Hessian-free metric-based mesh adaptation viageometry of interpolation error
Agouzal, Abdellatif, Konstantin Lipnikov, and Yuri Vassilevski
Research Notes, 17th International Meshing Roundtable, Springer-Verlag, pp.1-5, October 12-15 2008
17th International Meshing Roundtable
Pittsburgh, Pennsylvania, U.S.A.
October 12-15, 2008
Universite de Lyon 1, Laboratoire díAnalyse Numerique, firstname.lastname@example.org
Los Alamos National Laboratory, Theoretical Division, email@example.com
Institute of Numerical Mathematics, firstname.lastname@example.org
Generation of meshes adapted to a given function u requires a specially de-
signed metric. For metric derived from the Hessian of u, optimal error es-
timates for the interpolation error on simplicial meshes have been proved
in [2, 5, 8, 10, 11]. The Hessian-based metric has been successfully applied
to adaptive solution of PDEs [4, 7, 9]. However, theoretical estimates have
required to make an additional assumption that the discrete Hessian approx-
imates the continuous one in the maximum norm. Despite the fact that this
assumption is frequently violated in many Hessian recovery methods, the gen-
erated adaptive meshes still result in optimal error reduction.
In this article we continue the rigorous analysis [1, 3] of an alternative way
for generating a space tensor metric using the error estimates prescribed to
mesh edges. The new methodology produces meshes resulting in the optimal
reduction of the P1-interpolation error or its gradient. We define a tensor
metric M such that the volume and the perimeter of a simplex measured
in this metric control the norm of error or its gradient. The equidistribution
principle, which can be traced back to DíAzevedo , suggests to balance M-
volumes and M-perimeters. This leads to meshes that are quasi-uniform in
the metric M.
The paper outline is as follows. In Section 2, we derive appropriate metrics
from analysis of the interpolation errors. In Section 3, we present the algorithm
for generating adaptive meshes and its application to a model problem.
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