The adaptive Ciarlet-Raviart mixed method for biharmonic problems with simply supported boundary condition

In this paper, we study the adaptive fashion of the Ciarlet-Raviart mixed method for biharmonic equation/eigenvalue problem with simply supported boundary condition in Rd. We propose an a posteriori error indicator of the Ciarlet-Raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the Ciarlet-Raviart approximate eigenfuction, and prove the reliability and efficiency of the indicators. We also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability. We design an adaptive Ciarlet-Raviart mixed method with piecewise polynomials of degree less than or equal to m, and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order O(dof−2md)(d=2,m=2,3;d=3,m=3).