Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations

TL;DR

This paper introduces a spectral Variational Multi-Scale method for 1D parabolic problems, providing exact sub-grid scale calculation, error estimates, and an efficient offline/online solution strategy that improves accuracy over existing stabilized methods.

Contribution

The paper develops a spectral VMS approach for 1D advection-diffusion equations, with exact sub-grid scale computation and proven error estimates, enhancing numerical accuracy and efficiency.

Findings

Spectral VMS matches implicit Euler semi-discretisation at interpolation nodes.

Error estimates confirm method's accuracy for finite element discretisations.

Numerical tests show improved accuracy over stabilized methods.

Abstract

In this work, we introduce a Variational Multi-Scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements. We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity. We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods.