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In this paper we discuss the application of nonconforming virtual elementmethods(VEM) for the second order diffusion dominated convection diffusionreaction equation. Stability of the virtual element methods has been proved forthe symmetric bilinear form. But the same analysis cannot be carried out forthe non-symmetric case. In this work we present the external virtual elementmethods using $L^2$ projection operator and prove the well-posedness of VEM fornon symmetric bilinear form. We also proved polynomial consistency of discretebilinear form assuming $H^2$ regularity of approximate solution on eachtriangle. We have shown optimal convergence estimate in the broken sobolevnorm.

Analysis of nonconforming virtual element method for the convection diffusion reaction equation with polynomial coefficients