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ORIGINAL PAPER
An upwind DG scheme preserving the maximum
principle for the convective Cahn-Hilliard model
Daniel Acosta-Soba1,2 ·Francisco Guill´
en-Gonz´
alez3·
J. Rafael Rodr´
ıguez-Galv´
an1
Received:16 November 2021 / Accepted:13 June 2022 /
©The Author(s) 2022
Abstract
The design of numerical approximations of the Cahn-Hilliard model preserving the
maximum principle is a challenging problem, even more if considering additional
transport terms. In this work, we present a new upwind discontinuous Galerkin
scheme for the convective Cahn-Hilliard model with degenerate mobility which
preserves the maximum principle and prevents non-physical spurious oscillations.
Furthermore, we show some numerical experiments in agreement with the previous
theoretical results. Finally, numerical comparisons with other schemes found in the
literature are also carried out.
Keywords Discontinuous Galerkin ·Upwind scheme ·Diffuse interface ·
Convection ·Degenerate mobility
Daniel Acosta-Soba, Francisco Guill´
en-Gonz´
alez and J. Rafael Rodr´
ıguez-Galv´
an contributed
equally to this work.
J. Rafael Rodr´
ıguez-Galv´
an
rafael.rodriguez@uca.es
Daniel Acosta-Soba
daniel.acosta@uca.es
Francisco Guill´
en-Gonz´
alez
guillen@us.es
1Departamento de Matem´
aticas, Universidad de C´
adiz, Campus Universitario de Puerto Real
S/N, Puerto Real, 11510, C´
adiz, Spain
2Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Ave,
Chattanooga, 37403, TN, USA
3Departamento de Ecuaciones Diferenciales y An´
alisis Num´
erico & IMUS, Universidad de
Sevilla, Campus de Reina Mercedes S/N, Sevilla, 41012, Spain
Published online: 12 August 2022
Numerical Algorithms (2023) 92:1589–1619
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
1 Introduction
This paper is concerned with the development of discontinuous Galerkin (DG)
numerical schemes for the following convective Cahn-Hilliard (CCH) problem (writ-
ten as a second-order system): Given an incompressible velocity field v∈C()d
with ∇·v=0in, such that v·n=0on∂, find two real valued functions, the
phase uand the chemical potential μ, defined in ×[0,T]such that:
∂tu=1
Pe ∇·(M(u)∇μ)−∇·(uv)in ×(0,T), (1a)
μ=F(u) −ε2u in ×(0,T), (1b)
∇u·n=(M(u)∇μ−uv)·n=0on∂ ×(0,T), (1c)
u(0)=u0in . (1d)
The phase field variable ulocalizes the two different phases at the values u=0and
u=1, while the interface occurs when 0 <u<1. Here is a bounded polygonal
domain in Rd(d=2 or 3 in practice), with boundary ∂ whose outward-pointing
unit normal is denoted n. Parameters are Pe >0theP
´
eclet number of the flow and
ε>0 related to the width interface. For simplicity, we take Pe =1. The given initial
phase is denoted by u0. The classical Cahn–Hilliard equation (CH) corresponds to
the case without convection, i.e. v=0.
We consider the double-well Ginzburg-Landau potential
F (u) =1
4u2(1−u)2
and the degenerate mobility
M(u) =u(1−u).(2)
This type of degenerate mobility, vanishing at the pure phases u=0andu=1,
implies the following maximum principle property (see [16] for the CH equation): if
0≤u0≤1inthen 0 ≤u(·,t) ≤1infor t∈(0,T). This property does not
hold for constant mobility, and in general for fourth-order parabolic equations. We
are concerned in numerical schemes maintaining this property at the discrete level.
Cahn-Hilliard equation was originally introduced in [8,9] as a phenomenologi-
cal model of phase separation in a binary alloy and it has been successfully applied
in different contexts as a model which characterizes phase segregation and inter-
face dynamics. