 
Summary: A NUMERICAL STUDY OF SOME HESSIAN RECOVERY
TECHNIQUES ON ISOTROPIC AND ANISOTROPIC MESHES
MARCO PICASSO, FR´ED´ERIC ALAUZET, HOUMAN BOROUCHAKI, AND
PAULLOUIS GEORGE
Abstract. Spaces of continuous piecewise linear finite elements are considered to solve a Poisson
problem and several numerical methods are investigated to recover second derivatives. Numerical
results on 2D and 3D isotropic and anisotropic meshes indicate that the quality of the results is
strongly linked to the mesh topology and that no convergence can be insured in general.
1. Introduction. It is well known from standard finite element textbooks [8]
that optimal a priori error estimates for continuous piecewise linear finite elements in
the framework of elliptic problems involve the second derivatives of the exact solution:
(u  uh) L2() ChuH2(). (1.1)
Hereabove, u is the exact (unknown) solution of the elliptic problem in the domain
of R2
or R3
, h denotes the mesh size, uh is the finite element approximation, and C
is a positive value independent of h and u but dependent on the aspect ratio of the
finite elements, triangles or tetrahedra for instance.
Recently, adaptive finite elements with large aspect ratio have been used with
success for various engineering applications involving sharp boundary layers, see for
