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

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.

Appendix: Proof of Lemma 2.2

We came up with two independent proofs of Lemma 2.2, one based on local coordinates and the other one based on the formalism of Dziuk and Elliott in [22]. As should be, both approaches yield the same result as stated in Lemma 2.2. Here we present the more straightforward proof based on [22].

The following commutator formula for tangential differential operators ($D_i = (\nabla _{\varGamma })_i$) is shown in [22, Lemma 2.6]:

$$\begin{aligned} D_i D_j u - D_j D_i u = A_{jk} D_k u \nu _i - A_{ik} D_k u \nu _j \qquad (i , j = 1, 2, 3) , \end{aligned}$$

(7.1)

where we use the convention to sum over repeated indices. We also recall that $A_{ij} = \ D_i \nu _j = A_{ji}$ and $H = \mathrm{tr}(\nabla _{\varGamma }\nu ) = D_i \nu _i$.

Using (7.1) we obtain (cf. the proof of Lemma 3.2 in [22])

$$\begin{aligned} D_i D_i D_j u =&D_i \big ( D_j D_i u + A_{jk} D_k u \nu _i - A_{ik} D_k u \nu _j \big ) \\ =&D_i D_j D_i u + D_i \big ( D_k u (A_{jk} \nu _i - A_{ik} \nu _j) \big ) \\ =&D_i D_j D_i u + D_i D_k u \big ( A_{jk} \nu _i - A_{ik} \nu _j \big ) \\&+ D_k u D_i \big ( A_{jk} \nu _i - A_{ik} \nu _j \big ) . \end{aligned}$$

The second term is then rewritten as

$$\begin{aligned}&D_i D_k u \big ( A_{jk} \nu _i - A_{ik} \nu _j \big ) = A_{jk} D_i D_k u \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad = A_{jk} \big ( D_k D_i u + A_{kl} D_l u \nu _i - A_{il} D_l u \nu _k \big ) \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad =A_{jk} (D_k D_i u \nu _i) + A_{jk} A_{kl} D_l u \nu _i\nu _i - A_{jk} A_{il} D_l u \nu _k \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad = - A_{jk} A_{ki} D_i u + A_{jk} A_{kl} D_l u - A_{jk} A_{il} D_l u \nu _k \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad = - (A^2)_{ji} D_i u + (A^2)_{jl} D_l u - A_{jk} A_{il} D_l u \nu _k \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad = - A_{jk} A_{il} D_l u \nu _k \nu _i - A_{ik} D_i D_k u \nu _j \\&\quad = - A_{ik} D_i D_k u \nu _j . \end{aligned}$$

In particular for the last term we have:

$$\begin{aligned}&D_i \big ( A_{jk} \nu _i - A_{ik} \nu _j \big ) = D_i A_{jk} \nu _i - D_i A_{ik} \nu _j + A_{jk} D_i \nu _i - A_{ik} D_i \nu _j \\&\quad = D_i A_{jk} \nu _i - D_i A_{ik} \nu _j + A_{jk} H - A_{ki} A_{ij} \\&\quad = -\varDelta _{\varGamma }\nu _k \nu _j + (HA - A^2)_{jk} . \end{aligned}$$

The third-order term is again rewritten using (7.1) as

$$\begin{aligned} D_i D_j D_i u =&D_j D_i D_i u + A_{jk} D_k D_i u \nu _i - A_{ik} D_k D_i u \nu _j , \end{aligned}$$

which is further rewritten using

$$\begin{aligned} D_k D_i u \nu _i =&D_k (D_i u \nu _i) - D_k \nu _i D_i u = - A_{ki} D_i u , \\ D_k D_i u \nu _j =&D_k (D_i u \nu _j) - D_k \nu _j D_i u . \end{aligned}$$

Therefore,

$$\begin{aligned} D_i D_j D_i u =&D_j D_i D_i u - A_{jk} A_{ki} D_i u - A_{ik} D_k D_i u \nu _j \\ =&D_j D_i D_i u - (A^2)_{ji} D_i u - A_{ik} D_k D_i u \nu _j . \end{aligned}$$

Altogether, the above calculations yield

$$\begin{aligned}&D_i D_i D_j u \\&\quad = D_j D_i D_i u - (A^2)_{ji} D_i u - 2A_{ik} D_k D_i u \nu _j -\varDelta _{\varGamma }\nu _k \nu _j D_k u + (HA - A^2)_{jk} D_k u \\&\quad = D_j D_i D_i u - (A^2)_{ji} D_i u - 2D_k (A_{ik} D_i u) \nu _j + 2D_k A_{ik} D_i u \nu _j\\&\quad -\varDelta _{\varGamma }\nu _k \nu _j D_k u + (HA - A^2)_{jk} D_k u \\&\quad = D_j D_i D_i u - (A^2)_{ji} D_i u - 2D_k (A_{ik} D_i u) \nu _j + \varDelta _{\varGamma }\nu _k \nu _j D_k u + (HA - A^2)_{jk} D_k u . \end{aligned}$$

This implies

$$\begin{aligned} \varDelta _{\varGamma } \nabla _{\varGamma } u&= \nabla _{\varGamma } \varDelta _{\varGamma } u - A^2 \nabla _{\varGamma } u -2\nabla _{\varGamma }\cdot (A\nabla _{\varGamma }u) \nu \\&\quad + (\varDelta _{\varGamma }\nu \cdot \nabla _{\varGamma }u) \nu + (HA - A^2) \nabla _{\varGamma } u , \end{aligned}$$

which is the same as (2.6). $\square$