1 Introduction
In this paper we prove semi and fully discrete error bounds of a numerical algorithm for the evolution of a closed dimensional surface evolving under mean curvature flow of arbitrary codimension, with a particular interest for codimension at least .
The key idea is to derive nonlinear parabolic evolution equations for the mean curvature vector and the orthogonal projection onto the tangent space along the flow. Taking a similar approach as recent previous work for mean curvature flow [MCF], which first utilised such an approach using similar evolution equations for geometric quantities (mean curvature and surface normal), see [MCF_soldriven, Willmore, MCF_generalised] as well. Similarly as Huisken [Huisken1984] did for codimension 1 mean curvature flow, in higher codimension [AndrewsBaker] (or see [Smoczyk_survey]) have derived numerous geometric evolution equations for various geometric quantities. The evolution equations for the mean curvature vector and the orthogonal projection onto the tangent space is first derived here, and until the present work it was not evident that the evolution equations for and form a closed system that does not involve further geometric quantities.
We give a brief overview of mean curvature flow in higher codimension:
In two papers [Altschuler, AltschulerGrayson] Altschuler and Grayson have proved the first results for curves in , namely that unlike for planar curves singularities may occur in final time. Ambrosio and Soner [AmbrosioSoner_levelset, AmbrosioSoner_1] have studied high codimension mean curvature flow using a level set approach. Andrews and Baker [AndrewsBaker] have proved that submanifolds sufficiently close to the round sphere smoothly collapse to round points in finite time, and have proved pinching estimates. They derive evolution equations for geometric quantities along the flow, similar to [Huisken1984] in codimension 1, but have not derived the closed system of evolution equations derived and used in this paper. These pinching estimates were greatly refined by Naff in [Naff]. Ancient solutions were recently studied by Lynch and Nguyen [LynchNguyen]. In the survey article [Smoczyk_survey] Smoczyk presents results on: shorttime existence and uniqueness, longtime existence and convergence, and singularities. The survey type article by Wang [Wang_MCF_codim] collects several theorems on regularity, global existence and convergence, see also [Wang_lecturenotes].
We also give a literature overview on numerical methods for curve shortening and mean curvature flow in codimension at least (only giving a brief outlook on other flows):
Dziuk has proposed and analysed a finite element algorithm for curve shortening flow for curves in possibly higher codimension [Dziuk_CSF_1994], and proved semidiscrete  and norm error estimates. Carlini, Falcone and Ferretti [CarliniFalconeFerretti_CSF_codim_2] proposed a semiLagrangian scheme for curve shortening flow in codimension (i.e., closed curves in ), and have analysed its (conditional) consistency. Pozzi proposed a numerical method for anisotropic curve shortening and mean curvature flow in higher codimension in [Pozzi_CSF_codim, Pozzi_surf_codim], and proved semidiscrete error estimates for curves in arbitrary codimension. Barrett, Garcke and Nürnberg [BGN_curve_grad_flow] proposed numerical algorithms – allowing tangential movements – for gradient flows (including curve shortening and Willmore flow) for closed curves in (). In [BGN_curves_2012] they discretized highorder flows for plane and space curves. Dörfler and Nürnberg [DoerflerNuernberg] have proposed finite element discretizations for gradient flows for general curvature energies of space curves. A tangentially redistributing scheme for 3dimension curve evolutions was proposed in [MikulaUrban].
Apart from the convergence results by Dziuk [Dziuk_CSF_1994] and Pozzi [Pozzi_CSF_codim], both results for curves in , we are not aware of any convergence results for mean curvature flow in high codimension.
The geometric evolution equation system coupling the nonlinear evolution equations for the mean curvature vector and orthogonal projection , the velocity law
, and an ordinary differential equation (ODE) for the surface evolution is discretized using evolving surface finite elements (of degree at least 2) and using linearly implicit backward difference formulae (of order 2 to 5), under a mild step size restriction.
Utilising the derived geometric evolution equation system, we prove optimalorder timeuniform norm semi and fully discrete error estimates for the surface position and the velocity , and the geometric quantities . The convergence proofs are essentially based on the corresponding results of [MCF]. Since the matrix–vector formulation corresponding to the spatial semidiscretization are formally the same, using the same mass and stiffness matrices (of different blocksize), and with nonlinear terms sharing the same property (local Lipschitz continuity), the stability proofs of [MCF] directly apply for the present case as well. Consistency proofs can be also shown using analogous arguments.
