  The unique solvability and error analysis of the original Lagrange multiplier
approach proposed in [8] for gradient flows is studied in this paper. We
identify a necessary and sufficient condition that must be satisfied for the
nonlinear algebraic equation arising from the original Lagrange multiplier
approach to admit a unique solution in the neighborhood of its exact solution,
and propose a modified Lagrange multiplier approach so that the computation can
continue even if the aforementioned condition is not satisfied. Using
Cahn-Hilliard equation as an example, we prove rigorously the unique
solvability and establish optimal error estimates of a second-order Lagrange
multiplier scheme assuming this condition and that the time step is sufficient
small. We also present numerical results to demonstrate that the modified
Lagrange multiplier approach is much more robust and can use much larger time
step than the original Lagrange multiplier approach.
