A HIGH-ORDER FICTITIOUS-DOMAIN METHOD FOR THEADVECTION-DIFFUSION EQUATION ON TIME-VARYING DOMAIN

CHUWEN MA∗, QINGHAI ZHANG† , AND WEIYING ZHENG‡

Abstract. We develop a high-order finite element method to solve the advection-diffusion equation on a time-varying domain. The method is based on a characteristic-Galerkin formulation combined with the kth-order backwarddifferentiation formula (BDF-k) and the fictitious-domain finite element method. Optimal error estimates of thediscrete solutions are proven for 2 ≤ k ≤ 4 by taking account of the errors from interface-tracking, temporaldiscretization, and spatial discretization, provided that the (k+1)th-order Runge-Kutta scheme is used for interface-tracking. Numerical experiments demonstrate the optimal convergence of the method for k = 3 and 4.

Key words. Free-surface problem, fictitious-domain finite element method, high-order scheme, interface-trackingalgorithm, advection-diffusion equation.

AMS subject classifications. 65M60, 65L06, 76R99

1. Introduction. Multiphase flows with time-varying domains have an extremely wide appli-cation range in science and engineering. The deformation of fluid phases and consequent complexinteractions of multiple time and length scales pose great challenges to the design of accurate,efficient, and simple algorithms for these moving-boundary problems.

In terms of the relative position of the moving domain to the discretization mesh, currentnumerical methods for moving-boundary problems can be roughly classified into two regimes.

In body-fitted methods, the mesh is arranged to follow the moving phase, which makes it easyto implement the boundary conditions of governing equations. This mesh-domain alignment canalso be exploited in numerical analysis to provide thorough error estimates; see [6,7] for some earlyworks on elliptic and Maxwell interface problems. However, these conveniences incur the expenses ofmesh regeneration and data migration across the entire computational domain at each time step [1].Apart from the efficiency issue, another major concern of body-fitted methods is how to maintaincertain degree of mesh regularity to prevent ill-conditioning at the presence of abrupt movementsand/or large deformations of the phases. A typical example of body-fitted methods is the arbitraryLagrangian-Eulerian (ALE) approach; see [15,17,24] for some recent advancements of ALE.

In the other regime of unfitted methods, the mesh for the bulk phase is fixed while the movingboundary is allowed to cross the fixed mesh, resulting in irregular grids or cut cells near the boundarywhere the discretization of governing equations and the enforcement of boundary conditions haveto be adjusted appropriately. Popular methods in this regime include the immersed boundarymethod [28], the immersed interface method (IIM) [20,22], and the extended finite element method(XFEM). As the main ideas of XFEM, the degrees of freedom for the cut cells are increased andpenalty terms are added to enforce boundary conditions weakly. Fires and Zilian presented a first-order XFEM method by using backward Euler for time integration [10]. Based on a space-time

∗School of Mathematical Science, University of Chinese Academy of Sciences. Institute of Computational Mathe-matics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy ofSciences, Beijing, 100190, China. (chuwenii@lsec.cc.ac.cn)†School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang Province, 310027

China. The second author was supported in part by China NSF grant 11871429. (qinghai@zju.edu.cn)‡LSEC, NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of

Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China. School of MathematicalScience, University of Chinese Academy of Sciences. The third author was supported in part by the National ScienceFund for Distinguished Young Scholars 11725106 and by China NSF grant 11831016. (zwy@lsec.cc.ac.cn)

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discontinuous Galerkin discretization, Lehrenfeld and Reusken [18] proposed a two-dimensionalsecond-order XFEM scheme, which was extended to three dimensions by Christoph [9]. Recently,Guo [12] analyzed a backward Euler immersed finite element method for solving parabolic movinginterface problems. Apart from the aforementioned methods, XFEM ideas can also be found inCutFEM [2, 5, 13], the interface-penalty finite element method [25, 32], and the fictitious domainmethod [3, 4, 16].

The fidelity of simulating physical processes on time-varying domains is very much influencedby loci of the domain boundary at each time step. In order to achieve high-order accuracy for theentire solver, an interface-tracking algorithm with at least the same order of accuracy is needed totracking the moving boundary, even in the case of mild deformations. In the immersed boundarymethod, the moving boundary is represented by line segments of interface markers and tracked bya fronting tracking method. As such, the overall accuracy is at best second-order [33]. For IIMs,there exist fourth-order schemes for stationary interfaces [21], but we are not aware of any fourth-order IIMs capable of handling nontrivial deforming boundaries. Within the framework of XFEM,one can design high-order schemes for problems with stationary interfaces (cf. [25]), but still facegreat challenges in the case of moving boundaries. For other finite element methods of high-orderaccuracy [12,17–19,23], it is often assumed that the loci of the boundary curve are known a prioriat each time step. However, this assumption does not hold for problems with nontrivial boundarymovements, even when analytic expressions of these boundary movements are already given.

To the best of our knowledge, there exist no third- and fourth-order methods that can handlemoving-boundary problems with large nontrivial deformations of the domain boundaries. A mainreason for this absence is that high-order interface tracking has not been coupled to finite elementsmethods and finite different/volume methods. This is not surprising since third- and fourth-orderinterface tracking methods have not been available until recently [35]. On the other hand, as thescience of multiphase flows evolves toward more and more complex phenomena, there is a growingneed of higher-order methods for moving boundary problems.

We answer this need by developing in this work third-order and fourth-order methods for nu-merically solving the advection-diffusion equation (2.1) on time-varying domains. For the movingboundaries, we track their loci to fourth-order accuracy by the recent cubic MARS method [35],which features topological modeling of fluids [36] and a rigorous analytic framework for error es-timates [33]. For spatial discretization, we adopt a high-order fictitious-domain method on a fixedfinite element mesh that covers the full movement range of the deforming domain. For tempo-ral integration, we apply the BDF-k schemes [24] to a Lagrangian form of the advection-diffusionequation where the coordinates are defined by characteristic tracing along the driving velocity field.

The main contribution of this work is the development of a high-order method for solvingthe advection-diffusion equation on time-varying domains; our method is also established on solidground of numerical analysis as follows.

(a) We prove the stability of our method under weighted energy norm. As a main difficulty,the composite function un−jh Xn,n−j

τ does not belong to the finite element space at tnand thus can not serve as a test function, where un−jh is the discrete solution at tn−j and

Xn,n−jτ is the discrete flow map from tn to tn−j . We overcame this difficulty by introducing

a modified Ritz projection onto the finite element space.(b) We present thorough error analysis for our method by taking full consideration of all errors

from interface-tracking, spatial discretization, and temporal integration. Our main resultis that, if the discrete solution is obtained from the BDF-k scheme, the finite element dis-cretization with piecewise Qk-polynomials, the RK-k (kth-order Runge-Kutta) method for

2

interface-tracking, then the solution errors in approximating regular solutions is O(τk−1/2)under the energy norm, where τ = O(h) is the time step size and k = 2, 3, 4. In addi-tion, this error estimate can be improved to O(τk) if we use the RK-(k + 1) scheme forinterface-tracking.

The rest of this paper is organized as follows. In Section 2, we formalize the model problemand introduce some notations. In Section 3, we discuss the interface-tracking algorithms and deriveerror estimates between the exact boundary and the approximate boundary. Then we formulateour method as the weak discrete form (4.5) in Section 4 and prove its well-posedness in Section 5.In Section 6, we derive a priori error estimates for the entire solver. In Section 7, we present resultsof numerical experiments to demonstrate the third- and fourth-order convergence of our method.

2. The model problem. The advection-diffusion equation with initial and boundary condi-tions is proposed as follows

∂u

∂t+w · ∇u−∆u = f in Ωt, (2.1a)

u = 0 on Γt, (2.1b)

u(0) = u0 in Ω0, (2.1c)

where Ωt is a time-varying domain in R2, Γt = ∂Ωt is the boundary of Ωt, w(x, t) is the velocityof fluid which occupies Ωt, u(x, t) stands for the tracer transported by the fluid, and f(x, t) standsfor the source term distributed in R2 and has a compact support. The equation has been scaled sothat the diffusion coefficient before ∆u is unit. Since we focus on linear problems, the velocity w isassumed to be given and satisfies divw = 0. For convenience in analysis, we assume that Γt is Cr-smooth for r ≥ 4 and w can be extended smoothly to the exterior of Ωt, that is, w ∈ Cr(R2×[0, T ]).

We define the Lagrangian coordinates by the solution to the ordinary differential equations

dX(t)

dt= w(X(t), t) ∀ t > s ≥ 0; X(s) = xs. (2.2)

Since w is Cr-smooth, (2.2) has a unique solution for every s ≥ 0 and every xs ∈ R2. The graphof X is just the characteristic curve of u. To specify the dependence of X on the initial value, themapping from xs to X(t) is denoted by

X(t; s,xs) := X(t), t ≥ s.

The physical domain is driven by the fluid and can be defined by means of the flow map X

Ωt := X(t; 0,x) : x ∈ Ω0 . (2.3)

Since any flow map is a diffeomorphism, all topology features of Ωt stay the same as thoseof Ω0. In the cubic MARS method for interface tracking, a fluid phase with arbitrarily complextopology can be represented by a partially order set of oriented Jordan curves that are pairwisealmost disjoint [36]. Thanks to the generality of this representation, it suffices to only consider thecase of Ωt being simply connected, i.e. Γt is a positively oriented Jordan curve, since the algorithmsand analyses in this work extend straightforwardly to multi-connected domains.

Using (2.2), the material derivative of u is defined by

du

dt=∂u

∂t+w · ∇u. (2.4)

3

Then (2.1a) can be written into a compact form

du

dt−∆u = f in Ωt. (2.5)

We introduce some notations for Sobolev spaces and norms. For a domain Ω ⊂ R2, let L2(Ω) bethe space of square-integrable functions on Ω and L∞(Ω) the space of essentially bounded functions.For any integer m > 0, define

Hm(Ω) :=v ∈ L2(Ω) :

∂i+jv

∂xi1∂xj2

∈ L2(Ω), 0 ≤ i+ j ≤ m,

Wm,∞(Ω) :=v ∈ L∞(Ω) :

∂i+jv

∂xi1∂xj2

∈ L∞(Ω), 0 ≤ i+ j ≤ m.