The paper is organised as follows: Section 2 introduces basic notations for arbitrary codimension submanifolds, and contains the main technical results of the paper: deriving the evolution equations for and . Section 3 contains the evolving surface finite element spatial semidiscretization, and the corresponding matrix–vector formulation, discussing its relation to the matrix–vector formulation of mean curvature flow [MCF]. Section 4 presents the linearly implicit backward differentiation formulas. Section 5 contains the main results of the paper, semi and fully discrete error bounds. Section 6 reports on a large number of numerical experiments, to illustrate and complement our theoretical results, including convergence tests, comparisons with Dziuk’s algorithm [Dziuk_CSF_1994], and present some examples from the literature.
2 Evolution equations for mean curvature flow
2.1 Basic notions and notation
We start by introducing some basic concepts and notations.
We consider an evolving dimensional (with ) closed submanifold , in other words is an dimensional submanifold in of codimension . In this paper we allow submanifolds of arbitrary codimension .
The dimensional submanifold is given as the image
of a smooth mapping of an initial submanifold such that is an embedding for every , and . Here, the initial submanifold is smooth and of dimension . In view of the subsequent numerical discretization, it is convenient to think of as the position at time of a moving particle with label , and of as a collection of such particles. This approach is similar as in [MCF].
The velocity at a point equals
(1) 
For a known velocity field , the position at time of the particle with label is obtained by solving the ordinary differential equation (1) from to for a fixed .
For a function (, ) we denote the material derivative (with respect to the parametrization ) as
On a regular submanifold we denote by () the induced metric, and by its inverse. Moreover, we denote by
the second fundamental form. Here denotes the orthogonal projection to the orthogonal complement of the tangent space of at .
Further, the mean curvature vector is the trace of the Weingarten map, i.e.
i.e. we use the sign convention that for a sphere of radius the mean curvature vector points inwards and has length .
For every , we denote the orthogonal projection from to the tangent space of the submanifold , at the point , by
On any regular submanifold , the tangential gradient of a function is given by , and in the case of a vectorvalued function , we define componentwise , i.e. we use the convention that the gradient of has the gradient of the components as column vectors. We denote by the Laplace–Beltrami operator applied to , so that .
2.2 Evolution equations for orthogonal projection and mean curvature vector of a submanifold evolving under mean curvature flow
Mean curvature flow (in arbitrary codimension) sets the velocity (1) of the submanifold to
(2) 
For geometric surface flows, e.g. see [Huisken1984]
, it is known that the geometric quantities satisfy evolution equations along the flow. The algorithm here is based on parabolic partial differential equations for the mean curvature vector
and the projection , derived in the following result.Lemma 1
For a regular dimensional submanifold moving under mean curvature flow in codimension , the orthogonal projection and the mean curvature vector satisfy equationparentequation
(3a)  
(3b) 
where the Laplace–Beltrami operator is understood componentwise. The nonlinear terms are given componentwise, for , by
(4)  
Proof
These equations are formally the same as the analogous ones for mean curvature flow (the normal vector and mean curvature) see [Huisken1984], or [MCF, equations (2.4) and (2.5)], the nonlinear terms are however more complicated.
2.3 A coupled system for mean curvature flow
The evolution of a submanifold of dimension in codimension evolving by mean curvature flow is then governed by the coupled system (2), (3a)–(3b) together with the ODE (1). The numerical method is based on the weak form of the above coupled system which reads:
(5)  
for all test functions and . This system is complemented with the initial data for , and .
For simplicity, by we denote both the Euclidean scalar product for vectors, and the Frobenius inner product for matrices (i.e., the Euclidean product with an arbitrary vectorisation).
3 Evolving surface finite element semidiscretization
3.1 Evolving surface finite elements
We formulate the evolving surface finite element (ESFEM) discretization for the velocity law coupled with evolution equations on the evolving surface, following the description in [KLLP2017, MCF], which is based on [Dziuk88, Demlow2009, highorderESFEM]. We use simplicial finite elements and continuous piecewise polynomial basis functions of degree , as defined in [Demlow2009, Section 2.5].
We triangulate the given smooth initial surface by an admissible family of triangulations of decreasing maximal element diameter ; see [DziukElliott_ESFEM] for the notion of an admissible triangulation, which includes quasiuniformity and shape regularity. For a momentarily fixed , we denote by the vector in that collects all nodes
of the initial triangulation. By piecewise polynomial interpolation of degree
, the nodal vector defines an approximate surface that interpolates in the nodes . We will evolve the th node in time, denoted with , and collect the nodes at time in a column vectorWe just write for when the dependence on is not important.
By piecewise polynomial interpolation on the plane reference triangle that corresponds to every curved triangle of the triangulation, the nodal vector defines a closed surface denoted by . We can then define globally continuous finite element basis functions
which have the property that on every triangle their pullback to the reference triangle is polynomial of degree , and which satisfy at the nodes for all These functions span the finite element space on ,
For a finite element function , the tangential gradient is defined piecewise on each element.
