Two-grid variational multiscale method with bubble stabilization for convection diffusion equation

•We propose a stabilized FEMfor nonlinear convection diffusion equations.•We use the local Gauss integration to replace the projection operator.•We give the convergence order of our proposed scheme.•The proposed method is better than SUPG andVMS in CPU time and accuracy.

A two-grid variational multiscale (VMS) method based on two local Gauss integrations for the convection dominated nonlinear convection diffusion equation is investigated. This method combines the two-grid strategy with the variational multiscale method which chooses polynomial bubble functions as subgrid scale. Two local Gauss integrations are applied to replace the projection operator without adding any extra storage. Moreover, the error estimates for the algorithms of the two-grid method are obtained. Numerical examples validate the theoretical results of the presented methods.