Let Hm0 (Ω) be the completion of C∞0 (Ω) under the norm of Hm(Ω). Throughout this paper, vector-

valued quantities will be denoted by boldface symbols, such as L2(Ω) = L2(Ω)2, and matrix-valuedquantities will be denoted by blackboard bold symbols, such as L2(Ω) = L2(Ω)2×2. Their normsare defined by

‖v‖L2(Ω) =(‖v1‖2L2(Ω) + ‖v2‖2L2(Ω)

)1/2, ‖A‖L2(Ω) =

(∑2i,j=1 ‖Aij‖

2L2(Ω)

)1/2

,

‖v‖L∞(Ω) = max‖vi‖L∞(Ω) : i = 1, 2

, ‖A‖L∞(Ω) = max

‖Aij‖L∞(Ω) : 1 ≤ i, j ≤ 2

.

3. Interface-tracking algorithms. We first consider the tracking algorithm for Ωt whichis the closure of Ωt. The purpose is to find a discrete approximation to the continuous mapping[0, T ]→

Ωt : 0 ≤ t ≤ T

. Let tn = nτ , n = 0, 1, · · · , N , be a uniform partition of the interval [0, T ]

where T > 0 is the final time and τ = T/N . Write Xn−1,n := X(tn; tn−1, ·) for any n > 0. Theuniqueness of the solution to (2.2) implies that the mapping Xn−1,n: Ωtn−1

→ Ωtn is one-to-one.For any 1 ≤ i ≤ n, the multi-step mapping is defined by

Xn−i,n := Xn−1,n Xn−2,n−1 · · · Xn−i,n−i+1, Ωtn = Xn−i,n(Ωtn−i).

We introduce the shorthand notationXn,i := (Xi,n)−1. The interface-tracking problem is to seek anapproximate solution to (2.2) for each initial value x ∈ Γtn−1

. Throughout the following statement,the notation f . g means that f ≤ Cg with a constant C > 0 independent of sensitive quantities,such as the segment size η for interface-tracking, the spatial mesh size h, the time-step size τ , andthe number of time steps N . Moreover, f h g means that f . g and g . f hold simultaneously.

3.1. The approximate flow map. Given an approximate domain Ωn−1 of Ωtn−1, the ap-

proximate domain Ωn = Xn−1,nτ (Ωn−1) is defined by the RK-k scheme for (2.2) [33]. For any

xn−1 ∈ Ωn−1, the point xn = Xn−1,nτ (xn−1) ∈ Ωn is calculated as follows:

x(1) = xn−1,

x(i) = xn−1 + τ

i−1∑j=1

akijw(x(j), t(j)), t(j) = tn−1 + ckj τ, 2 ≤ i ≤ nk,

xn = xn−1 + τ

nk∑i=1

bkiw(x(i), t(i)).

(3.1)

4

Here akij , bki , c

ki are coefficients of the RK-k scheme, satisfying akij = 0 if j ≥ i, and nk is the

number of stages. We denote the mapping from xn−1 to x(i) by

φ(i)n−1(xn−1) := x(i), i = 1, · · · , nk.

Let I be the identity map. Then Xn−1,nτ can be represented explicitly as follows

Xn−1,nτ = I + τ

nk∑i=1

bkiw(φ

(i)n−1(·), t(i)

). (3.2)

The multi-step mapping from Ωn−i to Ωn is defined by

Xn−i,nτ = Xn−1,n

τ Xn−2,n−1τ · · · Xn−i,n−i+1

τ , 1 ≤ i ≤ n. (3.3)

The inverse of Xi,nτ is denoted by Xn,i

τ := (Xi,nτ )−1. We only consider the case of 1 ≤ k ≤ r − 1.

3.2. Useful estimates for Xn−i,n and Xn−i,nτ . For any t ≥ s ≥ 0 and x ∈ R2, the Jacobi

matrix of x→X(t; s,x) is given by

J(t; s,x) :=∂X(t; s,x)

∂x= I +

∫ t

s

∇w(X(ξ; s,x), ξ)J(ξ; s,x)dξ. (3.4)

Since divw = 0, it is well-known that det(J) ≡ 1 in R2 for all t ≥ s [8]. For convenience, we denotethe Jacobi matrices of the discrete flow maps by

Jn−i,n(x) := J(tn; tn−i,x), Jn−i,nτ (x) :=∂Xn−i,n

τ (x)

∂x.

The Jacobi matrices of Xn,n−i, Xn,n−iτ are denoted by Jn,n−i :=

(Jn−i,n

)−1and Jn,n−iτ :=(

Jn−i,nτ

)−1, respectively. Clearly det(Jn−i,n) = det(Jn,n−i) ≡ 1 in R2.

Lemma 3.1. For any 0 ≤ i ≤ k, 0 ≤ m ≤ n ≤ N , and 0 ≤ s ≤ t,∥∥Jn−i,n − I∥∥L∞(R2)

+∥∥Jn−i,nτ − I

∥∥L∞(R2)

. τ, (3.5)

‖J(t, s; ·)‖Wr−1,∞(R2) + ‖Jm,nτ ‖Wr−1,∞(R2) . 1. (3.6)

Proof. For any t ≥ s ≥ 0, from (3.4) we know that

‖J(t; s, ·)− I‖L∞(R2) . (t− s) ‖∇w‖L∞(R2×[0,T ]) + ‖∇w‖L∞(R2×[0,T ])

∫ t

s

‖J(ξ, s; ·)− I‖L∞(R2) dξ.

Then Gronwall’s inequality yields

‖J(t; s, ·)− I‖L∞(R2) . 1, ‖Jn−i,n − I‖L∞(R2) . τ.

The first-order derivatives of J(t; s, ·) are estimated as follows∥∥∥∥∂J(t; s, ·)∂xj

∥∥∥∥L∞(R2)

≤∫ t

s

∥∥∥∥∇ ∂w∂xj (X(ξ; s, ·), ξ)J(ξ; s, ·)2 +∇w

(X(ξ; s, ·), ξ

)∂J(ξ; s, ·)∂xj

∥∥∥∥L∞(R2)

dξ

. 1 +

∫ t

s

∥∥∥∥∂J(ξ; s, ·)∂xj

∥∥∥∥L∞(R2)

dξ, j = 1, 2.

5

Using Gronwall’s inequality again yields

∥∥∥∥∂J(t; s, ·)∂xj

∥∥∥∥L∞(R2)

. 1. High-order derivatives of J(t, s; ·)

can be estimated similarly.Next we estimate

∥∥Jn−i,nτ − I∥∥L∞(R2)

. From (3.1)–(3.2), it is easy to see

Jn−1,nτ (x) = I + τ

nk∑i=1

bki∇w(φ

(i)n−1(x), t(i)

)∇φ(i)

n−1(x), (3.7)

where ∇φ(1)n−1 = I and ∇φ(i)

n−1 = I + τ

i−1∑j=1

aij∇w(x(j), t(j)

)∇φ(j)

n−1 for i ≥ 2. The smoothness of w

implies ‖Jn−1,nτ − I‖L∞(R2) . τ . Using (3.3), we easily get ‖Jn−i,nτ − I‖L∞(R2) . τ .

The estimate for ‖Jm,nτ ‖Wr−1,∞(R2) is easy by using (3.3). In fact,

‖Jm,nτ ‖L∞(R2) ≤n∏

j=m+1

∥∥Jj−1,jτ

∥∥L∞(R2)

≤ (1 + Cτ)n−m . 1,

‖∇Jm,nτ ‖L∞(R2) ≤n∑

i=m+1

∥∥∇Ji−1,iτ

∥∥L∞(R2)

n∏j=m+1,j 6=i

∥∥Jj−1,jτ

∥∥L∞(R2)

.n∑

i=m+1

(1 + Cτ)τ . 1.

High-order derivatives of Jm,nτ can be estimated similarly.

Lemma 3.2. For 0 ≤ m ≤ n and µ = 0, 1,

‖Xm,nτ −Xm,n‖Wµ,∞(R2) . (n−m)τk+1−µ.

Proof. Since Xn−1,nτ is obtained by the RK-k scheme for (2.2), standard error estimates for RK

schemes yield∥∥Xn−1,n

τ −Xn−1,n∥∥L∞(R2)

. τk+1. Note that

Jn−1,n(x) = ∇Xn−1,n(x) = I +

∫ tn

tn−1

∇w(X(s; tn−1,x), s)J(s; tn−1,x)ds.

Using (3.7) and Taylor’s expansion of Jn−1,nτ (x) at x, we have

∥∥Jn−1,nτ − Jn−1,n

∥∥L∞(R2)

. τk. It

follows from (3.3) that

‖Xm,nτ −Xm,n‖L∞(R2) ≤

n−1∑j=m

∥∥(Xj+1,nτ Xj,j+1

τ −Xj+1,nτ Xj,j+1) Xm,j

∥∥L∞(R2)

≤n−1∑j=m

∥∥Jj+1,nτ

∥∥L∞(R2)

∥∥Xj,j+1τ −Xj,j+1

∥∥L∞(R2)

. (n−m)τk+1,

where we have applied the intermediate value theorem to each component of Xj+1,nτ in the second

inequality. Furthermore, ‖Jm,nτ − Jm,n‖L∞(R2) . (n−m)τk can be proven similarly.Lemma 3.3. For any 0 ≤ m ≤ n ≤ N and 0 ≤ i ≤ k,∥∥Jn,n−i − I

∥∥L∞(R2)

+∥∥Jn,n−iτ − I

∥∥L∞(R2)

. τ, ‖Jn,mτ ‖L∞(R2) . 1, (3.8)

‖Xn,mτ −Xn,m‖Wµ,∞(R2) . (n−m)τk+1−µ, µ = 0, 1. (3.9)

6

Proof. The identity det(Jn,n−i) = 1 and equation (3.6) indicate∥∥Jn,n−i∥∥L∞(R2)

. 1. Then∥∥Jn,n−i − I∥∥L∞(R2)

≤∥∥Jn,n−i(I− Jn−i,n)

∥∥L∞(R2)

.∥∥Jn−i,n − I

∥∥L∞(R2)

. τ.

