Variational multi-scale spectral solution of convection-dominated parabolic problems

TL;DR

This paper extends the variational multiscale spectral method to convection-dominated parabolic problems, providing error estimates and numerical validation for advection-diffusion equations.

Contribution

It introduces a spectral variational multiscale approach for parabolic problems with convection dominance, including error analysis and numerical tests.

Findings

Error estimates for convection-dominated flows

Validation on 1D advection-diffusion equations

Spectral method effectively stabilizes solutions

Abstract

In this work, we consider an extension to parabolic problems of the variational multiscale method with spectral approximation of the sub-scales. We first discretize in time using a finite difference scheme and second, apply the generalization of the spectral variational multi-scale method. To obtain error estimations in convection-dominated flows, we find a helpful link between the stabilized term expressed in terms of Green's functions and in terms of spectral functions. Finally, we present some numerical tests to show the reliability of the method. We consider the stationary one-dimensional advection-diffusion-reaction equation and the evolutive one-dimensional advection-diffusion equation.