Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

Abstract

We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a two-dimensional surface. Beside the well-known Willmore and Helfrich flows we will also consider flows involving the Gaussian curvature of the surface. Boundary conditions for these flows are highly nonlinear, and we use a variational approach to derive weak formulations, which naturally can be discretized with the help of a mixed finite element method. Our approach uses a parametric finite element method, which can be shown to lead to good mesh properties. We prove stability estimates for a semidiscrete (discrete in space, continuous in time) version of the method and show existence and uniqueness results in the fully discrete case. Finally, several numerical results are presented involving convergence tests as well as the first computations with Gaussian curvature and/or free or semi-free boundary conditions.

Finite element methods for fourth order axisymmetric geometric evolution equations

“…[10, p. 1706], that would need to be enforced for x t , cannot be enforced through this weak formulation in the open curve case. Instead, techniques as in [10] are needed here, and we will consider the details in the forthcoming paper [11]. Then we consider the following weak formulation of (2.30) and (2.6), on recalling (2.11).…”

Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

“…However, under discretization that formulation would lead to undesirable mesh effects. Hence, in line with the authors previous work in [10], we also allow θ ∈ [0, 1), which under discretization leads to an induced tangential motion and good meshes for θ = 0, in general. In rare cases we may need to dampen the tangential motion that occurs in the case θ = 0.…”

“…We will also show that for θ = 0 the scheme produces conformal polyhedral surfaces Γ 1 (t) and Γ 2 (t). Here we recall from [10], see also [3, §4.1], that the open surfaces Γ h i (t), i = 1, 2, are conformal polyhedral surfaces if…”

“…(4.41) We recall from [3,10] that conformal polyhedral surfaces exhibit good meshes. Moreover, we recall that in the case d = 2, conformal polyhedral surfaces are equidistributed polygonal curves, see [2,5].…”

“…The approach in this paper relies on a discretization of mean curvature leading to good mesh properties. This discretization was introduced by the present authors in [2,3] and has been previously used for closed and open membranes, see [8,10] and for elastic curvature flow of curves with junctions, see [6].…”