An adaptive div-curl-grad least-squares finite element method for the Cahn-Hilliard equation

Chad R. Westphal; Chad R. Westphal

doi:10.3934/mbe.2026077

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 2110-2131. doi: 10.3934/mbe.2026077

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An adaptive div-curl-grad least-squares finite element method for the Cahn-Hilliard equation

Chad R. Westphal ^,

Department of Mathematics and Computer Science, Wabash College, Crawfordsville, IN, USA

Received: 24 April 2026 Revised: 17 June 2026 Accepted: 02 July 2026 Published: 13 July 2026

This paper develops a new least-squares finite element approach for the efficient approximation of solutions of the Cahn-Hilliard equation. We discretize in time with either implicit backward Euler or Crank-Nicolson and use a regularized Newton iteration for the nonlinearity. The innermost iteration is reformulated as a coupled first-order div-curl-grad system, which is efficiently solved by a least-squares minimization approach using standard Lagrange finite elements for each unknown. Theoretical findings include results on the optimal finite element convergence of the variational problem, convergence of the nonlinear iteration, error bounds for the time integration schemes, and discrete asymptotic mass conservation and energy dissipation in the least-squares sense. The basic iterative approach is expanded to include adaptivity in both time and space. Numerical examples are provided to confirm the theoretical results and to demonstrate effectiveness of the proposed method for a range of test problems.
- Cahn-Hilliard equation,
- phase separation models,
- least-squares finite element methods,
- nonlinear partial differential equations
Citation: Chad R. Westphal. An adaptive div-curl-grad least-squares finite element method for the Cahn-Hilliard equation[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 2110-2131. doi: 10.3934/mbe.2026077

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Abstract

This paper develops a new least-squares finite element approach for the efficient approximation of solutions of the Cahn-Hilliard equation. We discretize in time with either implicit backward Euler or Crank-Nicolson and use a regularized Newton iteration for the nonlinearity. The innermost iteration is reformulated as a coupled first-order div-curl-grad system, which is efficiently solved by a least-squares minimization approach using standard Lagrange finite elements for each unknown. Theoretical findings include results on the optimal finite element convergence of the variational problem, convergence of the nonlinear iteration, error bounds for the time integration schemes, and discrete asymptotic mass conservation and energy dissipation in the least-squares sense. The basic iterative approach is expanded to include adaptivity in both time and space. Numerical examples are provided to confirm the theoretical results and to demonstrate effectiveness of the proposed method for a range of test problems.

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