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# Soon Capturing and Frequency Analysis for Mesh Adaptive Interpolation

1 SINUS - Numerical Simulation for the Engineering Sciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Let us call a {\bf highly heterogeneous function} a function that is either locally singular or a smooth function but, with too small details in comparison with domain size. We study the $L^2$ norm of the interpolation error $E_h$ between a function $u$ and $\Pi_h$ $u$ its $P1$ continuous interpolate: we use four examples of functions, that represent different cases of {\bf highly heterogeneous functions}. Comparing first the convergence of $E_h$ as a function of number of nodes on uniform or adaptive meshes, we observe a convergence of order 2, only for a smooth function when the number of nodes is sufficiently large, when an uniform sequence of mesh is choosen. Conversely, almost always holds second-order convergence when an adaptive mesh algorithm is applied. We give some theoretical arguments concerning this phenomenon. Following some ideas currently used in spectral methods, we consider the $P1$ approximation of $u$ on nested meshes and express the representation of $u_h$ as a {\bf series} with increasing fineness of its terms. The size of each terms as a function of the corresponding level number is examined.
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https://hal.inria.fr/inria-00073971
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Submitted on : Wednesday, May 24, 2006 - 2:12:25 PM
Last modification on : Friday, February 4, 2022 - 3:15:09 AM
Long-term archiving on: : Thursday, March 24, 2011 - 1:34:04 PM

### Identifiers

• HAL Id : inria-00073971, version 1

### Citation

Bernadette Palmerio, Alain Dervieux. Soon Capturing and Frequency Analysis for Mesh Adaptive Interpolation. RR-2722, INRIA. 1995. ⟨inria-00073971⟩

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