Numerical meshes ensuring uniform observability of one-dimensional waves: Construction and analysis

Ervedoza S, Marica A, Zuazua E (2016)

Publication Type: Journal article

Publication year: 2016

Journal

IMA Journal of Numerical Analysis Oxford University Press (OUP): Policy A - Oxford Open Option A

Book Volume: 36

Pages Range: 503-542

Journal Issue: 2

DOI: 10.1093/imanum/drv026

Abstract

We build nonuniform numerical meshes for the finite difference and finite element approximations of the one-dimensional wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a nonuniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without ever getting to the boundary. Our results show that this pathology can be avoided by taking suitable nonuniform meshes. This also allows us to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Université Fédérale de Toulouse Midi-Pyrénées FR France (FR) Politechnic University of Bucharest / Universitatea Politehnica din București RO Romania (RO) Basque Center of Applied Mathematics (BCAM) ES Spain (ES)

How to cite

APA:

Ervedoza, S., Marica, A., & Zuazua, E. (2016). Numerical meshes ensuring uniform observability of one-dimensional waves: Construction and analysis. IMA Journal of Numerical Analysis, 36(2), 503-542. https://dx.doi.org/10.1093/imanum/drv026

MLA:

Ervedoza, Sylvain, Aurora Marica, and Enrique Zuazua. "Numerical meshes ensuring uniform observability of one-dimensional waves: Construction and analysis." IMA Journal of Numerical Analysis 36.2 (2016): 503-542.

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