A New Edge Stabilization Method for the Convection-Dominated Diffusion-Convection Equations

We study a new edge stabilization method for the finite element discretization of the convection-dominated diffusion-convection equations. In addition to the stabilization of the jump of the normal derivatives of the solution across the inter-element-faces, we additionally introduce a SUPG/GaLS-like stabilization term but on the domain boundary other than in the interior of the domain. New stabilization parameters are also designed. Stability and error bounds are obtained. Numerical results are presented. Theoretically and numerically, the new method is much better than other edge stabilization methods and is comparable to the SUPG method, and generally, the new method is more stable than the SUPG method.