runge–kutta time discretization of parabolic diﬀerential ...mansour/papers/rksurf.pdf · time...

IMA Journal of Numerical Analysis (2005) of 22doi: 10.1093/imanum/dri017

Runge–Kutta time discretization ofparabolic differential equations on evolving surfaces

G. DziukAbteilung fur Angewandte Mathematik

University of FreiburgHermann-Herder-Strase 10D–79104 Freiburg i. Br.

Germany

Ch. Lubich and D. MansourMathematisches InstitutUniversity of TubingenAuf der Morgenstelle 10

D–72076 TubingenGermany

A linear parabolic differential equation on a moving surface is first discretized in space byevolving surface finite elements and then in time by an implicit Runge–Kutta method. For al-gebraically stable and stiffly accurate Runge–Kutta methods, unconditional stability of the fulldiscretization is proven and the convergence properties are analysed. Moreover, the implemen-tation is described for the case of the Radau IIA time discretization. Numerical experimentsillustrate the behaviour of the fully discrete method.

Keywords: parabolic PDE, evolving surface finite element method, implicit Runge–Kuttamethod

1. Introduction

Partial differential equations on surfaces appear in many applications. They arise in materialsciences, fluid mechanics and bio-physics [1], [9], [4], and it is important to discretize thesePDEs by efficient methods. The basic linear parabolic PDE on a moving surface is

u + u∇Γ (t) · v − ∆Γ (t)u = f on Γ (t). (1.1)

The moving surface Γ (t) with velocity v = v(x, t) is given and the solution u = u(x, t) (x ∈Γ (t), 0 6 t 6 T ) has to be computed. In [2] the evolving surface finite element method(ESFEM) was introduced in order to solve this model problem of diffusion and advection ona given moving surface. The work [2] develops a spatial discretization of (1.1) with piecewiselinear finite elements. The moving surface Γ (t) is approximated by a moving discrete surfaceΓh(t). An error analysis of this spatial discretization is given in [2] and [3].

The semi-discretization in space of (1.1) with piecewise linear surface finite elements leadsto an ODE system of the form

d

dt(M(t)U(t)) + A(t)U(t) = F (t), (1.2)

IMA Journal of Numerical Analysis c⃝ Institute of Mathematics and its Applications 2005; all rights reserved.

2 of 22 G. Dziuk, Ch. Lubich and D. Mansour

where M(t) is the evolving mass matrix and A(t) is the evolving stiffness matrix. U(t) denotesthe coefficient vector of the spatially discrete solution and F (t) is the discrete right-hand side.

Here we treat implicit Runge–Kutta time discretizations for the spatially discretized problem(1.2), aiming for temporal stability uniformly in the space discretization and for higher-orderbounds for the temporal error. Our key technical novelties are Lemma 4.1, which provides foran abstract framework in which we can treat the spatially discretized equation, and Lemma 7.1,which yields a stability estimate in the natural time-dependent norms for Runge–Kutta methodsthat are algebraically stable and stiffly accurate, such as the Radau IIA methods.

The paper is organized as follows: In Section 2 we recall the basic notation for PDEs onevolving surfaces. In Section 3 we describe the spatial discretization of (1.1) with piecewiselinear finite elements and derive the ODE system. Section 5 contains the numerical procedurefor the solution of the ODE system. Sections 7 and 8 are devoted to stability estimates anderror bounds. In Section 10 we describe the implementation of the time-stepping method andin Section 11 we give computational examples.

2. Parabolic equations on evolving surfaces

Assume that Γ (t), t ∈ [0, T ], is a smoothly evolving family of smooth d-dimensional hy-persurfaces in Rd+1. By v = v(x, t) we denote the vector valued given smooth velocity ofthe surface. Each surface is assumed to be compact. The conservation of a scalar quantityu = u(x, t), x ∈ Γ (t), t ∈ [0, T ] with a linear diffusive flux on Γ (t) can be modelled by the linearparabolic partial differential equation

u + u∇Γ · v − ∆Γ u = f on Γ (2.1)

together with the initial condition u = u0 on Γ0 = Γ (0). See [2] for a derivation of the PDE.By a dot we denote the material derivative

u =∂u

∂t+ v · ∇u, (2.2)

where here and in the following a · b =∑d+1

j=1 ajbj for vectors a and b in Rd+1. ∇ denotes theusual d + 1-dimensional gradient. The material derivative u only depends on the values of thefunction u on the space-time surface

GT =⋃

t∈(0,T )

Γ (t) × {t}.

Note that in general the quantities ∂u∂t or ∇u do not make sense on GT and only their combi-

nation in the material derivative (2.2) is well defined.By ∇Γ we denote the surface or tangential gradient on the surface Γ . This gradient is the

projection to the tangent space of the d + 1-dimensional gradient. For a smooth function gdefined in a neighbourhood of Γ we define

∇Γ g = ∇g −∇g · nn,

where n is a normal vector field to Γ . The components of the surface gradient are written as∇Γ g = ((∇Γ )1g, . . . , (∇Γ )d+1g). The tangential gradient only depends on the values of g on

Time Integration of parabolic PDEs on evolving surfaces 3 of 22

the surface Γ and not on the extension. For a more detailed discussion we refer to [5] and[2]. The Laplace-Beltrami operator on Γ then is defined as the tangential divergence of thetangential gradient:

∆Γ g = ∇Γ · ∇Γ g =d+1∑j=1

(∇Γ )j(∇Γ )jg.

Since Green’s formula holds on surfaces (here without boundary), a weak form of (2.1) isderived easily:

d

dt

∫Γ

uφ +∫

Γ

∇Γ u · ∇Γ φ =∫

Γ

uφ +∫

Γ

fφ. (2.3)

for all smooth φ : GT → R. Here one has used the Leibniz formula respectively the transporttheorem on surfaces,

d

dt

∫Γ

g =∫

Γ

g + g∇Γ · v.

We will use a similar rule for discrete surfaces, see Lemma 3.1.

3. The evolving surface finite element method

The weak form (2.3) serves as basis for a spatial finite element discretization of the PDE. Butin order to be able to compute all quantities, we first discretize the evolving surface itself.We approximate (interpolate) the smooth surface Γ (t) by a discrete polygonal surface Γh(t)homeomorphic to Γ (t), where h denotes the grid size. Then

Γh(t) =⋃

T (t)∈T (t)

T (t)

is the union of d-dimensional non degenerate simplices T which form an admissible triangulationT (t). Details of this construction can be found in [2]. We assume that the diameter of eachsimplex T , h(T ), satisfies the bounds c−1h 6 h(T ) 6 ch with a positive constant c uniformlyfor all times. Here h = maxT∈T h(T ).

On the discrete surface Γh we use a surface gradient, which has to be understood in apiecewise sense:

∇Γhg = ∇g −∇g · nhnh.

nh denotes the normal to the discrete surface.As finite element space on the discrete surface Γh(t) we choose

Sh(t) = {uh ∈ C0(Γh(t))|uh|T ∈ P1, T ∈ T (t)}.

Let ϕj(·, t) (j = 1, . . . , N) be the nodal basis of Sh(t), ϕj(ai(t), t) = δji, so that

Sh(t) = span{ϕ1(·, t), . . . , ϕN (·, t)}.

The vertices aj(t), j = 1, . . . , N of the simplices are taken to sit on the smooth surface Γ (t).The discrete surface has to be evolved by a piecewise linear velocity in order to stay a polygonalsurface. We define the discrete velocity of the discrete surface by

vh(x, t) =N∑

j=1

v(aj(t), t)ϕj(x, t)


Now define an adequate discrete material derivative on the discrete evolving surface,

uh =∂uh

∂t+ vh · ∇uh. (3.1)

Note that there is a slight clash of notation, since the dot is used for continuous and discretematerial derivative. But it will always be clear from the context which material derivative ismeant.

A very important property of the finite element space is that the (discrete) material deriva-tive of the basis functions vanishes:

ϕj = 0. (3.2)

We now discretize the PDE spatially by piecewise linear finite elements. For given initialvalue uh(·, 0) = uh0 ∈ Sh(0) we solve the system

d

dt

∫Γh

uhφh +∫

Γh

∇Γhuh · ∇Γh

φh =∫

Γh

uhφh +∫

Γh

f−lφh ∀φh ∈ Sh(t). (3.3)

Here by f−l : Γh → R we understand the extension of the function f : Γ → R constantly innormal direction to Γ . For a function fh : Γh → R, we let f l

h : Γ → R be such that (f lh)−l = fh.

Under suitable regularity assumptions an error estimate between continuous solution u andspatially discrete solution uh was proved in [2]:

sup06t6T

∥u(·, t) − ulh(·, t)∥2

L2(Γ (t)) +∫ T

0

∥∇Γ (t)(u(·, t) − ulh(·, t))∥2

L2(Γ (t)) dt 6 ch2.

The work [3] contains an optimal error estimate in the L2-norm

sup06t6T

∥u(·, t) − ulh(·, t)∥L2(Γ (t)) 6 ch2.

The discrete form (3.3) of PDE (2.1) is a system of ODEs. Let us derive a standard formfor this system. We define the evolving mass matrix M(t) and the stiffness matrix A(t) by

M(t)ij =∫

Γh(t)

ϕi(·, t)ϕj(·, t), A(t)ij =∫

Γh(t)

∇Γh(t)ϕi(·, t) · ∇Γh(t)ϕj(·, t)

for i, j = 1, . . . , N . The mass matrix is symmetric and positive definite. The stiffness matrix issymmetric and positive semidefinite only, because we are solving on closed surfaces. We denotethe discrete solution by

uh(x, t) =N∑

j=1

uj(t)ϕj(x, t)

and define u(t) ∈ RN as the column vector with entries uj(t). Then (3.3) can be written as

d

dt(M(t)u(t)) + A(t)u(t) = f(t), (3.4)

where we use f = (f1, . . . , fN ) with fj =∫

Γhf−lϕj .


For the convenience of the reader we give the arguments for the equivalence of (3.3) and (3.4).The ODE (3.4) follows from (3.3) by choosing φh = ϕj for j = 1, . . . , N and using (3.2). Forthe other direction let φh(x, t) =

∑Nj=1 αj(t)ϕj(x, t). Then with (3.2) we have φh =

∑Nj=1 αjϕj

and consequently

d

dt

∫Γh

uhφh +∫

Γh

∇Γhuh · ∇Γh

φh −∫

Γh

f−lφh

=d

dt

( N∑j=1

αj

∫Γh

uhϕj

)+

N∑j=1

αj

∫Γh

∇Γhuh · ∇Γh

ϕj −N∑

j=1

αj

∫Γh

f−lϕj

=N∑

j=1

αj

( d

dt

∫Γh

uhϕj +∫

Γh

∇Γhuh · ∇Γh

ϕj −∫

Γh

f−lϕj︸︷︷︸=0 by (3.4)

)+

N∑j=1

αj

∫Γh

uhϕj

=∫

Γh

uh

N∑j=1

αjϕj =∫

Γh

uhφh.

We will need the following formulae for the derivative of surface integrals with respect to aparameter.

Lemma 3.1 Let Γh(t), t ∈ [0, T ] be a discrete evolving surface and wh(·, t), zh(·, t) ∈ Sh(t) begiven functions. Then

d

dt

∫Γh

whzh =∫

Γh

whzh + whzh + whzh∇Γ · vh. (3.5)

With the tensorD(vh)ij = δij∇Γh

· vh − ((∇Γh)ivhj + (∇Γh

)jvhi)

(i, j = 1, . . . , n) we have for the derivative of Dirichlet’s integral

12

d

dt

∫Γh

|∇Γhwh|2 =

∫Γh

∇Γhwh · ∇Γh

wh +12

∫Γh

D(vh)∇Γhwh · ∇Γh

wh. (3.6)

A proof of this Lemma was given in [2] for smooth surfaces. The proof is easily adapted toevolving discrete surfaces.

4. The ODE system

We consider the large system of ordinary differential equations (3.4) on RN , viz.,

d

dt

(M(t)u(t)

)+ A(t)u(t) = f(t) , u(0) = u0, (4.1)

with symmetric positive definite matrix M(t) and symmetric positive semidefinite matrix A(t).With the finite element method these matrices are sparse.

We work with the norm

|w|2t = ⟨w |M(t) |w⟩ = w ·M(t)w, w ∈ RN ,


and the semi-norm∥w∥2

t = ⟨w |A(t) |w⟩ = w ·A(t)w, w ∈ RN .

