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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2011

Continuous Mesh Framework Part II: Validations and Applications

Résumé

This paper gives a numerical validation of the continuous mesh framework introduced in Part I [A. Loseille and F. Alauzet, SIAM J. Numer. Anal., 49 (2011), pp. 38-60]. We numerically show that the interpolation error can be evaluated analytically once analytical expressions of a mesh and a function are given. In particular, the strong duality between discrete and continuous views for the interpolation error is emphasized on two-dimensional and three-dimensional examples. In addition, we show the ability of this framework to predict the order of convergence, given a specific adaptive strategy defined by a sequence of continuous meshes. The continuous mesh concept is then used to devise an adaptive strategy to control the L p norm of the continuous interpolation error. Given the L p norm of the continuous interpolation error, we derive the optimal continuous mesh minimizing this error. This exemplifies the potential of this framework, as we use a calculus of variations that is not defined on the space of discrete meshes. Anisotropic adaptations on analytical functions correlate the optimal predicted theoretical order of convergence. The extension to a solution of nonlinear PDEs is also given. Comparisons with experiments show the efficiency and the accuracy of this approach.
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Dates et versions

hal-03167252 , version 1 (15-03-2021)

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Adrien Loseille, Frédéric Alauzet. Continuous Mesh Framework Part II: Validations and Applications. SIAM Journal on Numerical Analysis, 2011, 49 (1), pp.61-86. ⟨10.1137/10078654X⟩. ⟨hal-03167252⟩
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