A minimizing-movement framework for geometric gradient flows with admissible tangential motion

Mathematics > Numerical Analysis [Submitted on 16 Jun 2026] Title:A minimizing-movement framework for geometric gradient flows with admissible tangential motion View PDF HTML (experimental)Abstract:We develop a minimizing-movement framework for parametric finite element approximations of geometric gradient flows with admissible tangential motion. At each time step, the discrete variational problem combines a metric dissipation term for the normal displacement with a surface Dirichlet energy. The metric determines the normal geometric evolution: the $L^2(\Gamma)$ metric gives mean curvature flow, while the $H^{-1}(\Gamma)$ metric gives surface diffusion flow. Tangential velocity is selected independently through weak constraints on the deformation map. The central structural condition is admissibility, namely, that the identity map satisfies the constraint. This condition keeps the identity map available as a comparison function and yields the natural stability estimate. The framework recovers the classical Barrett--Garcke--Nürnberg (BGN) scheme from the unconstrained formulation and the dual minimal-deformation-rate (MDR) scheme from the MDR constraint. We further introduce two new admissible variants: an admissible BGN scheme and a relaxed MDR scheme. For the resulting fully discrete schemes, we prove existence and uniqueness under natural nondegeneracy assumptions and establish unconditional energy stability. Numerical experiments compare the admissible and classical schemes and illustrate their stability properties and mesh-quality behavior. Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.