  We develop several efficient numerical schemes which preserve exactly the
global constraints for constrained gradient flows. Our schemes are based on the
SAV approach combined with the Lagrangian multiplier approach. They are as
efficient as the SAV schemes for unconstrained gradient flows, i.e., only
require solving linear equations with constant coefficients at each time step
plus an additional nonlinear algebraic system which can be solved at negligible
cost, can be unconditionally energy stable, and preserve exactly the global
constraints for constrained gradient flows. Ample numerical results for
phase-field vesicle membrane and optimal partition models are presented to
validate the effectiveness and accuracy of the proposed numerical schemes.
