A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.

A kind of Keller Segel chemotaxis model has a wide range of applications, but its coupling relationship is very complex. The commonly used method of constructing the upper and lower solutions is no longer suitable for the model solution, which results in a long time for its analysis. In this paper, we propose a method to analyze the asymptotic behavior and stability of a Keller Segel chemotaxis model. The previous methods of first formally and then rigorously, the asymptotic expansion of these monotone steady states, and then we use this fine information on the spike to prove its local asymptotic stability. Moreover, we obtain the uniqueness of such steady states. The asymptotic behavior of the solution of a Keller Segel chemotaxis model is analyzed, and the asymptotic rate is calculated; According to the limitation of Neumann boundary condition, the complete blow up of chemotaxis model solution and the stability of the initial value of the complete blow up time are studied, and the asymptotic and stability analysis of a kind of Keller Segel chemotaxis model solution is completed. The experimental results show that the proposed method takes less time to solve a kind of Keller Segel chemotaxis model, improves the efficiency of the solution, and the accuracy of the solution is higher.

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