Since∥∥Jn−i,nτ − I

∥∥L∞(R2)

. τ , we also have∥∥Jn,n−iτ

∥∥L∞(R2)

. 1 and∥∥Jn,n−iτ − I∥∥L∞(R2)

=∥∥Jn,n−iτ (I− Jn−i,nτ )

∥∥L∞(R2)

. τ,

‖Jn,mτ ‖L∞(R2) ≤n∏

j=m+1

∥∥Jj,j−1τ

∥∥L∞(R2)

≤ (1 + Cτ)n−m . 1.

Finally, Lemma 3.2 and the boundedness of Jn,mτ indicate that

‖Xn,m −Xn,mτ ‖L∞(R2) = ‖(Xn,m Xm,n

τ −Xn,m Xm,n) Xn,mτ ‖L∞(R2)

= ‖Xn,m Xm,nτ −Xn,m Xm,n‖L∞(R2)

≤ ‖Jn,m‖L∞(R2) ‖Xm,nτ −Xm,n‖L∞(R2) . (n−m)τk+1,

where we have applied the intermediate value theorem again to each component of Xn,m in thelast inequality. The gradient of Xn,m

τ −Xn,m can be estimated similarly

‖∇(Xn,mτ −Xn,m)‖L∞(R2) = ‖Jn,mτ (Jm,n − Jm,nτ )Jn,m‖L∞(R2) . (n−m)τk.

The proof is finished.

3.3. The interface-tracking algorithm. In practice, it is unrealistic to track each point inΩn−1 to obtain Ωn. We adopt the interface-tracking algorithm in [35] which constructs a C2-smoothboundary with cubic spline interpolations. The purpose is to estimate the errors between the exactboundaries and the approximate boundaries.

The interface-tracking procedure starts from a partition P0 =p0j ∈ Γ0 : j = 0, 1, · · · , J

of the

initial boundary with p00 = p0

J . Let L be the arc length of Γ0 and suppose Γ0 has a parametrizationΓ0 = χ0(l) : l ∈ [0, L], where χ0 ∈ C

r([0, L]) satisfies

χ0(Lj) = p0j , Lj = jη, 0 ≤ j ≤ J.

Here η := L/J denotes the segment size for interface tracking. Clearly χ0(0) = χ0(L). The set ofinterpilation nodes is defined by L = Lj : j = 0, 1, · · · , J.

Algorithm 3.4. Given n ≥ 1, the interface-tracking algorithm for constructing Γnη from Γn−1η

consists of two steps.1. Trace forward each marker in Pn−1 to obtain the set of markers at t = tn,

Pn =pnj = Xn−1,n

τ (pn−1j ) : j = 0, · · · , J

.

2. Compute the cubic spline function χn ∈ C2([0, L]) based on L and Pn. Define

Γnη := χn(l) : l ∈ [0, L] . (3.10)

7

The exact boundary at tn is given by

Γtn = χn(l) : 0 ≤ l ≤ L , χn := X0,n χ0. (3.11)

For any 1 ≤ j ≤ J , by Lemmas 3.1 and 3.3, the arc length of Γtn between X0,n(p0j−1) and X0,n(p0

j )can be estimated formally as follows∫ Lj

Lj−1

∣∣χ′n∣∣ =

∫ Lj

Lj−1

∣∣J0,nχ′0∣∣ ≤ (1 + Cτ)n

∫ Lj

Lj−1

|χ′0| . η,

η =

∫ Lj

Lj−1

|χ′0| =∫ Lj

Lj−1

∣∣Jn,0χ′n∣∣ ≤ (1 + Cτ)n∫ Lj

Lj−1

∣∣χ′n∣∣ . ∫ Lj

Lj−1

∣∣χ′n∣∣ .This means that X0,n(P0) provides a quasi-uniform partition of Γtn . Heuristically, the convergencerate of the interface-tracking algorithm does not deteriorate in the time-evolution process if Γtnis smooth and max

0≤j≤J

∣∣pnj −X0,n(p0j )∣∣ = o(η). However, the global high accuracy may deteriorate

when Γtn suffers a largely deformation or a local C1 discontinuity. In order to alleviate the accuracydeterioration, we apply a more elaborate algorithm given by Zhang and Fogelson [35] in numericalexperiments. The algorithm adjusts the distance between adjacent markers by creating new markersor removing old markers. Therefore, it can be applied to largely deformed domains even at thepresence of dynamic C1 discontinuities.

3.4. Error estimates for Algorithm 3.4. Let Γnη be the approximate boundary formedwith Algorithm 3.4 and let Ωnη be the open domain surrounded by Γnη , namely, Γnη = ∂Ωnη . We aregoing to estimate the difference between the exact boundary and the tracked boundary. First weclarify three kinds of boundaries and their respective parametrizations

Γtn = χn(l) : l ∈ [0, L] , χn = X0,n χ0,

Γn = χn(l) : l ∈ [0, L] , χn = X0,nτ χ0,

Γnη = χn(l) : l ∈ [0, L] , χn ∈ C2([0, L]),

1 ≤ n ≤ N. (3.12)

Here Γtn is the exact material boundary and Γn is the approximate boundary obtained by the RK-kscheme (3.1). The starting point for high-order error estimates is the smoothness of Γtn . However,when t is large, the smoothness of Γtn and Γn is not guaranteed even w is regular [27, 29]. Sincethe objective of the paper is to study error estimates for numerical solutions, we do not elaborateon the smoothness of Γtn and Γn and simply assume

(A1) ‖χn‖Cr([0,L]) + ‖χn‖Cr([0,L]) . 1 for any 0 ≤ n ≤ N .

The assumption is reasonable if the deformation of Γtn is not large. Standard error estimates forcubic spline interpolations show that

‖χn − χn‖Cµ([0,L]) . η4−µ, µ = 0, 1. (3.13)

Theorem 3.5. For any 0 ≤ n ≤ N and 0 ≤ µ ≤ 2, there holds

‖χn − χn‖Cµ([0,L]) . η−µ(η4 + τk). (3.14)

8

Proof. Let χn,η be the cubic spline interpolation based on the set of nodes L and the set

of nodal parameters X0,n(P0). Step 2 of Algorithm 3.4 indicates that χn,η − χn is the cubic

spline interpolation based on L andX0,n(p0

j )−X0,nτ (p0

j ) : 0 ≤ j ≤ J

. Using Lemma 3.2 andthe stability and error estimates for cubic spline interpolations, we have∥∥χn,η − χn∥∥C([0,L])

. max0≤j≤J

∣∣X0,n(p0j )−X

0,nτ (p0

j )∣∣ . τk,∥∥χn,η − χn∥∥Cµ([0,L])

. η4−µ ‖χn‖C4([0,L]) . η4−µ.

The triangular inequality and the inverse estimates indicate that

‖χn − χn‖Cµ([0,L]) . η−µ∥∥χn,η − χn∥∥C([0,L])

+∥∥χn,η − χn∥∥Cµ([0,L])

. η−µ(τk + η4).

The proof is finished.Theorem 3.6. Suppose r ≥ max(k + 1, 4). Then for each n− k ≤ m ≤ n,

‖χn −Xm,nτ χm‖Cµ([0,L]) . η−µ

k−1∑i=0

[τ i+1ηmin(4,r−i) + τk+1

], µ = 0, 1. (3.15)

Proof. The cubic spline function χn is formed with L and Pn and has the explicit form

χn(l) =αnj−1

(Lj − l)3

6η+αnj

(l − Lj−1)3

6η+

(pnj−1 −

η2

6αnj−1

)Lj − lη

+

(pnj −

η2

6αnj

)l − Lj−1

η∀ l ∈ [Lj−1, Lj), (3.16)

where αn = [αn1 , · · · ,αnJ ]>

is the solution to the system of algebraic equations

Gαn = dn ≡ [dn1 , · · · ,dnJ ]>, dnj := 3(pnj+1 + pnj−1 − 2pnj )/η2, (3.17)

and the matrix is given by

G =

2 0 1/2 1/20 2 0 1/2 1/2

1/2 0 2 0 1/2. . .

. . .. . .

. . .. . .

1/2 0 2 0 1/21/2 1/2 0 2 0

1/2 1/2 0 2

(2J)×(2J)

.

It is well-known that∥∥G−1

∥∥∞ . 1. This yields ‖αn‖∞ . ‖dn‖∞.

Now we estimate αn −αm. For fixed s ≥ 0 and p ∈ R2, we define a univariate function of t

W (t; s,p) := w(X(t; s,p), t) ∀ t ≥ s.

LetW (i)(t; s,p) denote the ith-order derivative ofW (t; s,p) with respect to t. Sincew ∈ Cr(R2×I),the first-order derivative of W can be estimated by the chain rule∣∣∣W (1)(t; s,p)

∣∣∣ =

∣∣∣∣(w · ∇w)(t; s,p) +∂w

∂t(t; s,p)

∣∣∣∣ . 1.

9

High-order derivatives of W (·; s,p) can be estimated similarly. Together with (2.2), this shows

‖X(·; s,p)‖Cr+1([s,T ]) + ‖W (·; s,p)‖Cr([s,T ]) . 1.

By Lemma 3.1 and Taylor’s expansion of X(tn; tm,p) at tm, we have

Xm,nτ (p) = X(tn; tm,p) +O(τk+1) = p+

k−1∑i=0

W (i)(tm; tm,p)

(i+ 1)!(tn − tm)i+1 +O(τk+1). (3.18)

Since pnj = Xm,nτ (pmj ), this gives an explicit relation between dnj and dmj

dnj − dmj =

3

η2

[Xm,nτ (pmj+1)− pmj+1 − 2Xm,n

τ (pmj ) + 2pmj +Xm,nτ (pmj−1)− pmj−1

]=

k−1∑i=0

βi,j(i+ 1)!

