A variational multi-scale method with spectral approximation of the sub-scales: Application to the 1D advection-diffusion equations
KAUST DepartmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Applied Mathematics and Computational Science Program
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AbstractThis 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.
SponsorsThe research of T. Chacon Rebollo has been partially funded by Junta de Andalucia "Proyecto de Excelencia" Grant P12-FQM-454.