The unique solvability analysis and error estimate of the Lagrange multiplier approach for gradient flows is theoretically analyzed. We identify a necessary and sufficient condition that has to 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. In turn, a modified Lagrange multiplier approach is proposed so that the computation can continue even if the aforementioned condition is not satisfied. Using Cahn-Hilliard equation as an example, we rigorously establish the unique solvability analysis and optimal error estimates of a second-order Lagrange multiplier scheme assuming this condition and that the time step size is sufficient small.
