The Coupling Of Natural Boundary Element And Mini Element For Solving The Stokes Problem On Unbounded Domains

As a kind of partial differential equation boundary value problem, the Stokes problem is raised in the research of computation hydromechanics. People pay much attention to this problem, especially for the numerical solution of the Stokes problem in infinite domains. They try using different numerical methods to overcome the difficulties brought by the unbounded domain for many years. It is difficult to deal with the non-linear problem as well as the irregular boundary by the natural boundary element itself. At the same time, the unbounded domain brings much difficulty to Finite Elements. However the combination of the two methods can not only overcomes the respective shortcoming and forms natural and direct coupling based on the same variational principle, but also greatly extends the scope of its application.When the coupling of the natural boundary element method and the finite element method is applied to solve the Stokes problem on unbounded domains, it is still lack of the correlation analysis of the convergence analysis and the error estimate for the concrete numerical method research until now. Meanwhile, for solving the system of linear equations which the Stokes problem forms, as a result of refinement of mesh subdivision, a large-scale thin scatter formation will be generated. Therefore, it is essential to establish the effective iterative method for solving saddle point problems.In view of the above situation, this paper investigates the coupling of natural boundary element and Mini element for solving the Stokes problem. The original problem is turned into a variational coupling problem by making an artificial boundary and using a natural boundary reduction, and its uniqueness and existence are proved. The piecewise linear boundary element on the artificial boundary and the Mini element in the bounded domain are applied, the stiffness matrix of the mixed element method and the stiffness matrix of the natural boundary element method are obtained respectively, the total stiffness matrix and the coupling system of linear equations are established, and its convergence and error estimate are proved. Finally, Uzawa algorithm is used to solve this indefinite system of linear equations, and the corresponding numerical experiments are shown for the practical effectiveness of the proposed method and the correctness of its numerical analysis.This article also researches two kinds of Uzawa algorithm for solving the saddle point problem, namely the classical Uzawa algorithm and the preconditioned Uzawa algorithm, gives the necessary and sufficient condition or the sufficient condition of convergence and the spectral radius of the error propagation matrix. Finally, they are applied to the Mini element to solve the Stokes equation and the theoretical results given here will be verified by numerical calculation.