Lemma 4.1 There are constants µ, κ (independent of the discretization parameter h and thelength of the time interval T ) such that

w ·(M(s) − M(t)

)z 6 (eµ(s−t) − 1) |w|t |z|t (4.2)

w ·(A(s) − A(t)

)z 6 (eκ(s−t) − 1) ∥w∥t ∥z∥t (4.3)

for all w, z ∈ RN and 0 6 t 6 s 6 T .

We will apply this lemma with s close to t. Note that then eµ(s−t) − 1 6 2µ(s − t) andeκ(s−t) − 1 6 2κ(s − t). Apart from the fact that M(t) and A(t) are symmetric positive semi-definite, the inequalities (4.2)-(4.3) are the only properties of the evolving-surface finite-elementequations (4.1) that will be used in the stability analysis of their time discretizations.

Proof. For w, z ∈ RN we define the discrete functions wh(x, t) =∑N

j=1 wjϕj(x, t) and zh(x, t) =∑Nj=1 zjϕj(x, t). Note that wh = zh = 0. Then by the transport formula from Lemma 3.1 we

have

w · (M(s) − M(t)) z =∫

Γh(s)

wh(·, s)zh(·, s) −∫

Γh(t)

wh(·, t)zh(·, t)

=∫ s

t

d

dσ

∫Γh(σ)

wh(·, σ)zh(·, σ) dσ

=∫ s

t

∫Γh(σ)

wh(·, σ)zh(·, σ)∇Γh(σ) · vh dσ

6 µ

∫ s

t

∥wh∥L2(Γh(σ))∥zh∥L2(Γh(σ)) dσ

= µ

∫ s

t

|w|σ|z|σ dσ

where we have used that maxσ∈[t,s] ∥∇Γh(σ) · vh(·, σ)∥L∞(Γh(σ)) is bounded by a constant µindependent of h and s, t, since vh is the linear interpolant of the continuous velocity. Withz = w, this inequality implies

|w|2s 6 |w|2t + µ

∫ s

t

|w|2σ dσ, 0 6 t 6 s 6 T,

and hence the Gronwall inequality yields

|w|2s 6 eµ(s−t) |w|2t .

Inserting this bound for |w|σ and |z|σ in the above inequality yields the first inequality (4.2).With Lemma 3.1 we get for the matrix A

w · (A(s) − A(t)) z

=∫

Γh(s)

∇Γh(s)wh(·, s) · ∇Γh(s)zh(·, s) −∫

Γh(t)

∇Γh(t)wh(·, t) · ∇Γh(t)zh(·, t)

=∫ s

t

d

dσ

∫Γh(σ)

∇Γh(σ)wh(·, σ) · ∇Γh(σ)zh(·, σ) dσ.


Polarization of formula (3.6), keeping in mind that wh = zh = 0 here, gives

w · (A(s) − A(t)) z

=∫ s

t

∫Γh(σ)

D(vh(·, σ))∇Γh(σ)wh(·, σ) · ∇Γh(σ)zh(·, σ) dσ

6 κ

∫ s

t

∥w∥σ∥z∥σ dσ

since maxσ∈[t,s] ∥D(vh(·, σ)∥L∞(Γh(σ)) is uniformly bounded by a constant κ. Using this in-equality together with the Gronwall inequality as above yields (4.3). ¤

5. Runge–Kutta time discretization

Method description. We reformulate (4.1) as the system (˙ = d/dt)1

y(t) + A(t)u(t) = f(t)y(t) = M(t)u(t)

(5.1)

and apply an m-stage implicit RK method in the standard way for differential-algebraic equa-tions [6, 7] (for ease of notation we take constant step size τ , this is not essential): we determinethe approximation un+1 to u(tn+1) at tn+1 = tn + τ in a time step starting from un via internalstages Uni, Yni and increments Uni, Yni (here the dot is only suggestive notation) by equationsof the form (5.1),

Yni + AniUni = Fni

Yni = MniUni

(5.2)

for i = 1, . . . ,m with Ani = A(tn + ciτ), Mni = M(tn + ciτ), Fni = f(tn + ciτ), satisfying theRK relations

un+1 = un + τm∑

j=1

bjUnj

Uni = un + τ

m∑j=1

aijUnj (i = 1, . . . ,m)

and in the same way

yn+1 = yn + τm∑

j=1

bj Ynj

Yni = yn + τ

m∑j=1

aij Ynj (i = 1, . . . ,m)

1There should be no confusion about the different uses of the dot notation. The dot represents a materialderivative in the PDE context, it represents a time derivative in the ODE context, and it is merely suggestivenotation in the Runge–Kutta schemes.


Method assumptions. The method is characterized by its coefficients aij , bj , ci with 0 6ci 6 1. We assume that the method has stage order q > 1 and classical order p > q. The RKcoefficient matrix (aij) is assumed invertible, and we denote its inverse by (wij).

The method is algebraically stable: the m × m matrix

(biaij + bjaji − bibj) is positive semi-definite, and all bi > 0 , (5.3)

and it is stiffly accurate:

cm = 1 and bj = amj for j = 1, . . . ,m , (5.4)

which impliesun+1 = Unm, yn+1 = Ynm, yn+1 = Mn+1un+1

with Mn+1 = M(tn+1).Well-known examples are the collocation methods at Radau nodes, of stage order q = m

and classical order p = 2m − 1.

6. Defects and errors

Defects. The solution of (4.1) satisfies the RK relations up to a defect (quadrature error)

u(tn+1) = u(tn) + τm∑

j=1

bj u(tn + cjτ) + δn+1

u(tn + ciτ) = u(tn) + τ

m∑j=1

aij u(tn + cjτ) + ∆ni (i = 1, . . . ,m)

and

y(tn+1) = y(tn) + τm∑

j=1

bj y(tn + cjτ) + rn+1

y(tn + ciτ) = y(tn) + τm∑

j=1

aij y(tn + cjτ) + Rni (i = 1, . . . ,m)

By the assumption of stiff accuracy, we have

δn+1 = ∆nm, rn+1 = Rnm.

For smooth solutions, we have by Taylor expansion (in suitable norms!)

δn+1 = O(τp+1), ∆ni = O(τ q+1)rn+1 = O(τp+1), Rni = O(τ q+1).

Errors. We consider the errors

en = un − u(tn)Eni = Uni − u(tn + ciτ)Eni = Uni − u(tn + ciτ)


and

ℓn = yn − y(tn) = Mnen

Lni = Yni − y(tn + ciτ) = MniEni

Lni = Yni − y(tn + ciτ)

Error equations. We subtract to obtain

Lni + AniEni = 0 (6.1)Lni = MniEni (6.2)

and RK relations with defects:

en+1 = en + τm∑

j=1

bjEnj − δn+1 (6.3)

Eni = en + τ

m∑j=1

aijEnj − ∆ni (i = 1, . . . ,m) (6.4)

and

ℓn+1 = ℓn + τm∑

j=1

bjLnj − rn+1 (6.5)

Lni = ℓn + τm∑

j=1

aijLnj − Rni (i = 1, . . . ,m). (6.6)

We noteℓn+1 = Lnm = MnmEnm = Mn+1en+1. (6.7)

Modified error equations. It turns out to be favourable to work with modifications of Enj

in the stability and error analysis. We set

E′nj = Enj + τ−1

m∑j=1

wji(M−1n Rni − ∆ni) . (6.8)

Then (6.4) becomes

Eni = en + τ

m∑j=1

aijE′nj − Dni with Dni = M−1

n Rni, (6.9)

and, in view of (5.4), Eq. (6.3) turns into

en+1 = en + τ

m∑j=1

bjE′nj − dn+1 with dn+1 = M−1

n rn+1. (6.10)

With this modification, the defects ∆ni and δn+1 have disappeared from the error equationsand the defects in (6.5)-(6.6) and (6.9)-(6.10) are related in a way that is very helpful in thestability analysis.


7. Stability

The following is the technical key result of this paper.

Lemma 7.1 If the Runge–Kutta method is algebraically stable and stiffly accurate, then thereexist τ0 > 0 depending only on µ, κ of Lemma 4.1 and CT depending on µ, κ, T such that forτ 6 τ0 and tn 6 T , the errors are bounded by

|en|2tn+ τ

n∑k=1

∥ek∥2tk

6 CT τn∑

k=1

|dk/τ |2tk+ CT τ

n−1∑k=0

m∑i=1

(∥Dki∥2

tk+ |Dki|2tk

).

Proof. The proof uses algebraic stability and stiff accuracy similarly to the proof of Theorem1.1 in [11], but here works with the time-dependent norms and the modified error equations ofthe previous section.

(a) For brevity, we write | · |n instead of | · |tn . We start from (6.10), take the squared normat tn+1 and estimate the terms in

|en+1|2n+1 =∣∣∣en + τ

m∑j=1

bjE′nj

∣∣∣2n+1

− 2⟨en + τ

m∑j=1

bjE′nj

∣∣∣ Mn+1

∣∣∣ dn+1

⟩+ |dn+1|2n+1. (7.1)

Expressing en by (6.9), we obtain for the first term

∣∣en + τ

m∑j=1

bjE′nj

∣∣2n+1

= |en|2n+1 + 2τ

m∑j=1

bj

⟨E′

nj

∣∣ Mn+1

∣∣ Enj + Dnj

⟩+ τ2

m∑i=1

m∑j=1

(bibj − biaij − bjaji)⟨E′

ni

∣∣ Mn+1

∣∣ E′nj

⟩, (7.2)

where the last term is non-positive by the assumption of algebraic stability (5.3). In the secondand third term we write Mn+1 = Mn + (Mn+1 − Mn). By condition (4.2) we have

|en|2n+1 =⟨en

∣∣ Mn+1

∣∣ en

⟩=

⟨en

∣∣ Mn

∣∣ en

⟩+

⟨en

∣∣ Mn+1 − Mn

∣∣ en

⟩6 (1 + 2µτ) |en|2n . (7.3)

In the middle term we write⟨E′

nj

∣∣ Mn+1

∣∣ Enj + Dnj

⟩=

⟨E′

nj

∣∣ Mn

∣∣ Enj + Dnj

⟩+

⟨E′

nj

∣∣ Mn+1 − Mn

∣∣ Enj + Dnj

⟩(7.4)

and estimate the two terms on the right separately.(b) In the first term we express MnE′

nj in terms of Lnj as follows. By (6.6) we have

Lnj = τ−1m∑

i=1

wji(Lni − ℓn + Rni).

By (6.2) we have Lni = MniEni and, using once more (6.9),

ℓn = Mnen = MnEni − τm∑

k=1

aikMnE′nk + MnDni,


so that we obtain, recalling Rni = MnDni,

Lnj = MnE′nj + τ−1

m∑i=1

wji(Mni − Mn)Eni.

On the other hand, by (6.1) we have

Lnj = −AnjEnj = −AnEnj − (Anj − An)Enj .

Combining these equations yields⟨E′

nj

∣∣ Mn

∣∣ Enj + Dnj

⟩= −

⟨Enj

∣∣ An

∣∣ Enj + Dnj

⟩−

⟨Enj

∣∣ Anj − An

∣∣ Enj + Dnj

⟩− τ−1

m∑i=1

wji

⟨Eni

∣∣ Mni − Mn

∣∣ Enj + Dnj

⟩.

These terms are now estimated using the Cauchy-Schwarz inequality, Young’s inequality, and(4.2)-(4.3):

−⟨Enj

∣∣ An

∣∣ Enj + Dnj

⟩6 −∥Enj∥2

n + ∥Enj∥n ∥Dnj∥n

6 −34 ∥Enj∥2

n + ∥Dnj∥2n

−⟨Enj

∣∣ Anj − An

∣∣ Enj + Dnj

⟩6 2κτ ∥Enj∥n ∥Enj + Dnj∥n

6 3κτ ∥Enj∥2n + κτ ∥Dnj∥2

n∣∣⟨Eni

∣∣ Mni − Mn

∣∣ Enj + Dnj

⟩∣∣ 6 2µτ |Eni|n |Enj + Dnj |n6 2µτ |Eni|2n + µτ |Enj |2n + µτ |Dnj |2n.

