An Efficient Finite Element Method For Stokes Eigenvalue Problem

Stokes eigenvalue problem is widely used in many fields such as structural mechanics,electromagnetic field and fluid mechanics.It is considered to be one of the most important eigenvalue problems and plays an important role in the analysis of the stability of nonlinear partial differential equations.Although,some theories for solving eigenvalue problem are relatively mature,there are still a lot of difficulties in practical problems.So it is necessary to study a series of fast,stable and high precision numerical algorithm for solving eigenvalue problem.This paper mainly studies the high efficiency stabilizing finite element method for Stokes eigenvalue problem,including a stabilized method without parameter stabilization and a two-step algorithm based on gaussian stabilization.The main work is organized as follows: A parameterless stabilization method is given in the first part of this paper,and its stabilization term not only contains a momentum equation but also a continuous equation.The error estimation of eigenvalue and eigenfunction is deduced through the theoretical analysis for the Stokes source problem,and numerical examples verified the feasibility and effectiveness of this method by comparing five traditional stabilization methods.In the second part,we mainly study an acceleration algorithm for solving Stokes eigenvalue problem.As we known,many stabilization methods are validly used to solve Stokes eigenvalue problems,but it is so hard to find a fast effective and high precision method.The convergence rate of the solution obtained by two-step algorithm based on gaussian stabilization method keeps the same order as direct computation.Further,two-step algorithm can improve the computational efficiency so as to save a large amount of time.Numerical experiments show the efficiency of the algorithm.