(tn − tm)i+1 +O(η−2τk+1), (3.19)

where βi,j = 3η−2[W (i)(tm; tm,p

mj+1) + W (i)(tm; tm,p

mj−1) − 2W (i)(tm; tm,p

mj )]. Write βi =[

βi,1, · · · ,βi,J]>

and γi = G−1βi. It follows from (3.17) and (3.19) that

αn −αm = G−1(dn − dm) =

k−1∑i=0

(tn − tm)i+1

(i+ 1)!γi +O(η−2τk+1). (3.20)

Next we estimate χn − χm and χ′n − χ′m. Clearly (3.17) and the equality Gγi = βi indicatethat γi actually defines a cubic spline function

ζi(l) =γi,j−1

(Lj − l)3

6η+ γi,j

(l − Lj−1)3

6η+

(W (i)(tm; tm,p

mj−1)− η2

6γi,j−1

)Lj − lη

+

(W (i)(tm; tm,p

mj )− η2

6γi,j

)l − Lj−1

η∀ l ∈ [Lj−1, Lj). (3.21)

Substituting (3.18) and (3.20) into (3.16), we immediately get

χn − χm =

k−1∑i=0

(tn − tm)i+1

(i+ 1)!ζi +O(τk+1), µ = 0, 1. (3.22)

Taking derivatives of (3.16) and substracting the respective equations for n and m, we get

χ′n − χ′m =

k−1∑i=0

(tn − tm)i+1

(i+ 1)!ζ′i +O(η−µτk+1), µ = 0, 1. (3.23)

Now we are ready to prove (3.15). Since χm(Lj) = pmj for 0 ≤ j ≤ J , ζi is actually the cubic

spline interpolation ofW (i)(tm; tm, χm(l)), l ∈ [0, L]. Since r ≥ k+1 ≥ i+2 andW (i)(tm; tm, χm

)∈

Cr−i([0, L]). Standard error estimates for cubic spline interpolations show that∥∥∥ζi −W (i)(tm; tm, χm)∥∥∥Cµ([0,L])

. ηmin(4,r−i)−µ. (3.24)

10

Write Θ(τ, η) =∑k−1i=0 τ

i+1ηmin(4,r−i) + τk+1 for convenience. Combining (3.22)–(3.24) yields

χ(µ)n − χ(µ)

m =

k−1∑i=0

(tn − tm)i+1

(i+ 1)!

dµ

dlµW (i)(tm; tm, χm) +O

(η−µΘ(τ, η)

). (3.25)

Now we apply (3.18) to Xm,nτ χm. Together with (3.25) and (3.13), the result implies

|χn −Xm,nτ χm| .

k−1∑i=0

τ i+1|W (i)(tm; tm, χm)−W (i)(tm; tm,χm)|+ Θ(τ, η) . Θ(τ, η).

Since w ∈ Ck+1(R2 × [0, T ]), similar to (3.18), we also have

∇pXm,nτ (p) = I +

k−1∑i=0

(tn − tm)i+1

(i+ 1)!∇pW

(i)(tm; tm,p) +O(τk+1).

Using (3.25) and (3.13), the derivative of χn −Xm,nτ χm can be estimated similarly

|χ′n − (Xm,nτ χm)′| .

k−1∑i=0

τ i+1

∣∣∣∣ d

dlW (i)(tm; tm, χm)− d

dlW (i)(tm; tm,χm)

∣∣∣∣+ η−1Θ(τ, η)

.k−1∑i=0

τ i+1∣∣χ′m − χ′m∣∣+ η−1Θ(τ, η) . η−1Θ(τ, η).

The proof is finished.

4. The fictitious-domain finite element method. The purpose of this section is to proposehigh-order fictitious-domain finite element methods for solving (2.1) on a fixed mesh. The BDF-kschemes for 2 ≤ k ≤ 4 and the characteristics-based discretization will be used.

4.1. Finite element spaces. First we take an open squareD ⊂ R2 such that Ωt∪Ωn∪Ωnη ⊂ Dfor all 0 ≤ t ≤ T and 0 ≤ n ≤ N . Let Th be the uniform partition of D into closed squares of side-length h. It generates a cover of Ωnη and a cover of Γnη , respectively, given by

T nh :=K ∈ Th : area(K ∩ Ωnη ) > 0

, T nh,B :=

K ∈ T nh : length(K ∩ Γnη ) > 0

.

The cover T nh generates a fictitious domain which is denoted by

Ωn := interior(∪K∈T nh K

), Γn := ∂Ωn.

Let Eh be the set of all edges in Th. The set of interior edges of boundary elements is denoted by

Enh,B =E ∈ Eh : ∃K ∈ T nh,B s.t. E ⊂ ∂K\Γn

.

The finite element spaces on D and on the fictitious domain are, respectively, defined by

V (k, Th) :=v ∈ H1(D) : v|K ∈ Qk(K), ∀K ∈ Th

, V (k, T nh ) :=

v|Ωn : v ∈ V (k, Th)

,

where Qk is the space of polynomials whose degrees are no more than k for each variable. The spaceof piecewise regular functions over T nh is defined by

Hm(T nh ) :=v ∈ L2(Ωn) : v|K ∈ Hm(K), ∀K ∈ T nh

, m ≥ 1.

11

Ωnη

Γnη

Fig. 4.1: Left: the approximate domain Ωnη with a curved boundary Γnη and the sub-mesh T nh which

consists of red squares and the squares inside Ωnη . Right: the fictitious domain Ωn whose closure isthe union of red and white squares, and Enh,B which consists of blue edges.

4.2. The discrete problem. For an interior edge E ∈ Eh, letK1,K2 ∈ Th be the two elementssharing E. The jump of a function v across E is defined by

JvK (x) = limε→0+

[v(x− εnK1)nK1

+ v(x− εnK2)nK2

] ∀x ∈ E,

where nK1, nK2

are the unit outward normals on ∂K1 and ∂K2, respectively. We define four bilinearforms on Hk+1(T nh ) ∩H1(Ωn) as follows

A nh (w, v) :=

∫Ωnη

∇w · ∇v + S nh (w, v) + J n

0 (w, v) + J n1 (w, v), (4.1)

S nh (w, v) := −

∫Γnη

(v∂nw + w∂nv) , (4.2)

J n0 (w, v) :=

γ0

h

∫Γnη

wv, (4.3)

J n1 (w, v) := γ1

∑E∈Enh,B

k∑l=1

h2l−1

∫E

q∂lnw

y q∂lnv

y, (4.4)

where γ0 and γ1 are positive constants, J n0 is called boundary penalty, J n

1 is called boundary-zonepenalty, and ∂lnv denotes the l-th order derivative of v in the normal direction n of E. Here J n

0 isused to impose the Dirichlet boundary condition of unh weakly, J n

1 is used to enhance the stabilityof unh (see section 5). In (4.2), ∂nv denotes the directional derivative of v and n is the unit outwardnormal to Γnη .

The discrete approximation to problem (2.1) is to seek unh ∈ V (k, T nh ) such that

1

τ

(ΛkUn

h, vh)

Ωnη+ A n

h (unh, vh) = (fn, vh)Ωnη∀ vh ∈ V (k, T nh ), (4.5)

where fn = f(tn), Unh =

[Un−k,nh , · · · , Un,nh

]>, Um,nh := umh X

n,mτ for any n ≥ m ≥ k, and (·, ·)Ωnη

stands for the inner product on L2(Ωnη ). Moreover, τ−1Λk stands for the BDF-k finite-difference

12

K

K2K3

Fig. 5.1: An illustration of (A2) with I = 3: K and K3 are connected with K2.

operator which is defined by (cf. [24])

ΛkUnh =

k∑i=0

λki Un−i,nh . (4.6)

The coefficients λki for k = 1, · · · , 4 are listed in Table 4.1.

Table 4.1: Coefficients of the BDF-k scheme (cf. e.g. [24]).

k λki

i0 1 2 3 4 5

1 1 −1 0 0 0 0

2 3/2 −2 1/2 0 0 0

3 11/6 −3 3/2 −1/3 0 0

4 25/12 −4 3 −4/3 1/4 0

Throughout the paper, we extend each vh ∈ V (k, T nh ) to the exterior of Ωn such that theextension, denoted still by vh, belongs to V (k, Th) and vanishes at all degrees of freedom outside ofΓn ∪ Ωn. It is easy to verify that

‖vh‖L2(D) . ‖vh‖L2(Ωn) , ‖vh‖H1(D) . ‖vh‖H1(Ωn). (4.7)

5. The well-posedness of the discrete problem. First- and second-order methods havebeen studied extensively in the literature. To study higher-order methods (k ≥ 3), first we make amild assumption on the finite element mesh.(A2) There exist an integer I > 0 and a constant γ > 0 independent of h and τ such that, for any

K ∈ T nh,B , one can find at most I elements KjIj=1 ⊂ T nh satisfying that K1 = K, Kj∩Kj+1

is an interior edge of T nh , and that KI ∩ Ωnη contains a disk of radius γh (see Fig. 5.1).

We remark that the assumption dose not require how boundary intersects Th and is less re-strictive than generally used in the literature [13, 14, 25, 26]. Since the domain is time-varying, toostrong assumptions are inappropriate. In fact, (A2) can be satisfied if Th is fine and the deformationof the domain is moderate.

5.1. Trace inequalities and norm equivalence. Now we give some useful results on traceinequalities and norm equivalence. Similar results can be found in [13,14,26] under various stronger

13

assumptions.Lemma 5.1. Suppose that Ωnη is a Lipschitz domain. For any K ∈ Th, we have

‖v‖L2(∂K) + ‖v‖L2(K∩Γnη ) . h−1/2 ‖v‖L2(K) + h1/2|v|H1(K) ∀ v ∈ H1(K). (5.1)

The upper bound for ‖v‖L2(∂K) is a standard result. The proof for the upper bound for‖v‖L2(K∩Γnη ) can be found in [11, Lemma 1 and Appendix A]. To study the well-posedness of

the discrete problems, we define the mesh-dependent norms

‖|v|‖Ωnη =(|v|2H1(Ωnη ) + h−1 ‖v‖2L2(Γnη ) + h ‖∂nv‖2L2(Γnη )

)1/2

,

‖|v|‖T nh =(|v|2H1(Ωnη ) + J n

0 (v, v) + J n1 (v, v)

)1/2

,

‖v‖∗,Ωn =(|v|2H1(Ωn) + h−1 ‖v‖2L2(Γnη )

)1/2

.

Clearly ‖|·|‖T nh is a norm on H1(Ωnη ) ∩Hk+1(T nh ).

