Metric Generation for a Given Error Estimation
Hecht, Frederic and Raphael Kuate
Proceedings, 17th International Meshing Roundtable, Springer-Verlag, pp.569-584, October 12-15 2008
17th International Meshing Roundtable
Pittsburgh, Pennsylvania, U.S.A.
October 12-15, 2008
Laboratoire Jacques-Louis Lions, Universit¥e Pierre et Marie Curie
The mesh adaptation is a classical method for accelerating and improving
the PDE finite element computation. Two tools are generally used: the metric  to
define the mesh size and the error indicator to know if the solution is accurate enough.
A lot of algorithms used to generate adapted meshes suitable for a PDE numerical
solution for instance [15, 1] for discrete metrics, or [15, 2] for the continuous one, use
those tools for the local specification of the mesh size.
Lot of PDE softwares like Freefem++  use metrics to build meshes, the edges
sizes of which are equal with respect to the metric field. The construction of metrics
from the hessian matrix [16, 18, 13] is only justify for the piecewise linear Lagrange
So there is the problem of metric generation when we have another interpolation
error estimator [6, 11] that could be used for instance when the Lagrange interpolation
needed is a k degree polynomial, k > 1.
To answer that question, we propose in this paper an algorithm whose complexity
is quasi-linear, in two spacial dimensions; assuming that the error is locally described
by a closed curve representing the error level set. Some efficient numerical examples
are given. This algorithm allows us to obtain the analytical metric , when the error
indicator is based on the hessian matrix.
We have also done one comparison in the software Freefem++ of mesh adaptation
with metrics computed using this algorithm with respect to the interpolation error
estimation described in , and the method with metrics based on the hessian. The
results seem to be better for the maximal error.
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