A New Adaptive Mixed Finite Element Method Based On Residual Type A Posterior Error Estimates For The Stokes Eigenvalue Problem

The a posterior error estimates and adaptive algorithms of finite element methodsare the main streams in the academic circle. From the perspective of methodology, onehas concluded that adaptive methods are the optimal discretization methods when usingfinite element methods to solve differential equations. The a posterior error estimates arethe theoretical foundations of adaptive finite element methods. The idea of a posteriorerror estimates and adaptive finite element methods was first proposed by Babuska andRheinboldt in1978, and was great developed so far and thus produced many importantapproaches of a posterior error estimates such as residual types and recovery types. Par-tial differential equation eigenvalue problems are also hot topics focused on by numerousscholars. In recent years, they attached much importance to adaptive finite element meth-ods for partial differential equation eigenvalue problems. Under the above background, inthis paper we aim to discuss a new adaptive mixed finite element algorithm for the Stokeseigenvalue problem.In this paper, we combine mixed finite element method, multi-scale discretizationand Rayleigh quotient iteration to propose a new adaptive algorithm based on residualtype a posterior error estimates for the Stokes eigenvalue problem. Both reliability andefficiency of the error indicator are proved. In addition, we combine higher order finiteelements and the shifted inverse power method to propose a modified algorithm. Theefficiency of both algorithms are also investigated by using Chen’s iFEM package[28].Numerical results are satisfying.