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 Cp-1, and finally we address the case of finite-elements with global regularity C0 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.

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

Bianchi D.
;
2018

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 Cp-1, and finally we address the case of finite-elements with global regularity C0 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.
Boundary control and observation
Finite-differences
Finite-elements
Isogeometric analysis
Non-uniform grids versus approximately weakly regular grids
Spectral gap
Spectral symbol
Velocity of propagation
Wave equation
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/338911
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