An approximation of anisotropic metrics from higher order interpolation error for triangular mesh adaptation.

Frédéric Hecht 1, 2, 3 Raphaël Kuate 3
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, INSMI - Institut National des Sciences Mathématiques et de leurs Interactions
Abstract : For any positive integer k, the (k+1)st-order tensor for the partial derivatives of a given order of a function at a point has its anisotropic behaviour characterized by a positive definite matrix, which involves a nonlinear minimization with respect to the matrix. Using the error estimates described in [W. M. Cao, SIAM J. Numer. Anal. 45 (2007), no. 6, 2368–2391; MR2361894 (2008k:65264)] and [F. Hecht, in Numerical analysis and scientific computing for partial differential equations and their challenging applications, 108–120, CIMNE, 2008; per bibl.], a formulation of the optimization problem is given and an algorithm is presented for its resolution [cf. J.-M. Mirebeau, Constr. Approx. 32 (2010), no. 2, 339–383; MR2677884 (2011g:65278)]. The main purpose of this paper is to perform anisotropic mesh adaptation for numerical simulations with a Lagrange finite element approximation of degree k, k>1. Numerical experiments of metrics computed, and examples of mesh adaptation for a function with metrics generated by the algorithm presented, are also given.
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Frédéric Hecht, Raphaël Kuate. An approximation of anisotropic metrics from higher order interpolation error for triangular mesh adaptation.. Journal of Computational and Applied Mathematics, Elsevier, 2014, 258, pp.99-115. ⟨hal-01105158⟩

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