[Gmsh] Curved high order gmsh meshes are not really high order

[Pierre Saramito]

Dear gmsh developers and users,

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 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.

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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