The discrete surface at time is parametrized by the initial discrete surface via the map defined by
which has the properties that for , that for all , and
The discrete velocity at a point is given by
In view of the transport property of the basis functions [DziukElliott_ESFEM], the discrete velocity equals, for ,
where the dot denotes the time derivative . Hence, the discrete velocity is in the finite element space , with nodal vector .
The discrete material derivative of a finite element function with nodal values is
3.2 ESFEM spatial semidiscretizations
Now we will describe the semidiscretization of the coupled system for mean curvature flow in arbitrary codimension.
The finite element spatial semidiscretization of the weak coupled parabolic system (5) reads as follows: Find the unknown nodal vector and the unknown finite element functions and , and satisfying the coupled semidiscrete system: equationparentequation
(6a)  
(6b)  
(6c) 
where and are the spatially discrete analogons of the nonlinear expressions (4), for all and , with the surface given by the differential equation
(7) 
The initial values for the nodal vector are taken as the positions of the nodes of the triangulation of the given initial surface . The initial data and are determined by componentwise Lagrange interpolation of and .
3.3 Matrix–vector formulation
The nodal values of the unknown semidiscrete functions , , and are collected into column vectors , , and , respectively. We furthermore set
and set to be a block matrix extracting the component of , that is .
We define the surfacedependent mass matrix and stiffness matrix :
for . The nonlinear terms are defined by
for and .
We further let, for (with the identity matrices )
When no confusion can arise, we will write for , and for .
The above matrix–vector formulation (8) for mean curvature flow in arbitrary codimension is almost identical to the same formulas for mean curvature flow in codimension [MCF, equation (3.4)–(3.5)]:
In the two above ODE systems the equations for and are formally the same, the equation for is even simpler here. Note that here collects and , whereas for mean curvature flow in codimension it collects , the nodal values of the approximations to the normal vector and scalar mean curvature. Naturally, the blocksize of the matrices in the two equations for are greatly different (respectively, and ).
It is crucial to notice that, thanks to the coinciding matrix–vector formulations many results from [MCF], most notably the stability results Proposition 7.1 and Proposition 10.1 therein, hold directly for the present case as well.
3.4 Lifts
As in [KLLP2017] and [MCF, Section 3.4], we compare functions on the exact surface with functions on the discrete surface , via functions on the interpolated surface , where denotes the nodal vector collecting the grid points on the exact surface, where are the nodes of the discrete initial triangulation .
Any finite element function on the discrete surface, with nodal values , is associated with a finite element function on the interpolated surface with the exact same nodal values. This can be further lifted to a function on the exact surface by using the lift operator , mapping a function on the interpolated surface to a function on the exact surface , via the closest point projection. Provided that the two surfaces are sufficiently close, for find such that is minimal, i.e.
This definition is consistent with the lift operator in codimension , see [Dziuk88, DziukElliott_ESFEM, Demlow2009], using the signed distance function . The standard normequivalence results ([Demlow2009, equation (2.15)–(2.17)]) hold for this definition as well.
Then the composed lift maps finite element functions on the discrete surface to functions on the exact surface via the interpolated surface , and it is defined by
4 Linearly implicit full discretization
Similarly as for mean curvature flow [MCF], for the time discretization of the system of ordinary differential equations (8) we use a step linearly implicit backward difference formula (BDF method). For a step size , and with , we determine the approximations to all variables to , to , and to by the fully discrete system of linear equations equationparentequation
(9a)  
(9b)  
(9c) 
where denote the discretized time derivatives
(10) 
and where and are the extrapolated values
(11) 
The starting values and () are assumed to be given. They can be precomputed using either a lower order method with smaller step sizes or an implicit Runge–Kutta method.
The method is determined by its coefficients, given by and . The classical BDF method is known to be zerostable for and to have order ; see [HairerWannerII, Chapter V]. This order is retained, for , by the linearly implicit variant using the above coefficients ; cf. [LubichMansourVenkataraman_bdsurf, AkrivisLiLubich_quasilinBDF].
We again point out that the fully discrete system (9)–(12) is formally the same as the fully discrete system for the mean curvature flow for surfaces [MCF, equations (5.1)–(5.4)]. Theorem 6.1 in [MCF] proves optimalorder error bounds for the combined ESFEM–BDF full discretization of the mean curvature flow system, for finite elements of polynomial degree and BDF methods of order .
We note that in the th time step, the method decouples and hence only requires solving a linear system with the symmetric positive definite matrix .
From the vectors and matrices , , and with , where and , we obtain position approximations to , , velocity approximations to , and approximations to the orthogonal projection and the mean curvature vector, respectively, at time as
(12)  
In the semidiscrete case, the approximations of the same quantities are given analogously.
