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<p>This paper examines a local discontinuous Galerkin (LDG) method for the Cahn–Hilliard equation, specifically focusing on scenarios where the nonlinear term is twice continuously differentiable, with its convex and concave components of the first derivatives being Lipschitz continuous functions. Our primary focus is on analyzing the errors in an unconditionally energy-stable fully discrete LDG scheme where the temporal derivative is approximated by the backward Euler method. The main challenge in error estimation stems from the difficulty of handling specific jump terms at cell boundaries in the LDG discretization. We prove the unconditional energy stability of the scheme using a convex-concave decomposition for the nonlinear term. Additionally, we establish the existence and uniqueness of the initial numerical solutions using the standard fixed point theorem in finite-dimensional spaces. Optimal error estimates are demonstrated for the initial numerical solutions, which are crucial to deriving the main error estimates. Furthermore, we derive a priori error estimates of optimal order in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^\infty (L^2)\)</EquationSource> </InlineEquation>-norm for the fully discrete LDG scheme. Numerical experiments with elements <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P^1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P^2\)</EquationSource> </InlineEquation> validate our theoretical convergence rates. Moreover, numerical verifications confirm that the discrete energy consistently decreases and the mass remains constant throughout the simulation.</p>

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Analysis of a local discontinuous Galerkin method for the Cahn–Hilliard equation using convex-concave decomposition

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Monirul Islam,
  • Rajen Kumar Sinha

摘要

This paper examines a local discontinuous Galerkin (LDG) method for the Cahn–Hilliard equation, specifically focusing on scenarios where the nonlinear term is twice continuously differentiable, with its convex and concave components of the first derivatives being Lipschitz continuous functions. Our primary focus is on analyzing the errors in an unconditionally energy-stable fully discrete LDG scheme where the temporal derivative is approximated by the backward Euler method. The main challenge in error estimation stems from the difficulty of handling specific jump terms at cell boundaries in the LDG discretization. We prove the unconditional energy stability of the scheme using a convex-concave decomposition for the nonlinear term. Additionally, we establish the existence and uniqueness of the initial numerical solutions using the standard fixed point theorem in finite-dimensional spaces. Optimal error estimates are demonstrated for the initial numerical solutions, which are crucial to deriving the main error estimates. Furthermore, we derive a priori error estimates of optimal order in the \(L^\infty (L^2)\) -norm for the fully discrete LDG scheme. Numerical experiments with elements \(P^1\) and \(P^2\) validate our theoretical convergence rates. Moreover, numerical verifications confirm that the discrete energy consistently decreases and the mass remains constant throughout the simulation.