Geuzaine
University of Liege, Electrical Engineering and Computer Science 
http://www.montefiore.ulg.ac.be/~geuzaine