Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation
Xiao Li, Zhonghua Qiao, Cheng Wang
Abstract
In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript negative 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">H^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for the error function to establish the convergence result. In addition, the energy stability obtained by Du et al., [J. Comput. Phys. 363 (2018), pp. 39–54] requires an assumption on the uniform <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bound of the numerical solution, and such a bound is figured out in this paper by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded uniformly under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution.