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

Qing Cheng, Jie Shen, Cheng Wang

TL;DR

The paper tackles the challenge of unique solvability and error control in the original Lagrange multiplier approach for gradient flows, using the Cahn–Hilliard equation as a representative model. It identifies a precise condition on $S_n = \frac{d}{dt}\int_\Omega F(\Phi)dx|_{t=t^n}$ that governs local solvability of the nonlinear multiplier equation and introduces a robust modified LM scheme that remains usable when this condition is not met. A rigorous solvability result shows unique roots near $\eta=1$ under a quantified bound, and a comprehensive error analysis yields optimal second-order convergence for small time steps, alongside energy stability and uniform $H^2$ bounds. Numerical experiments corroborate the theoretical findings, demonstrating enhanced robustness and the ability to take significantly larger time steps with the modified LM approach. The results provide a foundational understanding of LM-based gradient-flow solvers and pave the way for extending the framework to more complex constrained LM formulations.

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.

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

TL;DR

The paper tackles the challenge of unique solvability and error control in the original Lagrange multiplier approach for gradient flows, using the Cahn–Hilliard equation as a representative model. It identifies a precise condition on that governs local solvability of the nonlinear multiplier equation and introduces a robust modified LM scheme that remains usable when this condition is not met. A rigorous solvability result shows unique roots near under a quantified bound, and a comprehensive error analysis yields optimal second-order convergence for small time steps, alongside energy stability and uniform bounds. Numerical experiments corroborate the theoretical findings, demonstrating enhanced robustness and the ability to take significantly larger time steps with the modified LM approach. The results provide a foundational understanding of LM-based gradient-flow solvers and pave the way for extending the framework to more complex constrained LM formulations.

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.
Paper Structure (13 sections, 13 theorems, 150 equations, 1 figure)

This paper contains 13 sections, 13 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Lagrange multiplier approach and a modified version
  3. The unique solvability of \ref{['nonlinear-2']}
  4. The main result
  5. Some preliminary estimates
  6. Proof of Theorem \ref{['thm: solvability']}
  7. Error analysis
  8. Energy stability and the $H^1$ estimate
  9. The $\ell^\infty(0,T; H^2)$ estimate for the numerical solution
  10. Estimate for $\| \phi^{m+1} - \phi^m \|_{H^\ell}$
  11. A refined estimate for $\eta^{n+1/2}$
  12. Proof of Theorem 4.1
  13. Concluding remarks

Key Result

Theorem 2.1

The numerical scheme gflow-SAVN-1-gflow-SAVN-3 is unconditional stable and satisfies the following energy dissipative law where the energy $E^{n+1}$ is defined as

Figures (1)

  • Figure 1: (a-c): The evolution of energy, iterations and $\eta$ using the original Lagrange multiplier approach with $\Delta t=10^{-7}$. (d-f): The evolution of energy, iterations and $\eta$ using modified Lagrange multiplier approach with $\Delta t=10^{-3}$.

Theorems & Definitions (23)

  • Theorem 2.1
  • Proof 1
  • Theorem 3.1
  • Lemma 3.1
  • Proof 2
  • Lemma 3.2
  • Proof 3
  • Lemma 3.3
  • Proof 4
  • Theorem 4.1
  • ...and 13 more