Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

The uniform bounds on eigenvalues of ？？122 are shown both analytically and numerically by the 1 finite element preconditioner ？？12 for the Legendre spectral element system 2= which is arisen from a coupled elliptic system occurred by an optimal control problem. The finite element preconditioner is corresponding to a leading part of the coupled elliptic system.