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.

翻译：本文研究文献[8]中提出的原始拉格朗日乘子方法在梯度流问题中的唯一可解性与误差分析。我们识别了原始拉格朗日乘子方法中非线性代数方程在其精确解邻域内存在唯一解所必须满足的充分必要条件，并提出了一种修正的拉格朗日乘子方法，使得即使前述条件不满足时计算仍能继续进行。以Cahn-Hilliard方程为例，我们严格证明了在满足该条件且时间步长足够小的前提下，一种二阶拉格朗日乘子格式的唯一可解性，并建立了最优误差估计。同时，数值结果表明，修正的拉格朗日乘子方法较原始方法具有更强的鲁棒性，且可采用显著更大的时间步长。