Unique Solvability and Error Analysis of a Scheme Using the Lagrange Multiplier Approach for Gradient Flows

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 772-799, April 2025.
Abstract. The unique solvability and error analysis of a scheme using the original Lagrange multiplier approach proposed in [Q. Cheng, C. Liu, and J. Shen, Comput. Methods Appl. Mech. Engrg., 367 (2020), 13070] 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. Then we find that the unique solvability of the original Lagrange multiplier approach depends on the aforementioned condition and may be valid over a finite time period. Afterward, we propose a modified Lagrange multiplier approach to ensure that the computation can continue even if the aforementioned condition was not satisfied. Using the 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 sufficiently small. We also present numerical results to demonstrate that the modified Lagrange multiplier approach is much more robust and can use a much larger time step than the original Lagrange multiplier approach.