5 Main results: error estimates
We are now in the position to state the main results of this paper, time uniform optimalorder semi and fully discrete norm error estimates for the position, velocity, orthogonal projection, and mean curvature vector obtained, respectively, by the semidiscretization (6) (or (8)), or the linearly implicit BDF full discretization (9), using evolving surface finite elements of polynomial degree at least , and step BDF method with .
5.1 Convergence of the semidiscretization
Theorem 1
Consider the semidiscretization (6) of the mean curvature flow (2) in arbitrary codimension , using evolving surface finite elements of polynomial degree . Suppose that the mean curvature flow problem in arbitrary codimension admits an exact solution that is sufficiently smooth on the time interval , and that the flow map is nondegenerate so that is a regular surface on the time interval .
Then there exist constants and such that
for all . The constant is independent of and , but depends on the norms of the exact solution and on the final time .
Proof
The result essentially follows from the proof of Theorem 4.1 in [MCF].
The stability is shown following the proof of Proposition 7.1 in [MCF], since (as we have pointed out above) the matrix–vector formulation (8) is (almost) identical to the matrix–vector formulation of [MCF, equation (3.4)–(3.5)] (recalling that here is in the role of in [MCF]). The system uses the same mass and stiffness matrices (but of different size), while the proof therein only uses the local Lipschitz continuity of the nonlinear terms, which holds here as well. The bounded operator in the velocity equation even simplifies part (B) of the stability proof of Proposition 7.1 in [MCF].
The consistency errors for are shown by the exact techniques of the consistency analysis [MCF, Lemma 8.1].
The uniformintime norm error bounds are proved by combining stability and consistency, verbatim as in [MCF, Section 9].
5.2 Convergence of the full discretization
Theorem 2
Consider the full discretization (9) of the mean curvature flow (2) in arbitrary codimension , using evolving surface finite elements of polynomial degree and linearly implicit BDF time discretization of order with . Suppose that the mean curvature flow problem in arbitrary codimension admits an exact solution that is sufficiently smooth on the time interval , and that the flow map is nondegenerate so that is a regular surface on the time interval .
Then, there exist constants , and such that for all mesh sizes and time step sizes satisfying the mild step size restriction
(13) 
(where can be chosen arbitrarily), the following error bounds for the lifts of the discrete position, velocity, tangential projection and mean curvature vector hold over the exact surface: provided that the starting values are ) accurate in the norm at time for , we have at time
for all . The constant is independent of , and , but depends on bounds of higherderivatives of the exact solution , on the final time and on .
Proof
Similarly to the semidiscrete error bounds: Since the ESFEM / linearly implicit BDF discretization (9) is (almost) identical to equation (5.1) in [MCF], the proof of this result directly follows as the proof of Theorem 6.1 in [MCF].
Remark 1
The stability and convergence results readily extend to higher dimensional submanifolds of dimension (of arbitrary codimension ), cf. Section 14 of [MCF], provided that optimalorder quasiinterpolation is used instead of the nodal interpolation (cf. [DziukElliott_acta, Lemma 4.3]), and requires evolving surface finite elements of degree and BDF methods of order . For the sixstep BDF method a new multiplierbased energy technique was developed in [Akrivisetal_BDF6]. The fully discrete stability proof in [MCF] should generalise to this approach.
6 Numerical examples for curves in
We performed the following numerical experiments for mean curvature flow of curves in :

A convergence test using planar curves where the exact solution is known.

A comparison test with Dziuk’s algorithm for curves [DeckelnickDziukElliott_acta] using circles and Angenent ovals [Angenent_ovals].

Experiments for space curves using established examples from the literature [BGN_curve_grad_flow, Pozzi_CSF_codim], e.g. a trefoil knot, still comparing with Dziuk’s algorithm.
All our numerical experiments were carried out in Matlab, using quadratic evolving surface finite elements, and BDF methods of various order specified in the experiments.
6.1 Convergence test
We are reporting on the errors of our algorithm for mean curvature flow in codimension for flat space curves. Simple test examples are constructed in this setting, by using the fact that the evolution of flat space curves evolving under the flow (2) is equivalent to their evolution under curve shortening flow.
Let the curve be a circle of initial radius in an arbitrary plane.
We consider the mean curvature flow of with initial value . Using the rotational symmetry of along flow, we obtain that its radius satisfies the ODE:
(14) 
The above initial value problems has the solution until final time . Therefore, the curvature of is given by .
We computed numerical approximations to the flow using quadratic finite elements () and using the step linearly implicit BDF method () for a circle of radius which lies in the plane rotated by . The starting values and for were computed as the interpolations of the exact values.
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