Taken all together, the first term on the right-hand side of (7.4) is bounded, for sufficientlysmall τ , by

⟨E′

nj

∣∣ Mn

∣∣ Enj + Dnj

⟩6 −2

3 ∥Enj∥2n + C

m∑i=1

|Eni|2n (7.5)

+ C ∥Dnj∥2n + C |Dnj |2n.

(c) To bound the last term in (7.4), we rewrite (6.9) as

E′nj = τ−1

m∑i=1

wji(Eni − en + Dni)

and use (4.2) to estimate

⟨E′

nj

∣∣ Mn+1 − Mn

∣∣ Enj + Dnj

⟩6 C|en|2n + C

m∑i=1

|Eni|2n + Cm∑

i=1

|Dni|2n. (7.6)


(d) The second term on the right-hand side of (7.1) is estimated using (5.4) as

en + τm∑

j=1

bjE′nj = en + τ

m∑j=1

amjE′nj = Enm + Dnm = Enm + dn+1

and the Cauchy-Schwarz inequality and (4.2) to obtain

2⟨en + τ

m∑j=1

bjE′nj

∣∣ Mn+1

∣∣ dn+1

⟩6 2 |Enm + dn+1|n+1 |dn+1|n+1

6 τ(1 + 2µτ) |Enm|2n + (1 + 2τ)τ |dn+1/τ |2n+1. (7.7)

(e) The above bounds contain terms |Eni|2n which we further estimate. With (6.4), werewrite

|Eni|2n =⟨en

∣∣ Mn

∣∣ Eni

⟩+ τ

m∑j=1

aij

⟨E′

nj

∣∣ Mn

∣∣ Eni

⟩−

⟨Dni

∣∣ Mn

∣∣ Eni

⟩.

The first term on the right-hand side is estimated as⟨en

∣∣ Mn

∣∣ Eni

⟩6 |en|n |Eni|n 6 1

2δ|Eni|2n + 12δ−1|en|2n

with a small constant δ > 0. Similarly, the last term is bounded by

−⟨Dni

∣∣ Mn

∣∣ Eni

⟩6 1

2δ|Eni|2n + 12δ−1|Dni|2n .

As in part (b), the expressions⟨E′

nj

∣∣ Mn

∣∣ Eni

⟩are rewritten as⟨

E′nj

∣∣ Mn

∣∣ Eni

⟩= −

⟨Enj

∣∣ An

∣∣ Eni

⟩−

⟨Enj

∣∣ Anj − An

∣∣ Eni

⟩− τ−1

m∑k=1

wjk

⟨Enk

∣∣ Mnk − Mn

∣∣ Eni

⟩and bounded by ∣∣⟨E′

nj

∣∣ Mn

∣∣ Eni

⟩∣∣ 6 Cm∑

k=1

∥Enk∥2n + C

m∑k=1

|Enk|2n.

Hence, choosing δ sufficiently small (but independent of τ), we obtain the bound

|Eni|2n 6 C|en|2n + Cτm∑

k=1

∥Enk∥2n + C

m∑k=1

|Dnk|2. (7.8)

(f) Combining (7.1)–(7.8), we thus have

|en+1|2n+1 − |en|2n + 12τ

m∑j=1

bj∥Enj∥2n 6 Cτ |en|2n + Cτ |dn+1/τ |2n+1

+ Cτ

m∑i=1

(∥Dni∥2

n + |Dni|2n)

.

Summing over n and using a discrete Gronwall inequality finally gives the stated result. ¤


8. Error bounds

If it holds that

sup06t6T

∣∣∣M(t)−1 dk(Mu)dtk

(t)∣∣∣t6 α for k = q + 1, q + 2 (8.1)∫ T

0

∥∥∥M(t)−1 dq+1(Mu)dtq+1

(t)∥∥∥2

tdt 6 α2, (8.2)

then Lemma 7.1 yields the following error bound of order q + 1, where q is the stage order.

Theorem 8.1 Under the regularity conditions (8.1)-(8.2), the error of the algebraically stableand stiffly accurate Runge–Kutta discretization of stage order q and classical order p > q + 1 isbounded by

|un − u(tn)|tn +(

τn∑

k=1

∥uk − u(tk)∥2tk

)1/2

6 CT α τ q+1

for tn = nτ 6 T .

Proof. It suffices to note that under conditions (8.1)-(8.2), the Peano representation of thequadrature errors Rni shows that the defects Dni = M−1

n Rni are bounded by

sup06nτ6T

|Dni|tn 6 Cατ q+1 , sup06nτ6T

|dn|tn 6 Cατ q+2,∑06nτ6T

∥Dni∥2tn

6(Cατ q+1

)2,

and to apply Lemma 7.1. ¤The classical order p is obtained if stronger regularity conditions are satisfied. Suppose that∥∥∥∥M(t)−1 dkj−1

dtkj−1

(A(t)M(t)−1

). . .

dk1−1

dtk1−1

(A(t)M(t)−1

) dl

dtl(M(t)u(t)

)∥∥∥∥t

6 β

for all k1 > 1, . . . , kj > 1 and l > q + 1 with k1 + . . . + kj + l 6 p, (8.3)and the same bounds in the norm | · |t for k1 + . . . + kj + l 6 p + 1.

(The 0-th derivative of AM−1 is just AM−1 itself.) Then, there is the following error bound offull order p.

Theorem 8.2 Under conditions (8.3), the error of the Runge–Kutta discretization is boundedby

|un − u(tn)|tn +(

τ

n∑k=1

∥uk − u(tk)∥2tk

)1/2

6 CT β τp

for tn = nτ 6 T .

Proof. The proof adapts that of Theorem 1 in [12] to the present situation. The trick is tomodify the error equations such that they have smaller defects. This will be achieved by aniterative procedure. Add the defect in (6.6) to Lni to obtain

L(1)ni = Lni + Rni


and define E(1)ni such that

L(1)ni = MniE

(1)ni .

By (6.2), we thus haveE

(1)ni = Eni + M−1

ni Rni.

We define L(1)ni such that (6.1) holds for the modified errors:

L(1)ni = −AniE

(1)ni = Lni − AniM

−1ni Rni.

We determine E(1)ni such that no defect appears in formula (6.4) for the modified errors:

E(1)nj = E′

nj = Enj + τ−1m∑

i=1

wji(M−1ni Rni − ∆ni).

