Abstract
We present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail. Although numerical methods for geometric flows have been used frequently in axisymmetric settings, numerical analysis results so far are rare. In this paper, we present stability, equidistribution, existence and uniqueness results for the introduced approximations. Numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one-dimensional problem has to be solved. The good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries.
Highlights
Numerical approximations of curvature flows such as the mean curvature flow and the Gauss curvature flow have been studied intensively during the last 30 years
In many situations the axisymmetry of these geometric flows can be used to reduce the dimension of the governing equations, and so numerical methods have been used frequently in such axisymmetric settings
In this paper we present parametric finite element approximations for axisymmetric curvature flows, and carefully analyse their properties
Summary
Numerical approximations of curvature flows such as the mean curvature flow and the Gauss curvature flow have been studied intensively during the last 30 years. Graph formulations for axisymmetric geometric evolution laws have been considered in [15,17,19], while a finite difference approximation of a parametric description for the evolution of general axisymmetric surfaces has been studied in [39]. The latter is closely related to the presented work, we stress that it does not contain any numerical analysis. In this paper we introduce a parametric finite element method for the axisymmetric formulations of the surface evolution equations discussed above relying on ideas of our earlier work. A precise derivation of (2.17) in the context of a weak formulation of (2.14) will be given in the “Appendix A”
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