The Adaptive Finite Element Method Based On Multi-scale Discretizations For Eigenvalue Problems With Homogeneous Mixed Boundary Conditions

Adaptive finite element methods are a fundamental numerical instrument in science and engineering to solve partial differential equations. These methods have been extensively studied theoretically since Babuska and have also been successful in practice.In this paper, adaptive finite element methods for eigenvalue problems with homogeneous mixed boundary value conditions are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration(see Scheme 3 in SIAM J Numer Anal49(2011), pp. 1602-1624), a posteriori error indicator is given. Both reliability and efficiency of the error indicator are proved. In addition, we give two adaptive finite element methods based on multiscale discretization schemes for eigenvalue problems with homogeneous mixed boundary conditions. The algorithms are performed under the package of Chen, and the associated numerical results are provided.