A two-level variational multiscale meshless local Petrov–Galerkin (VMS-MLPG) method for convection-diffusion problems with large Peclet number

•VMS method is constructed in the MLPG method.•Utilize the high and low orders Gauss integration to instead the large scale and small scale.•The numerical solution is accuracy and stability. It is challengeable to obtain the stable and accurate solutions of convection-diffusion problems with large Peclet number (Pe) since the convection term may cause oscillation solutions at large Pe. In this paper, a unit operator (first level) and an orthogonal project operator (second level) are constructed to act as the stability terms for meshless local Petrov-Galerkin (MLPG) method, which is called a two-level variational multiscale MLPG (VMS-MLPG) method. The VMS-MLPG method is applied to eliminate oscillation, overshoots and undershoots of MLPG method at large Pe. The prediction accuracy and the numerical stability of the proposed method for the Smith-Hutton and the Brezzi problems are analyzed and validated by comparing with the MLPG method and the finite volume method (FVM) with various difference schemes. It is showed that the present VMS-MLPG method can guarantee the stable and reasonable solutions of convection-diffusion problems with large Peclet number.