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

[Pierre.Saramito at imag.fr]

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
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