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This article concerns the construction of high-order energy-decaying numerical methods for gradient flows of evolving surfaces with curvature-dependent energy functionals. The semidiscrete evolving surface finite element method is derived based on the calculus of variation of the semidiscrete surface energy functional. This makes the semidiscrete problem naturally inherit the energy decay structure. With this property, the semidiscrete problem is furthermore formulated as a gradient flow system of ODEs. The averaged vector-field collocation method is used for time discretization of the ODEs to preserve energy decay at the fully discrete level while achieving high-order accuracy in time. Extensive numerical examples are provided to illustrate the accuracy and energy diminishing property of the proposed method, as well as the effectiveness of the method in capturing singularities in the evolution of closed surfaces.
}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2021-0007}, url = {http://global-sci.org/intro/article_detail/aam/20091.html} }This article concerns the construction of high-order energy-decaying numerical methods for gradient flows of evolving surfaces with curvature-dependent energy functionals. The semidiscrete evolving surface finite element method is derived based on the calculus of variation of the semidiscrete surface energy functional. This makes the semidiscrete problem naturally inherit the energy decay structure. With this property, the semidiscrete problem is furthermore formulated as a gradient flow system of ODEs. The averaged vector-field collocation method is used for time discretization of the ODEs to preserve energy decay at the fully discrete level while achieving high-order accuracy in time. Extensive numerical examples are provided to illustrate the accuracy and energy diminishing property of the proposed method, as well as the effectiveness of the method in capturing singularities in the evolution of closed surfaces.