We then get for these modified errors in the internal stages (but unmodified en and ℓn) equations(6.3)–(6.6) with modified defects

d(1)n+1 = M−1

n+1rn+1, D(1)ni = 0

r(1)n+1 = rn+1 − τ

m∑j=1

bjAnjM−1nj Rnj , R

(1)ni = −τ

m∑j=1

aijAnjM−1nj Rnj .

Here we used (5.4) in the formula for d(1)n+1, in the form that

∑mj=1 bjwji = 1 for i = m and 0

else and using (6.7).The construction is such that the defects R

(1)ni in the internal stages carry an additional

factor τ compared to Rnj , at the price that AM−1 is applied to Rnj . The relation (6.7) is nolonger fully satisfied for the modified errors, but we have

en+1 = E(1)nm + d

(1)n+1, ℓn+1 = L(1)

nm − rn+1.

By the order conditions for Runge–Kutta methods, the defects dn+1, rn+1 and d(1)n+1, r

(1)n+1 are

all O(τp+1) in the norm ∥ · ∥tn+1 under the regularity conditions (8.3). The defects R(1)ni are

O(τ q+2), one order higher than Rni.The construction can now be continued iteratively until, after p − q steps, all defects are of

size O(τp+1). With the stability lemma we then conclude to the full order O(τp) of the errors.We omit the technical details that ensure that condition (8.3) is indeed sufficient to achieve thefull order p; cf. [12] for a related situation. ¤

9. Regularity estimates

We show that under suitable conditions on the evolving surfaces the assumptions (8.1) and(8.2) can be fulfilled for the time discretization of (3.3). We do this for f = 0.

In the following we denote time derivatives of k-th order by the superscript (k). We alsouse this notation for k-th order discrete material derivatives according to (3.1). We omit theomnipresent argument t in all appearing functions and surfaces.

We start with formulae for higher order Leibniz rules.


Lemma 9.1 Assume that the following quantities exist and set a = ∇Γh· vh. Then there exist

polynomials gkl = gkl(a, a, . . . , a(l)), l = 1, . . . , k so that

dk

dtk

∫Γh

f =∫

Γh

f (k) +k∑

l=1

∫Γh

gklf(k−l). (9.1)

Similarly there exist polynomials Gkl = Gkl(B, B, . . . , B(l)) with the matrix B = D(vh), sothat

dk

dtk

∫Γh

∇Γhf · ∇Γh

φh =∫

Γh

∇Γhf (k) · ∇Γh

φh +k∑

l=1

Gkl∇Γhf (k−l) · ∇Γh

φh (9.2)

for any function φh with φh = 0.

Proof. One easily proves this by induction with the help of the Leibniz rules from Lemma 3.1.The rule for Dirichlet’s integral is used in a polarized form. ¤

In the following we assume that a and B are sufficiently often continuously differentiablewith respect to time. Then gkl and Gkl are bounded independently of the grid size h and wecan prove the following lemma.

Lemma 9.2 Let uh =∑N

j=1 ujϕj be the solution of (3.3). Then

M−1(Mu)(k) · (Mu)(k) 6 ck∑

l=0

∥u(l)h ∥2

L2(Γh) (9.3)

and

M−1(Mu)(k) · AM−1(Mu)(k) 6 ck∑

l=0

(∥u(l)

h ∥2L2(Γh) + ∥∇Γh

u(l)h ∥2

L2(Γh)

). (9.4)

Proof. We set w = M−1(Mu)(k) and wh =∑N

j=1 wjϕj . We then observe∫Γh

whϕj =N∑

k=1

wk

∫Γh

ϕkϕj = (Mw)j = (Mu)(k)j =

dk

dtk

∫Γh

uhϕj .

Then by Lemma 9.1 we have that∫Γh

whφh =∫

Γh

u(k)h φh +

k∑l=1

∫Γh

gklu(k−l)h φh ∀φh ∈ Sh(t)

This means that

wh = u(k)h +

k∑l=1

Ph

(gklu

(k−l)h

)with the L2-projection Ph onto Sh. Here we used the fact that the material derivatives ofuh ∈ Sh again are elements of Sh, since ϕi = 0. Then,

√w · Mw = ∥wh∥L2(Γh) 6 ∥u(k)

h ∥L2(Γh) +k∑

l=1

∥gklu(k−l)h ∥L2(Γh),


which yields (9.3). We write similarly

√w · Aw = ∥∇Γh

wh∥L2(Γh) 6 ∥∇Γhu

(k)h ∥L2(Γh) +

k∑l=1

∥∇ΓhPh

(gklu

(k−l)h

)∥L2(Γh). (9.5)

Here, gkl = gkl(a, a, . . . , a(l)) are piecewise constant functions on the discrete surface Γh, sincea = ∇Γh

· vh.We now show the proof of (9.4) for the case k = 1 and discuss the general case later. For

k = 1 we estimate the last term on the right hand side of (9.5) as follows:

∥∇ΓhPh(auh)∥L2(Γh)

6 ∥∇ΓhPh(uh(∇Γh

· vh − (∇Γ · v)−l))∥L2(Γh)

+∥∇Γh

(Ph(uh(∇Γ · v)−l) − uh(∇Γ · v)−l

)∥L2(Γh)

+∥∇Γh(uh(∇Γ · v)−l)∥L2(Γh)

6 c

h∥Ph(uh(∇Γh

· vh − (∇Γ · v)−l))∥L2(Γh)

+c

h∥Ph(uh(∇Γ · v)−l) − uh(∇Γ · v)−l∥L2(Γh)

+c∥uh∥L2(Γh) + c∥∇Γhuh∥L2(Γh).

By interpolation estimates from [2] and since uh is piecewise linear on Γh and v is sufficientlysmooth we have that

∥Ph(uh(∇Γ · v)−l) − uh(∇Γ · v)−l∥L2(Γh) 6 ch2(∥uh∥L2(Γh) + ∥∇Γh

uh∥L2(Γh)

).

For the remaining term we observe:

∥Ph(uh(∇Γh· vh − (∇Γ · v)−l))∥L2(Γh)

6 ∥uh∥L2(Γh)∥∇Γh· vh − (∇Γ · v)−l∥L∞(Γh) 6 ch∥uh∥L2(Γh),

since vh is the linear interpolant of v−l.Thus we have proved the estimate

∥∇ΓhPh(auh)∥L2(Γh) 6 c

(∥uh∥L2(Γh) + ∥∇Γh

uh∥L2(Γh)

)and with (9.5) for k = 1 we arrive at the inequality

√w · Aw 6 ∥∇Γh

uh∥L2(Γh) + c(∥uh∥L2(Γh) + ∥∇Γh

uh∥L2(Γh)

).

