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

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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.

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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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Authors and Affiliations

  1. Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100, Como, Italy

    Davide Bianchi & Stefano Serra-Capizzano

  2. Department of Information Technology, Uppsala University, Box 337, SE-751 05, Uppsala, Sweden

    Stefano Serra-Capizzano

Authors
  1. Davide Bianchi
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  2. Stefano Serra-Capizzano
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Correspondence to Davide Bianchi.

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This work was supported by INdAM-GNCS and the Grant FAR 2015/16 from the University of Insubria.

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Bianchi, D., Serra-Capizzano, S. Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids. Calcolo 55, 47 (2018). https://doi.org/10.1007/s10092-018-0288-x

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