Lemma 5.2. Let assumption (A2) be satisfied. Then for any vh ∈ V (k, T nh ),

‖vh‖2L2(Ωn) . ‖vh‖2L2(Ωnη ) + h2J n1 (vh, vh), (5.2)

‖|vh|‖Ωnη . ‖|vh|‖T nh h ‖vh‖∗,Ωn . (5.3)

Proof. For each K ∈ T nh,B , by assumption (A2), there exist (at most) I elements K1 =K,K2, · · · ,KI such that Ej = Kj ∩ Kj+1 is an interior edge of T nh and KI ∩ Ωnη contains adisk of radius γh. From [26, Lemma 5.1], we have

‖∇µvh‖2L2(Kj). ‖∇µvh‖2L2(Kj+1) +

k∑l=1

h2(l−µ)+1

∫Ej

q∂lnvh

y2, j = 1, · · · , I − 1, µ = 0, 1.

Since KJ ∩ Ωnη contains a disk of radius γh, the norm equivalence shows

‖∇µvh‖2L2(K) . ‖∇µvh‖2L2(KI) +

I−1∑j=1

k∑l=1

h2(l−µ)+1

∫Ej

q∂lnvh

y2

. ‖∇µvh‖2L2(KI∩Ωnη ) +

I−1∑j=1

k∑l=1

h2(l−µ)+1

∫Ej

q∂lnvh

y2, µ = 0, 1. (5.4)

Now we sum up (5.4) for all K ∈ T nh,B . Letting µ = 0 yields (5.2) and letting µ = 1 yields∑K∈T nh,B

|vh|2H1(K) . |vh|2H1(Ωnη ) + J n

1 (vh, vh). (5.5)

This shows ‖vh‖∗,Ωn . ‖|vh|‖T nh .

14

For any E ∈ Enh,B and 1 ≤ l ≤ k, the norm equivalence and the inverse estimate show

h2l−1

∫E

q∂lnvh

y2. h2l

∥∥∇lvh∥∥2

L∞(KE). h2l−2

∥∥∇lvh∥∥2

L2(KE). ‖∇vh‖2L2(KE) ,

where KE ∈ T nh,B satisfies E ⊂ ∂KE . This shows J n1 (vh, vh) . |vh|2H1(Ωn). Then we have ‖|vh|‖T nh h

‖vh‖∗,Ωn . The proof for ‖|vh|‖Ωnη . ‖vh‖∗,Ωn is similar and omitted here.

5.2. The well-posedness of the discrete problem. First we prove the coercivity andcontinuity of the discrete bilinear form A n

h .

Lemma 5.3. Suppose γ0 is large enough and γ1 > 0. Then for any uh, vh ∈ V (k, T nh ),

A nh (vh, vh) & ‖|vh|‖2T nh , |A n

h (uh, vh)| . ‖|uh|‖T nh ‖|vh|‖T nh .

Proof. The definition of A nh shows

A nh (vh, vh) = |vh|2H1(Ωnη ) + S n

h (vh, vh) + J n0 (vh, vh) + J n

1 (vh, vh). (5.6)

For any ε > 0, the Cauchy-Schwarz inequality shows

|S nh (vh, vh)| 6 εh

∑K∈T nh,B

‖∂nvh‖2L2(ΓK) +1

εh

∑K∈T nh,B

‖vh‖2L2(ΓK) . (5.7)

From Lemma 5.1, the inverse inequality and (5.5), we know that

h∑

K∈T nh,B

‖∂nvh‖2L2(ΓK) ≤ C∑

K∈T nh,B

|vh|2H1(K) ≤ C0

[|vh|2H1(Ωnη ) + J n

1 (vh, vh)]. (5.8)

Taking ε = 1/(2C0) in (5.7) and inserting the result into (5.6), we get

A nh (vh, vh) ≥ 1

2|vh|2H1(Ωnη ) +

(γ0 − ε−1

)J n

0 (vh, vh) +1

2J n

1 (vh, vh).

The coercivity of A nh is obtained by setting γ0 ≥ 2ε−1 = 4C0.

Similarly, the Cauchy-Schwarz inequality shows

|A nh (uh, vh)| ≤ ‖|uh|‖T nh ‖|vh|‖T nh + γ

−1/20

(h∑

K∈T nh,B

‖∂nuh‖2L2(ΓK)

)1/2

J n0 (vh, vh)1/2

+ γ−1/20

(h∑

K∈T nh,B

‖∂nvh‖2L2(ΓK)

)1/2

J n0 (uh, uh)1/2.

Using (5.8), we obtain the continuity of A nh .

Since (4.5) is a linear problem for givenUn−i,nh : i = 1, · · · , k

, by Lemma 5.3 and the

Lax-Milgram lemma, problem (4.5) has a unique solution unh in each time step.

15

5.3. The modified Ritz projection. Since the computational domain is varying, in general,we have to deal with the problem that un−jh /∈ V (k, T nh ) for 1 ≤ j ≤ k when proving the stability andconvergence of discrete solutions. To overcome this difficulty, we define a modified Ritz projectionoperator Pnh : Y (Ωnη )→ V (k, T nh ) as follows

A nh (Pnhw, vh) = anh(w, vh) ∀ vh ∈ V (k, T nh ), (5.9)

where anh(w, v) := A nh (w, v)−J n

1 (w, v) and Y (Ωnη ) :=v ∈ H1(Ωnη ) : ‖|v|‖Ωnη <∞

.

Lemma 5.4. For any v ∈ Y (Ωnη ), there holds

‖|Pnhw|‖Ωnη + ‖|Pnhw|‖T nh . ‖|w|‖Ωnη ∀w ∈ Y (Ωnη ).

Proof. Taking vh = Pnhw in (5.9) and using Lemma 5.3, we find that

‖|Pnhw|‖2T nh

. A nh (Pnhw,Pnhw) = anh(w,Pnhw) . ‖|w|‖Ωnη ‖|P

nhw|‖Ωnη .

Together with Lemma 5.2, this shows ‖|Pnhw|‖T nh . ‖|w|‖Ωnη . The estimate for ‖|Pnhw|‖Ωnη is easily

obtained by using Lemma 5.2 again.To estimate the error w − Pnhw, we shall use the the Scott-Zhang interpolation operator Ih:

H1(D)→ V (k, Th) (cf. [30]). For any element K ∈ Th and any edge E ∈ Eh,

‖Ihv − v‖Hl(K) . hk−l+1 |v|Hk+1(DK) , (5.10)

‖Ihv − v‖Hl(E) . hk−l+1/2 |v|Hk+1(DE) , 0 ≤ l ≤ k, (5.11)

where DA is the union of all elements having non-empty intersection with A = K or A = E.Lemma 5.5. For any w ∈ Hk+1(D), there holds

‖|w − Ihw|‖Ωnη + ‖|w − Ihw|‖T nh . hk|w|Hk+1(D). (5.12)

Proof. Write wh = Ihw for convenience. From (5.10)–(5.11), we have

J1(w − wh, w − wh) . h2k∑

E∈Enh,B

|w|2Hk+1(DE) . h2k |w|2Hk+1(Ωn) . (5.13)

For any K ∈ T nh,B and ΓK = Γnη ∩K, from Lemma 5.1 and inequality (5.10), we deduce that

‖w − wh‖L2(ΓK) . h−1/2 ‖w − wh‖L2(K) + h1/2|w − wh|H1(K) . hk+1/2 |w|Hk+1(DK) ,

‖∂n(w − wh)‖L2(ΓK) . h−1/2|w − wh|H1(K) + h1/2 |w − wh|H2(K) . hk−1/2 |w|Hk+1(DK) .

This shows ∑K∈T nh,B

(h−1 ‖w − wh‖2L2(ΓK) + h ‖∂n(w − wh)‖2L2(ΓK)

). h2k |w|2Hk+1(D) . (5.14)

The proof is finished by inserting (5.10), (5.13)–(5.14) into the definitions of ‖|·|‖T nh and ‖|·|‖Ωnη .

Lemma 5.6. For any w ∈ Hk+1(D), there holds

‖|w − Pnhw|‖Ωnη + ‖|w − Pnhw|‖T nh . hk |w|Hk+1(D) .

16

Proof. By the definition of Pnh and the fact J n1 (w,Pnhw − Ihw) = 0, we have

‖|Pnhw − Ihw|‖2T nh

. A nh (Pnhw − Ihw,Pnhw − Ihw) = A n

h (w − Ihw,Pnhw − Ihw).

Then Lemma 5.3 and (5.3) imply ‖Pnhw − Ihw‖Ωnη . ‖|Pnhw − Ihw|‖T nh . ‖w − Ihw‖T nh . The proof

is finished by using the triangle inequality and Lemma 5.5.

Lemma 5.7. Let assumptions (A2) be satisfied. Then

‖w − Pnhw‖L2(Ωnη ) . h ‖|w|‖Ωnη ∀w ∈ Y (Ωnη ), (5.15)

‖w − Pnhw‖L2(Ωnη ) . hk+1 |w|Hk+1(Ωnη ) ∀w ∈ Hk+1(Ωnη ). (5.16)

Proof. We will prove the lemma by the duality technique. Consider the auxiliary problem

−∆z = w − Pnhw in Ωnη , z = 0 on Γnη . (5.17)

From Lemma 3.3 and Theorem 3.5, we know that Γnη is C2-smooth and its parametrization satisfies‖χn‖C2([0,L]) . 1. The regularity result for elliptic equations indicates that

‖z‖H2(Ωnη ) ≤ C ‖w − Pnhw‖L2(Ωnη ) . (5.18)

Multiplying both sides of the elliptic equation with w − Pnhw and integrating by parts, we have

‖w − Pnhw‖2L2(Ωnη ) =

∫Ωnη

∇z · ∇(w − Pnhw)−∫

Γnη

∂z

∂n(w − Pnhw) = ah(w − Pnhw, z). (5.19)

Let z ∈ H2(D) be the Sobolev extension of z to the exterior of Ωnη [31]. There exists a constantC depending only on Ωnη such that

‖z‖H2(D) ≤ C ‖z‖H2(Ωnη ) ≤ C ‖w − Pnhw‖L2(Ωnη ) .