This gives (9.4) in the case k = 1.The case k > 1 is similar but more technical. We only give the basic ingredients for the proof.

In this case one has to deal with polynomials of the time derivatives a, . . . , a(k) of a = ∇Γh· vh.

The most important formula is the material derivative of the tangential gradient. For a vectorvalued function z one has the identity

(∇Γh· z)· = ∇Γh

· z − trace(A∇Γh

z)

(9.6)


with the matrix Alr =(∇Γh

)lvhr −

∑d+1s=1 nhsnhl

(∇Γh

)rvhs (l, r = 1, . . . , d + 1). One then has

to use this formula for z = vh and follow the ideas of the case k = 1.¤

According to the previous lemma we now have to prove a priori estimates for the materialtime derivatives of the discrete solution uh of the problem (3.3).

Theorem 9.1 Assume that the given evolution of Γh(t), t ∈ [0, T ], is sufficiently smooth. Thenwe have the a priori estimate

sup(0,T )

∥u(k)h ∥2

L2(Γh) +∫ T

0

∥∇Γhu

(k)h ∥2

L2(Γh) dt 6 ck∑

l=0

∥u(l)h (·, 0)∥2

L2(Γh0). (9.7)

Proof. The discrete PDE (3.3) is equivalent to the system

d

dt

∫Γh

uhϕj +∫

Γh

∇Γhuh · ∇Γh

ϕj = 0 j = 1, . . . , N. (9.8)

We differentiate this equation k times with respect to time and use Lemma 9.1. Then∫Γh

u(k+1)h ϕj +

k+1∑l=1

gk+1,lu(k+1−l)h ϕj

+∫

Γh

∇Γhu

(k)h · ∇Γh

ϕj +k∑

l=1

Gkl∇Γhu

(k−l)h · ∇Γh

ϕj = 0.

We multiply this equation with u(k)j and sum over j = 1, . . . , N and get∫

Γh

u(k+1)h u

(k)h +

∫Γh

|∇Γhu

(k)h |2

= −k+1∑l=1

∫Γh

gk+1,lu(k+1−l)h u

(k)h −

k∑l=1

∫Γh

Gkl∇Γhu

(k−l)h · ∇Γh

u(k)h .

Standard arguments then lead to the estimate

d

dt∥u(k)

h ∥2L2(Γh) + ∥∇Γh

u(k)h ∥2

L2(Γh)

6 c∥u(k)h ∥2

L2(Γh) + ck−1∑l=0

(∥u(l)

h ∥2L2(Γh) + ∥∇Γh

u(l)h ∥2

L2(Γh)

).

From this we get inequality (9.7).¤

Corollary 9.1 Under suitable smoothness assumptions on Γ (t), t ∈ [0, T ], conditions (8.1)and (8.2) are fulfilled with

α = ck∑

l=0

∥u(l)h (·, 0)∥L2(Γh0).


Proof. This follows by a combination of Lemma 9.2 and Theorem 9.1. ¤Without proof, we expect that condition (8.3) is satisfied for smooth solutions of equations onsmooth closed surfaces. As already the case of plane domains shows, condition (8.3) fails tohold uniformly in the mesh size on surfaces with boundary; cf. the discussion of order reductionphenomena in [10, 12, 13].

10. Implementation

Implementation of ESFEM. The implementation of the evolving surface finite elementmethod is analoguous to a Cartesian finite element method. The only differences are that thevertices of the elements are (d + 1)-dimensional, i.e., the d-dimensional simplices making upthe discrete surface live in Rd+1 and that the matrices M and A have to be assembled in eachtime step anew. See also Section 7.2 in [2]. The program is set up in such a way that for giventime t it provides the sparse matrices M(t) and A(t) in suitable form.

Implementation in the RADAU code. We indicate how to implement the method in thecode RADAU5 or RADAU of Hairer and Wanner [7, 8]. That code is written for problems ofthe form

Bx = f(t, x)

with a constant, not necessarily invertible square matrix B. Our problem (5.1) is of this formwith

x =(

yu

), B =

(I 00 0

), f(t, x) =

(0 −A(t)I −M(t)

)(yu

).

The code requires the user to specify a Jacobian at each step, which here is chosen as

Jn =(

0 −A(tn+1)I −M(tn+1)

).

It also requires a routine for solving linear systems with the matrices

τ−1λiB − Jn =(

τ−1λiI A(tn+1)−I M(tn+1)

)for i = 1, . . . ,m,

where λi are the eigenvalues of the inverse coefficient matrix (wij) (which are real or come incomplex conjugate pairs, with positive real part). By straightforward block Gaussian elimina-tion, this reduces to solving linear systems with the m matrices

τ−1λiM(tn+1) + A(tn+1). (10.1)

Building the mass and stiffness matrices and solving linear systems with the matrices (10.1)are the main computational efforts in the method.

Iteration for the internal stages. Let us describe the approach in more detail. The equations(5.2)-(5.3) can be combined into a linear system for the internal stages Uni:

MniUni + τ

m∑j=1

aijAnjUnj = Mnun. (10.2)


We solve this iteratively. Given an iterate U(k)ni (i = 1, . . . ,m), we determine

U(k+1)ni = U

(k)ni + δU

(k)ni

from the equations

Mn+1δU(k)ni + τ

m∑j=1

aijAn+1δU(k)nj = ρ

(k)ni

with

ρ(k)ni = Mnun − MniU

(k)ni − τ

m∑j=1

aijAnjU(k)nj .

Equivalently, with δU(k)n =

(δU

(k)ni

)m

i=1and Oι = (aij), this is rewritten as

(Im ⊗ Mn+1 + τOι ⊗ An+1)δU (k)n = ρ(k)

n .

If we diagonalize Oι−1 = TΛT−1 with Λ = diag(λi) and set

δV (k)n = (T−1 ⊗ I)δU (k)

n , σ(k)n = τ−1(ΛT−1 ⊗ I)ρ(k)

n ,

then the system becomes

(τ−1Λ ⊗ Mn+1 + Im ⊗ An+1)δV (k)n = σ(k)

n ,

which decouples into the m equations

(τ−1λiMn+1 + An+1)δV(k)ni = σ

(k)ni (i = 1, . . . ,m).

