A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector a...

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Veröffentlicht in:	Numerische Mathematik Jg. 149; H. 3; S. 595 - 643
Hauptverfasser:	Kovács, Balázs, Li, Buyang, Lubich, Christian
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021 Springer Nature B.V
Schlagworte:	Algorithms Consistency Convergence Curvature Error analysis Estimates Evolution Finite element method Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Numerical methods Polynomials Simulation Stability analysis Surface diffusion Theoretical Two dimensional flow 65M15 35R01 65M60 65M12
ISSN:	0029-599X, 0945-3245
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Abstract	A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order H 1 -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.
AbstractList	A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order H 1 -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results. A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order H1-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.
Author	Kovács, Balázs Li, Buyang Lubich, Christian
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SubjectTerms	Algorithms Consistency Convergence Curvature Error analysis Estimates Evolution Finite element method Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Numerical methods Polynomials Simulation Stability analysis Surface diffusion Theoretical Two dimensional flow
Title	A convergent evolving finite element algorithm for Willmore flow of closed surfaces
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