Unique solvability and error analysis of the Lagrange multiplier approach for gradient flows

TL;DR

This paper analyzes the unique solvability and error properties of a Lagrange multiplier method for gradient flows, proposing modifications to improve robustness and allow larger time steps, demonstrated through the Cahn-Hilliard equation.

Contribution

It identifies a key condition for solvability, proposes a modified approach, and provides rigorous error analysis and numerical validation for gradient flow schemes.

Findings

The original Lagrange multiplier approach's solvability depends on a specific condition.

The modified approach remains robust even when the original condition fails.

Numerical results show the modified method allows larger time steps and improved stability.

Abstract

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.