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 stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.
%0 Journal Article
%1 10.1007/s00211-022-01280-5
%A Beschle, Cedric Aaron
%A Kovács, Balázs
%D 2022
%J Numerische Mathematik
%K ians uq
%P 1--48
%R 10.1007/s00211-022-01280-5
%T Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces
%X 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 stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.
@article{10.1007/s00211-022-01280-5,
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 stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.}},
added-at = {2022-04-06T11:03:15.000+0200},
author = {Beschle, Cedric Aaron and Kovács, Balázs},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2837cd72b5b09dfbe8d846ab1f19b080a/mathematik},
doi = {10.1007/s00211-022-01280-5},
interhash = {7d864fbce1f93726be014b0b9ac82335},
intrahash = {837cd72b5b09dfbe8d846ab1f19b080a},
issn = {0029-599X},
journal = {Numerische Mathematik},
keywords = {ians uq},
note = {\url{https://rdcu.be/cKKTQ}},
pages = {1--48},
timestamp = {2023-02-23T14:12:27.000+0100},
title = {{Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces}},
year = 2022
}