This yields δU(k)n = (T ⊗ I)δV (k)

n and the new iterate U(k+1)n = U

(k)n + δU

(k)n .

Convergence of the iteration. We show that under the given conditions on M(t) and A(t)and on the Runge–Kutta method, the above iteration converges linearly with rate O(τ). In thefollowing result we denote the iteration errors as E(k)

ni = U(k)ni − Uni.

Lemma 10.1 There exist τ0 > 0 and C, both depending only on µ, κ of Lemma 4.1, such thatfor τ 6 τ0,

m∑j=1

(∣∣E(k+1)nj

∣∣2tn+1

+ τ∥∥E(k+1)

nj

∥∥2

tn+1

)6 Cτ2

m∑j=1

(∣∣E(k)nj

∣∣2tn+1

+ τ∥∥E(k)

nj

∥∥2

tn+1

).

Proof. We find

Mn+1E(k+1)ni + τ

m∑j=1

aijAn+1E(k+1)nj

= (Mn+1 − Mni)E(k)ni + τ

m∑j=1

aij(An+1 − Anj)E(k)nj .


We multiply this equation with biE(k+1)ni , sum over i and use the algebraic stability condition

(5.3) on the left-hand side and conditions (4.2)-(4.3) on the right-hand side to conclude

m∑i=1

bi

∣∣E(k+1)ni

∣∣2n+1

+ 12τ

∥∥∥ m∑i=1

biE(k+1)ni

∥∥∥2

n+1

6 2µτ

m∑i=1

∣∣E(k+1)ni

∣∣n+1

·∣∣E(k)

ni

∣∣n+1

+ Cκτ2m∑

i=1

m∑j=1

∥∥E(k+1)ni

∥∥n+1

·∥∥E(k)

nj

∥∥n+1

.

This inequaltiy does not yet bound∑m

i=1 bi

∥∥E(k+1)ni

∥∥2

n+1. For this purpose we rewrite the error

equation equivalently as

τ−1m∑

j=1

wjiMn+1E(k+1)ni + An+1E(k+1)

nj

= τ−1m∑

j=1

wji(Mn+1 − Mni)E(k)ni + (An+1 − Anj)E(k)

nj .

We multiply by bjE(k+1)nj , sum over j and use the algebraic stability condition (5.3) and the stiff

accuracy condition (5.4) on the left-hand side and again conditions (4.2)-(4.3) on the right-handside to obtain

12τ−1

∣∣E(k+1)nm

∣∣2n+1

+m∑

j=1

bj

∥∥E(k+1)nj

∥∥2

n+1

6 Cµm∑

j=1

m∑i=1

∣∣E(k+1)nj

∣∣n+1

·∣∣E(k)

ni

∣∣n+1

+ 2κτm∑

j=1

∥∥E(k+1)nj

∥∥n+1

·∥∥E(k)

nj

∥∥n+1

.

Combining the two inequalities and using Young’s inequality on the right-hand side yields theresult. ¤

11. Numerical experiments

We take up Example 7.3 from [2], which is a PDE on a moving surface with time-dependentcurvature. The surface is given as the level set

Γ (t) = {x ∈ R3 : d(x, t) = 0} with d(x, t) =x2

1

a(t)+ x2

2 + x23 − 1,

which is an ellipsoid with time-dependent axis. We have chosen a(t) = 1 + 14 sin(t). As exact

continuous solution, we choose u(x, t) = e−6tx1x2 and compute a right-hand side for the PDEfrom the equation f = ut + v · ∇u + u∇Γ · v − ∆Γ u.

In a first numerical experiment we consider the evolving surface finite element equations (3.4)for four spatial refinement levels, with 22ℓ+6 + 2 meshpoints, for ℓ = 1, 2, 3, 4. We integrate thedifferential equations in time numerically using the Radau IIA method with three stages with


10−2 10−1 10010−7

10−6

10−5

10−4

10−3

10−2

10−1

step size

erro

r(M

)

10−2 10−1 10010−7

10−6

10−5

10−4

10−3

10−2

10−1

step size

erro

r(A

)

Fig. 1. Errors in L2-norm (left) and energy semi-norm (right) vs. time step size for four spatial refinements att = 1.

fixed time step size τ . At time t = 1, we collect the errors e(x, t) = un(x)−u(x, t) (with nτ = t)at the mesh points of the surface into a vector e ∈ RN and consider the norm and semi-normdefined by the mass and stiffness matrix, respectively, at time t,

|e|t =(e · M(t)e

)1/2 and ∥e∥t =(e · A(t)e

)1/2.

In Figure 1 we plot |e|t and ∥e∥t at t = 1 to the left and right, respectively, versus the timestep size τ . We observe the full order of temporal convergence p = 5 and error independenceof the spatial refinement level in the regime where the temporal error dominates the spatialerror. In the complementary regime, for smaller time steps, we clearly observe a faster rateof spatial convergence in the L2-norm than in the energy seminorm, in accordance with theconvergence theory for the evolving surface finite element method in [2, 3]. Higher refinementlevels correspond to lower-lying error curves.

10−7

10−6

10−5

10−4

10−3

10−2

100 101 102 103 104CPU time

erro

r(M

)

10−7

10−6

10−5

10−4

10−3

10−2

100 101 102 103 104CPU time

erro

r(A

)

Fig. 2. Errors in L2-norm (left) and energy semi-norm (right) vs. CPU time for four spatial refinements andten tolerances at t = 1.

In a second numerical experiment we use the time integrator with variable time steps asprovided by the RADAU5 code of [7]. In Figure 2 we plot the errors in the M -norm andA-seminorm versus computing time in seconds on a standard PC for ten local error tolerancesranging from Atol = Rtol = 10−1 to Atol = Rtol = 10−10. Here, each curve correponds to the


errors of one spatial mesh for the various tolerance parameters, and higher spatial refinementcorresponds to smaller errors.

Acknowledgement: This work was supported by the Deutsche Forschungsgemeinschaft DFGvia the SFB/TR 71 ”Geometric Partial Differential Equations”.

References

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