Variational Multi-Scale method with spectral approximation of the sub-scales: Application to 1-d convection-diffusion equation
Bibliography:
Bibliography
Tomas Chacon Rebollo, Ben Mansour Dia, "A Variational Multi-Scale method with spectral approximation of the sub-scales: Application to 1-d convection-diffusion equation", Comput. Methods Appl. Mech. Engrg 285 (2015) 406-426.
Authors:
Tomas Chacon Rebollo, Ben Mansour Dia
Keywords:
Variational Multiscale, Convection-Diffusion, Stabilization, Spectral Approximation
Abstract:
This paper introduces a variational multi-scale method where the sub-grid scales
are computed by spectral approximations. It is based upon an extension of the
spectral theorem to non necessarily self-adjoint elliptic operators that have an associated base of eigenfunctions which are orthonormal in weighted L2 spaces. This
allows to element-wise calculate the sub-grid scales by means of the associated spectral expansion. We propose a feasible VMS-spectral method by truncation of this
spectral expansion to a finite number of modes. We apply this general framework to
the convection-diffusion equation, by analytically computing the family of eigenfunctions. We perform a convergence and error analysis. We also present some numerical
tests that show the stability of the method for an odd number of spectral modes,
and an improvement of accuracy in the large resolved scales, due to the adding of
the sub-grid spectral scales.
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