
Metric Generation for a Given Error Estimation
Hecht, Frederic and Raphael Kuate
Proceedings, 17th International Meshing Roundtable, SpringerVerlag, pp.569584, October 1215 2008

INTERNATIONAL MESHING ROUNTABLE

17th International Meshing Roundtable
Pittsburgh, Pennsylvania, U.S.A.
October 1215, 2008
Laboratoire JacquesLouis Lions, UniversitĀ„e Pierre et Marie Curie
hecht@ann.jussieu.fr,kuate@ann.jussieu.fr
Abstract
The mesh adaptation is a classical method for accelerating and improving
the PDE finite element computation. Two tools are generally used: the metric [10] 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++ [14] 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
finite element.
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 quasilinear, 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 [1], 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 [11], and the method with metrics based on the hessian. The
results seem to be better for the maximal error.
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