A Quadratic Stabilization Finite Element Method For Solving Advection-Dominated Equation

In the numerical solution of the finite element method, the convection - diffusion equation is an important mathematical model in the neighborhood of fluid mechanics.Which convection dominance is a difficult problem to solve such equations many scien-tists put forward a number of stable finite element method to achieve the purpose of elim-inating numerical oscillations to improve numerical accuracy.For example exponentially fitted finite volume scheme, streamline-upwind Petrov-Galerkin, spurious oscillations at layers diminishing finite element method, continuous interior penalty stabilization,discontinuous Galerkin(DG)finite element method and a total variation diminishing finite dement method(FEMTVD). These methods are mainly for the convection-dominated of non-reaction term convection-diffusion equations. A stable method for solving convection-diffusion reaction equation proposed in [2],mainly for the case of convection dominant and the reaction coefficient is large. In [1] the author designed a new stabilization pa-rameter regardless of whether the coefficient of reaction is zero or very small. The new stable finite element method proposed in [1] for the convection-diffusion equation is a case where the mesh is a linear element.In this paper, the mesh is a quadratic element, Therefore, the stabilization param-eters are redefined by the stabilization method proposed in [1], where the stabilization parameters depend not only on the diffusion coefficient, the convective coefficient, the re-action coefficient coefficient and the mesh size, but also on the constant C in the inverse inequality. Finally, numerical results show theL2 norm error order reaches O(h3)and the H1 norm error order reaches O(h2),which is consistent with the theoretical results.