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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Bibliographic Details
Published in:Numerische Mathematik Vol. 149; no. 3; pp. 595 - 643
Main Authors: Kovács, Balázs, Li, Buyang, Lubich, Christian
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-021-01238-z