Let zh ∈ V (1, Th) be the Scott-Zhang interpolation of z. The arguments similar to Lemma 5.5 show

‖|z − zh|‖2Ωnη +∑

E∈Enh,B

h

∫E

J∂n(z − zh)K2 . h2 |z|2H2(D) . h2 ‖w − Pnhw‖2L2(Ωnη ) . (5.20)

Inserting ah(w − Pnhw, zh) = J n1 (Pnhw, zh) into (5.19) leads to

‖w − Pnhw‖2L2(Ωnη ) = ah(w − Pnhw, z − zh) + J n

1 (Pnhw, zh). (5.21)

By (5.20), the first term on the right-hand side is estimated as follows

ah(w − Pnhw, z − zh) . ‖|w − Pnhw|‖Ωnη ‖|z − zh|‖Ωnη . h ‖|w − Pnhw|‖Ωnη ‖w − Pnhw‖L2(Ωnη ) . (5.22)

17

Note that each edge E ∈ Enh,B is parallel either to the x-axis or the y-axis so that zh|E is a linearfunction of either x or y. Using (5.20) and Lemma 5.4, we deduce that

|J n1 (Pnhw, zh)| =

∣∣∣ ∑E∈Enh,B

h

∫E

J∂n(Pnhw)K J∂nzhK∣∣∣ =

∣∣∣ ∑E∈Enh,B

h

∫E

J∂n(Pnhw)K J∂n(zh − z)K∣∣∣

≤( ∑E∈Enh,B

h ‖J∂n(Pnhw)K‖2L2(E)

)1/2( ∑E∈Enh,B

h ‖J∂n(z − zh)K‖2L2(E)

)1/2

.h ‖|Pnhw|‖T nh ‖w − Pnhw‖L2(Ωnη ) . h ‖|w|‖Ωnη ‖w − P

nhw‖L2(Ωnη ) . (5.23)

Inserting (5.22) and (5.23) into (5.21) and using Lemma 5.4, we get (5.15).

Next let w ∈ Hk+1(D) be the Sobolev extension of w ∈ Hk+1(Ωnη ). Since J∂nwK = 0 on anyE ∈ Enh,B , using (5.21), (5.20), and Lemma 5.6, we find that

‖w − Pnhw‖2L2(Ωnη ) ≤ ‖|w − P

nhw|‖Ωnη ‖|z − zh|‖Ωnη +

∣∣∣ ∑E∈Enh,B

h

∫E

J∂n(w − Pnh w)K J∂n(z − zh)K∣∣∣

.hk+1 |w|Hk+1(D) ‖w − Pnhw‖L2(Ωnη ) .

This finishes the proof.

5.4. The stability of the discrete solutions. First we cite the telescope formulas of BDFschemes from [24, Section 2 and Appendix A].

Lemma 5.8. Let 1 ≤ k ≤ 4 and α = δk,3 + δk,4 where δij is the Kronecker delta. Then

(ΛkUn

h

)(unh + αΛ1Un

h

)=

k+1∑i=1

(Ψki (Un

h))2

−k∑i=1

(Φki (Un

h))2

,

where Ψki

(Unh

)=∑ij=1 c

ki,jU

n+1−j,nh , Φk

i

(Unh

)=∑ij=1 c

ki,jU

n−j,nh , and the parameters cki,j, 1 ≤

j ≤ i ≤ k + 1, are given in [24, Table 2.2].

Theorem 5.9. Suppose 2 ≤ k ≤ 4, O(η4/k) = O(h) = τ ≤ h, and that the penalty parameterγ0 in A n

h is large enough. Let unh, n ≥ k, be the solution to the discrete problem (4.5) based on thepre-calculated initial values u0

h, · · · , uk−1h . There is an h0 > 0 small enough such that, for any

h ∈ (0, h0] and m ≥ k,

‖umh ‖2L2(Ωnη ) +

m∑n=k

τ ‖|unh|‖2T n .

m∑n=k

τ ‖fn‖2L2(Ωnη ) +

k−1∑i=0

(∥∥uih∥∥2

L2(Ωiη)+ τ∥∥uih∥∥2

H1(Ωi)

). (5.24)

Proof. Without loss of generality, we fix the k in (5.24) to be 4 in the following discussion. Theproof for other cases are similar. For convenience, we write Un−i,nh = Pnh (Un−i,nh ) ∈ V (k, T nh ) fori > 1. The discrete problem for k = 4 has the form(

Λ4Unh, vh

)Ωnη

+ τA nh (unh, vh) = τ (fn, vh)Ωnη

∀ vh ∈ V (k, T nh ). (5.25)

18

Since Un−1,nh /∈ V (k, T nh ), we choose vh = 2unh − U

n−1,nh as a test function in (5.25). From equation

(4.6), we know that Λ1Unh = unh − U

n−1,nh . An application of Lemma 5.8 shows

5∑i=1

∥∥∥Ψ4i (U

nh)∥∥∥2

L2(Ωnη )−

4∑i=1

∥∥∥Φ4i (U

nh)∥∥∥2

L2(Ωnη )+ τA n

h (unh, 2unh − U

n−1,nh ) = τAn1 +An2 , (5.26)

where An1 =(fn, 2unh − Un−1,n

h

)Ωnη

and An2 =(Λ4Un

h, Un−1,nh − Un−1,n

h

)Ωnη

. Here we take it for

granted that un−jh ∈ V (k, Th) as in (4.7).Note that Φ4

i

(Unh

)= Ψ4

i

(Un−1h

)Xn,n−1

τ . By Lemmas 3.1 and A.2 , we find that

∥∥∥Φ4i

(Unh

)∥∥∥2

L2(Ωnη )≤ (1 + Cτ)

∥∥∥Ψ4i

(Un−1h

)∥∥∥2

L2(Ωn−1η )

+ Cτ5h−1i∑

j=1

∥∥∥un−jh

∥∥∥2

L2(Ωn−j)

≤∥∥∥Ψ4

i

(Un−1h

)∥∥∥2

L2(Ωn−1η )

+ Cτi∑

j=1

∥∥∥un−jh

∥∥∥2

L2(Ωn−j). (5.27)

From (A.4) and (4.7) we have∣∣Un−1,nh

∣∣H1(Γnη )

. h−1/2∥∥un−1

h

∥∥H1(Ωn−1)

, and from (5.1) we have

‖∂nunh‖L2(Γnη ) . h−1/2|unh|H1(Ωn). For any ε ∈ (0, 1), by the definitions of Pnh and the Cauchy-

Schwarz inequality, we deduce that

A nh (unh, 2u

nh − U

n−1,nh ) = 2A n

h (unh, unh)− anh(unh, U

n−1,nh )

≥ 3

2‖|unh|‖

2T nh− 5

2εh‖unh‖

2L2(Γnη ) − Cε|u

nh|

2H1(Ωn) −

1

2

∣∣∣Un−1,nh

∣∣∣2H1(Ωnη )

Cε∥∥un−1

h

∥∥2

H1(Ωn−1)− γ0 + ε−1

2h

∥∥∥Un−1,nh

∥∥∥2

L2(Γnη ). (5.28)

Since Un−1,nh ∈ Y (Ωnη ), we infer from (5.15) and the Cauchy-Schwarz inequality that

An1 ≤ 2 ‖fn‖2L2(Ωnη ) + ‖unh‖2L2(Ωnη ) + 2

∥∥∥Un−1,nh

∥∥∥2

L2(Ωnη )+ Ch2

∥∥∥∣∣∣Un−1,nh

∣∣∣∥∥∥2

Ωnη

, (5.29)

An2 ≤Ch

ε

4∑j=0

∥∥∥Un−j,nh

∥∥∥2

L2(Ωnη )+ εh

∥∥∥∣∣∣Un−1,nh

∣∣∣∥∥∥2

Ωnη

. (5.30)

Substituting (5.27)–(5.30) into (5.26) and using O(h) = τ ≤ h ε 1 lead to

4∑i=1

∥∥∥Ψ4i (U

nh)∥∥∥2

L2(Ωnη )−

4∑i=1

∥∥∥Ψ4i (U

n−1h )

∥∥∥2

L2(Ωn−1η )

+3

2τ ‖|unh|‖

2T nh

≤ 2τ ‖fn‖2L2(Ωnη ) + Cτ

4∑j=0

∥∥∥un−jh

∥∥∥2

L2(Ωn−j)+

5τ

2εh‖unh‖

2L2(Γnη ) + Cετ

(|unh|

2H1(Ωn) +

∥∥un−1h

∥∥2

H1(Ωn−1)

)+

1 + Cε

2τ∣∣∣Un−1,nh

∣∣∣2H1(Ωnη )

+(γ0 + 3ε−1)τ

2h

∥∥∥Un−1,nh

∥∥∥2

L2(Γnη )+Cτ

ε

4∑j=0

∥∥∥Un−j,nh

∥∥∥2

L2(Ωnη ). (5.31)

19

Using Lemmas A.1–A.2 for k = 4, we deduce from the inverse estimates and (4.7) that∥∥∥∇µUn−j,nh

∥∥∥2

L2(Ωnη )≤ (1 + Cτ)

∥∥∥∇µun−jh

∥∥∥2

L2(Ωn−jη )+ Cτ4/h

∥∥∥∇µun−jh

∥∥∥2

L2(Ωn−j), µ = 0, 1,∥∥∥Un−1,n

h

∥∥∥2

L2(Γnη )≤ (1 + Cτ)

∥∥un−1h

∥∥2

L2(Γn−1η )

+ Cτ4/h2∥∥un−1

h

∥∥2

H1(Ωn−1).

Substitute the estimates into (5.31), apply Lemma 5.2 and take the sum of the inequalities over4 ≤ n ≤ m. After proper arrangements and combinations, we end up with

4∑i=1

∥∥∥Ψ4i

(Umh

)∥∥∥2

L2(Ωmη )+

3

2τ

m∑n=4

‖|unh|‖2T nh

≤m∑n=4

τ(C ‖unh‖

2L2(Ωnη ) +

1 + Cε

2|unh|

2H1(Ωnη ) +

1 + 10/(γ0ε)

2J n

0 (unh, unh) + CεJ n

1 (unh, unh))

+ 2m∑n=4

τ ‖fn‖2L2(Ωnη ) + C

3∑i=0

τ∥∥uih∥∥2

H1(Ωi)+

4∑i=1

∥∥∥Ψ4i

(U3h

)∥∥∥2

L2(Ω3η). (5.32)

From [24, Table 2.2], we know Ψ41

(Umh

)= c41,1u

mh = 0.06umh . Finally, we choose ε small enough and

γ0 large enough such that Cε + 10/(εγ0) ≤ 1. Then the proof is finished by using Lemma 5.2 andGronwall’s inequality.

