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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

IMR
PROCEEDINGS

17th International Meshing Roundtable
Pittsburgh, Pennsylvania, U.S.A.
October 12-15, 2008

Universite de Lyon 1, Laboratoire díAnalyse Numerique, agouzal@univ-lyon1.fr
Los Alamos National Laboratory, Theoretical Division, lipnikov@lanl.gov
Institute of Numerical Mathematics, vasilevs@dodo.inm.ras.ru

Introduction
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 [6], 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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