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Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids

  • Davide Bianchi
  • Stefano Serra-Capizzano
Article
  • 39 Downloads

Abstract

We study the gap of discrete spectra of the Laplace operator in 1d for non-uniform meshes by analyzing the corresponding spectral symbol, which allows to show how to design the discretization grid for improving the gap behavior. The main tool is the study of a univariate monotonic version of the spectral symbol, obtained by employing a proper rearrangement via the GLT theory. We treat in detail the case of basic finite-difference approximations. In a second step, we pass to precise approximation schemes, coming from the celebrated Galerkin isogeometric analysis based on B-splines of degree p and global regularity \(C^{p-1}\), and finally we address the case of finite-elements with global regularity \(C^0\) and local polynomial degree p. The surprising result is that the GLT approach allows a unified spectral treatment of the various schemes also in terms of the preservation of the average gap property, which is necessary for the uniform gap property. The analytical results are illustrated by a number of numerical experiments. We conclude by discussing some open problems.

Keywords

Wave equation Boundary control and observation Finite-differences Finite-elements Isogeometric analysis Velocity of propagation Non-uniform grids versus approximately weakly regular grids Spectral symbol Spectral gap 

Mathematcis Subject Classification

65F10 65N22 15A18 15A12 47B65 

Notes

Acknowledgements

We are grateful to Professor Enrique Zuazua for the time he dedicated to us and to this work, for discussions and illuminating advices. Without his help this paper would have never came to light. A special thank goes to Carlo Garoni for his careful reading and his numerous appropriate suggestions. Finally, we thank a lot the Editor and the Reviewers for their comments and criticisms which helped us both in improving the quality of the presentation and in making the content more complete.

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Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Department of Science and High TechnologyUniversity of InsubriaComoItaly
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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