6. A priori error estimates. The purpose of this section is to prove the error esti-mates between the exact solution and the finite element solution. For convenience, we defineQT = (x, t) : x ∈ Ωt, t ∈ [0, T ], and

L∞(0, T ;Hm(Ωt)) =v ∈ L2(QT ) : esssup

t∈[0,T ]

‖v(X(t; 0, ·), t)‖Hm(Ω0) < +∞, m ≥ 0.

Since Ωnη\Ωtn 6= ∅ in general, we follow Lehrenfeld and Olshanaskii [19] to extend the solution u to

the exterior of Ωt. By [31, Chapter 6], there is an extension operator E0: Hk+1(Ω0) → Hk+1(R2)such that

(E0w) |Ω0= w, ‖E0w‖Hk+1(R2) . ‖w‖Hk+1(Ω0) ∀w ∈ Hk+1(Ω0).

Since X(t; s, ·) is one-to-one, its inverse is denoted by X(s; t, ·). Then Ω0 = X(0; t,Ωt). We candefine an extension operator from Hk+1(Ωt) to Hk+1(R2) by Etw :=

[E0

(w X(t; 0, ·)

)]X(0; t, ·).

The global extension operator E: L∞(0, T ;Hk+1(Ωt))→ L∞(0, T ;Hk+1(R2)) is defined by

(Ev)(·, t) = Etv(·, t) ∀ t ∈ [0, T ].

By Lemma 3.3 and arguments similar to [19], we have the following lemma. The proof is omitted.Lemma 6.1. There is a constant C > 0 depending only on Ω0 and ‖w‖Wk,∞(R2×[0,T ]) such that,

for any v ∈ L∞(0, T ;Hk+1(Ωt)) ∩Hk+1(QT ),

‖Ev‖Hm(R2) ≤ C‖v‖Hm(Ωt), 1 ≤ m ≤ k + 1, ‖Ev‖Hk+1(R2×[0,T ]) ≤ C‖v‖Hk+1(QT ).

Furthermore, for v ∈ L∞(0, T ;Hm(Ωt)) satisfying ∂tv ∈ L∞(0, T ;Hm−1(Ωt)), it holds

‖∂t(Ev)‖Hm−1(R2) ≤ C(‖v‖Hm(Ωt) + ‖∂tv‖Hm−1(Ωt)

), 1 ≤ m ≤ k + 1.

20

Let u be the exact solution to (2.1). For convenience, we abuse the notation and denote theextension Eu still by u throughout this section. Write un := u(tn), Um,n := um Xn,m, and

Un :=[Un−k,n, · · · , Un,n

]>. By (2.5), un satisfies the semi-discrete equation

1

τΛkUn −∆un = fn +Rn in Ωnη ,

where fn =du

dt(tn)−∆un and Rn =

1

τΛkUn − du

dt(tn). Multiplying both sides of the equation by

vh ∈ V (k, Th) and using integration by parts, we obtain

1

τ

(ΛkUn, vh

)Ωnη

+ anh(un, vh) = (fn +Rn, vh)Ωnη+

∫Γnη

un(γ0

hvh − ∂nvh

). (6.1)

Now we present the main theorem of this section.

Theorem 6.2. Let the assumptions in Theorem 5.9 be satisfied. Suppose that the exact solutionsatisfies u ∈ L∞(0, T ;Hk+1(Ωt)) ∩Hk+1(QT ) and ∂tu ∈ L∞(0, T ;H2(Ωt)). Moreover, suppose thepre-calculated initial solutions satisfy∥∥ui − uih∥∥2

L2(Ωiη)+ τ

∥∥∣∣ui − uih∣∣∥∥2

Ωiη. τ2k, i = 0, 1, · · · , k − 1. (6.2)

Then for any k ≤ m ≤ N ,

‖um − umh ‖L2(Ωmη ) +( m∑n=k

τ ‖|un − unh|‖2T nh

)1/2

. τk−1/2.

Proof. We only prove the theorem for k = 4. The proofs for other cases are similar. Writeρn := un − Pnhun and θnh := Pnhun − unh. By Lemmas 5.6–5.7, it suffices to estimate θnh .

Define Θnh :=

[Θn−k,nh , · · · ,Θn,n

h

]>and ζn :=

[ζn−k,n, · · · , ζn,n

]>where Θm,n

h = θmh Xn,mτ

and ζm,n = (um Xn,mτ − um Xn,m)− ρm Xn,m

τ . Subtracting (4.5) from (6.1) and using (5.9),we find that

1

τ(Λ4Θn

h, vh)Ωnη+ A n

h (θnh , vh) = A1 +A2, (6.3)

where

A1 =1

τ

(Λ4ζn, vh

)Ωnη

+(Rn + fn − fn, vh

)Ωnη, A2 =

∫Γnη

(γ0h−1vh − ∂nvh

)un.

Taking vh = 2θnh − PnhΘn−1,nh and using arguments similar to (5.31), we find that, for any ε > 0,

1

τ

4∑i=1

∥∥∥Ψ4i (Θ

nh)∥∥∥2

L2(Ωnη )− 1

τ

4∑i=1

∥∥∥Ψ4i (Θ

n−1h )

∥∥∥2

L2(Ωn−1η )

+3

2‖|θnh |‖

2T nh

≤A1 +A2 +εh

τ

∥∥∥∣∣∣Θn−1,nh

∣∣∣∥∥∥2

Ωnη

+Ch

ετ

4∑j=0

∥∥∥Θn−j,nh

∥∥∥2

L2(Ωnη )+

5

2εh‖θnh‖

2L2(Γnη ) +

5εh

2‖∂nθnh‖

2L2(Γnη )

+ C

4∑j=1

∥∥∥θn−jh

∥∥∥2

L2(Ωn−j)+

1

2

∣∣∣Θn−1,nh

∣∣∣2H1(Ωnη )

+εh

2

∣∣∣Θn−1,nh

∣∣∣2H1(Γnη )

+γ0 + ε−1

2h

∥∥∥Θn−1,nh

∥∥∥2

L2(Γnη ).

(6.4)

21

By Lemma A.1, Lemma 5.7, and Taylor’s formula, we have∥∥Λ4ζn∥∥L2(Ωnη )

. τ5|un|H1(D) +

4∑i=0

∥∥ρn−i∥∥L2(Ωn−iη )

. (τ5 + h5)‖un‖H5(D),

‖Rn‖2L2(Ωnη ) =

∫Ωnη

∣∣∣∣ 4∑i=1

λ4i

4!τ

∫ tn

tn−i

(tn − ξ)4 d5

dt5u(X(ξ; tn,x), ξ)dξ

∣∣∣∣2dx . τ7 ‖u‖2H5(D×(tn−i,tn)) .

Moreover, using Lemma A.3 and the identity fn = fn in Ωtn , we find that∣∣∣(fn − fn, vh)Ωnη

∣∣∣ =∣∣∣(fn − fn, vh)Ωnη\Ωtn

∣∣∣ . τ4h−1/2‖fn − fn‖H2(D) ‖vh‖L2(Ωn) (6.5)

Since un = 0 on Γtn , we have un χn ≡ 0 and deduce from Theorem 3.5 that

‖un‖2L2(Γnη ) =

∫ L

0

(un χn − un χn)2 |χ′n| .∫ L

0

|χn − χn|2∫ 1

0

|∇un(ξχn + (1− ξ)χn)|2 dξ

. τ8 |un|2H1(D) . (6.6)

The inverse estimate and the assumption τ = O(h) show

|A1| .(τ4‖un‖H5(D) + τ7/2 ‖u‖H5(D×(tn−i,tn)) + τ7/2‖fn − fn‖H2(D)

)‖vh‖L2(Ωn) , (6.7)

|A2| .h−1 ‖un‖L2(Γnη ) ‖vh‖L2(Γnη ) . h−1τ4 |un|H1(D) ‖vh‖L2(Γnη ) . (6.8)

Moreover, vh can be estimated by using Lemmas 5.2, 5.3, and 5.4 and satisfies

‖vh‖2L2(Ωnη ) . ‖vh‖2L2(Ωnη ) + h2J n

1 (vh, vh)

. ‖θnh‖2L2(Ωnη ) + h2J n

1 (θnh , θnh) +

∥∥∥Θn−1,nh

∥∥∥2

L2(Ωnη )+ h2

∥∥∥∣∣∣Θn−1,nh

∣∣∣∥∥∥2

Ωnη

, (6.9)

‖vh‖2L2(Γnη ) .hJ n0 (θnh , θ

nh) + h

∥∥∥∣∣∣Θn−1,nh

∣∣∣∥∥∥2

Ωnη

. (6.10)

Inserting est-A1–(6.10) into (6.4) yields

4∑i=1

∥∥∥Ψ4i (Θ

nh)∥∥∥2

L2(Ωnη )−

4∑i=1

∥∥∥Ψ4i (Θ

n−1h )

∥∥∥2

L2(Ωn−1η )

+3

2τ ‖|θnh |‖

2T nh

≤Cτ8(‖u‖2H5(D×(tn−i,tn)) + τ ‖un‖2H5(D) + ‖un‖2H1(D) + ‖fn − fn‖2H2(D)

)+ C[ε+ (γ0ε)

−1]τ ‖|θnh |‖2T nh

+ ετ∥∥∥∣∣∣Θn−1,n

h

∣∣∣∥∥∥2

Ωnη

+ Cτ

4∑j=0

(‖Θn−j,n

h ‖2L2(Ωnη ) + ‖θn−jh ‖2L2(Ωn−j)

)+τ

2

∣∣∣Θn−1,nh

∣∣∣2H1(Ωnη )

+εhτ

2

∣∣∣Θn−1,nh

∣∣∣2H1(Γnη )

+γ0 + ε−1

2hτ∥∥∥Θn−1,n

h

∥∥∥2

L2(Γnη ).

