Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces

Beschle, Cedric Aaron; Kovács, Balázs

doi:10.1007/s00211-022-01280-5

Numer. Math.

2022

DOI: 10.1007/s00211-022-01280-5

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Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces

Cedric Aaron Beschle

Balázs Kovács

Abstract: In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stab… Show more

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Cited by 2 publications

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On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

Andreïanov

Quenjel²

2022

Vietnam J. Math.

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We highlight the interest and the limitations of the L 1 -based Young measure technique for studying convergence of numerical approximations for diffusion problems of the variable-exponent p(x)and p(u)laplacian kind. CVFE (Control Volume Finite Element) and DDFV (Discrete Duality Finite Volume) schemes are analyzed and tested. In the situation where the variable exponent is log-Hölder continuous, convergence is proved along the guidelines elaborated in [Andreianov, Bendahmane, Ouaro, Nonlinear Analysis, 2010, Vol. 72&73] while investigating the structural stability of weak solutions for this class of PDEs. In general, the lack of density of the smooth functions in the energy space, related to the Lavrentiev phenomenon for the associated variational problems, makes it necessary to distinguish two notions of solutions, the narrow On numerical approximation of variable-exponent problems ones (the H-solutions) and the broad ones (the W-solutions). Some situations where approximation methods "select" the one or the other of these two solution notions are described and illustrated by numerical tests.

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On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

Andreïanov

Quenjel²

2022

Vietnam J. Math.

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Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

Kovács

2022

IMA Journal of Numerical Analysis

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Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.

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Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces

Cited by 2 publications

References 38 publications

On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

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