Finite element preconditioning on spectral element discretizations for coupled elliptic equations

The uniform bounds on eigenvalues of B̂ ^-1 _h2  _N2 are shown both analytically and numerically by the P ₁ finite element preconditioner B̂ ^-1 _h2 for the Legendre spectral element system  _N2u = f 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.