Leting 1 ε−1 γ0 and using arguments similar to (5.32), we obtain

‖θmh ‖2L2(Ωmη ) + τ

m∑n=4

‖|θnh |‖2Tn . τ7

(τ ‖u‖2H5(D×[0,T ]) + (1 + τ) ‖u‖2L∞(0,T ;H5(D)) + ‖∂tu‖2L∞(0,T ;H2(D))

)+

3∑i=0

(‖θih‖2L2(Ωiη) + τ

∥∥∣∣θih∣∣∥∥2

T ih

).

22

The proof is finished by using Lemmas 5.6, 5.7, 6.1, and the assumption (6.2).

Remark 6.3. The error estimate in Theorem 6.2 is sub-optimal. In view of (6.5)–(6.7), wefind that the loss of one half order is due to the kth-order interface-tracking algorithm. In fact, theoptimal error estimate can be recovered if we use the RK-(k + 1) scheme for interface-tracking.

7. Numerical experiments. Now we use two numerical experiments with k = 3, 4, respec-tively, to validate the theoretical analysis. The exact solution and the flow velocity are set by

u(x, t) = e−t sin(πx1) sin(πx2), w = cosπt

4

(sin2(πx1) sin(2πx2),− sin2(πx2) sin(2πx1)

)>.

The initial domain Ω0 is a disk of unit radius, and the final domain ΩT is stretched into a snake-likedomain (see Fig. 7.1). In real computations, we apply the cubic MARS algorithm in [35] to trackthe boundary. The algorithm is a slight modification of Algorithm 3.4 by creating new markers orremoving old markers when necessary. Throughout the section, we choose γ0 = 800 and γ1 = 1/γ0.

Fig. 7.1: The approximate domains Ωnη at tn = 0, 1/2, 1, and 2, respectively (h = 1/16).

The solution errors are measured by the norm

eN =

(‖u(·, T )− uNh ‖2L2(ΩNη ) + τ

N∑n=k

|u− unh|2H1(Ωnη )

)1/2

.

To simplify the computation, we set the pre-calculated initial values by the exact solution, namely,

h = τ eN rate eN rate1/16 3.47e-05 - 2.43e-6 -1/32 4.31e-06 3.01 9.90e-8 4.621/64 4.25e-07 3.34 4.56e-9 4.441/128 4.93e-08 3.11 2.34e-10 4.29

Table 7.1: Convergence rates for k = 3 (the middle column) and k = 4 (the right column).

ujh = u(tj), for 0 ≤ j ≤ k−1. Numerical results for k = 3, 4 are shown in Tables 7.1. They show thatoptimal convergence rates, eN ∼ τk, are obtained for both the third- and fourth-order methods.

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23

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25

Appendix A. Useful estimates for the pull-back maps.In this appendix, we prove some useful estimates concerning the pull-back maps v → vXn,n−i

and v → v Xn,n−iτ for a funciton v. Assume 1 ≤ n ≤ N and 0 ≤ i ≤ k throughout the appendix.

Lemma A.1. Let Ω ⊂ D satisfy Xn,n−iτ (Ω) ⊂ D and Xn,n−i(Ω) ⊂ D. There exists a constant

C > 0 independent of n and τ such that, for any v ∈ H1(D),∥∥∇µ(v Xn,n−iτ )

∥∥2

L2(Ω)≤ (1 + Cτ) ‖∇µv‖2L2(Xn,n−i

τ (Ω)) , µ = 0, 1, (A.1)∥∥v Xn,n−i − v Xn,n−iτ

∥∥L2(Ω)

. τk+1|v|H1(D). (A.2)

Moreover, if η = O(τk/4), then for any vh ∈ V (k, Th),∥∥vh Xn,n−iτ

∥∥2

L2(Γnη )≤ (1 + Cτ) ‖vh‖2L2(Γn−iη ) + Cτk+1h−2‖vh‖2H1(D), (A.3)∣∣vh Xn,n−i

τ

∣∣2H1(Γnη )

.h−1|vh|2H1(D). (A.4)

Proof. Changing variables of integration and using Lemma 3.1, we have, for µ = 0, 1,∥∥∇µ(v Xn,n−iτ )

∥∥2

L2(Ω)=

∫Xn,n−iτ (Ω)

∣∣(Jn,n−iτ ∇)µv∣∣2 det(Jn−i,nτ ) ≤ (1 + Cτ) ‖∇µv‖2L2(Xn,n−i

τ (Ω)) .

To prove (A.2), we use Lemma 3.1 and get∥∥v Xn,n−i − v Xn,n−iτ

∥∥L2(Ω)

≤ ‖∇v‖L2(D)

∥∥Xn,n−iτ −Xn,n−i∥∥

L∞(R2). τk+1|v|H1(D).

Suppose η = O(τk/4) and note from Theorem 3.6 that∣∣χ′n − (Xn−i,n

τ χn−i)′∣∣ . τ1+3k/4.

Together with Lemma 3.1, this shows

|χ′n|∣∣χ′n−i∣∣−1

= |χ′n|∣∣Jn,n−iτ (Xn−i,n

τ χn−i)′∣∣−1 ≤ 1 + Cτ.

We deduce that then we have∫ L

0

∣∣vh χn−i∣∣2 |χ′n| ≤ (1 + Cτ)

∫ L

0

∣∣vh χn−i∣∣2 ∣∣χ′n−i∣∣ = (1 + Cτ) ‖vh‖2L2(Γn−i) . (A.5)

Note from (3.13) that |χ′n| ≤∣∣χ′n∣∣ +

∣∣χ′n − χ′n∣∣ . 1 + η3 . 1. By Theorem 3.6 and the inverseestimate, we deduce that∫ L

0

(∣∣vh Xn,n−iτ χn

∣∣2 − ∣∣vh χn−i∣∣2).∫ L

0

∣∣Xn,n−iτ (χn −X

n−i,nτ χn−i)

∣∣ ∫ 1

0

∣∣∇v2h

(θXn,n−i

τ χn + (1− θ)χn−i)∣∣dθ

. τk+1 ‖vh‖L∞(D) ‖∇vh‖L∞(D) . τk+1h−2‖vh‖2H1(D). (A.6)

Then (A.3) follows from (A.5), (A.6), and the following identity

∥∥vh Xn,n−iτ

∥∥2

L2(Γnη )=

∫ L

0

∣∣vh χn−i∣∣2 |χ′n|+ ∫ L

0

( ∣∣vh Xn,n−iτ χn

∣∣2 − ∣∣vh χn−i∣∣2 ) |χ′n| .26

The proof of (A.4) is easy. Using Lemma 3.1 and Lemma 5.1, we immediately get∣∣vh Xn,n−iτ

∣∣2H1(Γnη )

=

∫Xn,n−iτ (Γnη )

∣∣(Jn,n−iτ Xn−i,nτ

)∇vh

∣∣2 ≤ (1 + Cτ)

∫Xn,n−iτ (Γnη )

|∇vh|2 .

Using the scaling technique and the norm equivalence shows (A.4).Lemma A.2. Assume η = O(τk/4). There exists a constant C > 0 independent of n and τ such

that, for any 1 ≤ i ≤ l ≤ k and µ = 0, 1,∥∥∥∇µ(vh Xn,n−lτ )

∥∥∥2

L2(Ωnη )≤ (1 + Cτ)

∥∥∥∇µvh Xn−i,n−lτ

∥∥∥2

L2(Ωn−iη )+ Cτk+1h−1 ‖∇µvh‖2L2(D) .

(A.7)

Proof. Using Lemma 3.1 and (A.1), we have∥∥∥vh Xn,n−lτ

∥∥∥2

L2(Ωnη )=∥∥∥vh Xn,n−l

τ

∥∥∥2

L2(Xn−i,nτ (Ωn−iη ))

+∥∥∥vh Xn,n−l

τ

∥∥∥2

L2(Ωnη\Xn−i,nτ (Ωn−iη ))

≤ (1 + Cτ)( ∥∥∥vh Xn−i,n−l

τ

∥∥∥2

L2(Ωn−iη )+ ‖vh‖2L2(Xn,n−l

τ (Ωnη )\Xn−i,n−lτ (Ωn−iη ))

).

By Theorem 3.6 and the inverse inequality, we know that

‖vh‖2L2(Xn,n−lτ (Ωnη )\Xn−i,n−l

τ (Ωn−iη )) =∑K∈Th

∫[Xn,n−l

τ (Ωnη\Xn−i,nτ (Ωn−iη ))]∩K

|vh|2

.hτk+1∑

K∩Xn−i,n−lτ (Γn−iη )6=∅

‖vh‖2L∞(K) + hτk+1∑

K∩Xn,n−lτ (Γnη )6=∅

‖vh‖2L∞(K)

. τk+1h−1 ‖vh‖2L2(D) . (A.8)

This yields (A.7) for µ = 0. The proof for the case of µ = 1 is similar.Lemma A.3. Assume η = O(τk/4) and r ≥ 1. For any vh ∈ V (k, Th) and v ∈ Hr(D),

‖vh‖2L2(Ωnη\Ωtn ) . τkh−1 ‖vh‖2L2(D) , ‖v‖2L2(Ωnη\Ωtn ) . σ(r)‖v‖2Hr(D),

where σ(r) = max(h, τk/h) if r = 1 and σ(r) = τk for r ≥ 2.Proof. It is easy to obtain the first inequality from the proof of (A.8). Now we prove the second

one for different values of r. If r ≥ 2, Theorem 3.6 and the injection Hr(D) → L∞(D) shows

‖v‖2L2(Ωnη\Ωtn ) ≤ area(Ωnη\Ωtn)‖v‖2L∞(D) . τk‖v‖2Hr(D).

If r = 1, the Scott-Zhang interpolation in (5.10) and the first inequality lead to

‖v‖2L2(Ωnη\Ωtn ) ≤ ‖v − Ihv‖2L2(D) + ‖Ihv‖2L2(Ωnη\Ωtn ) . σ(1)‖v‖2H1(D).

The proof is finished.

27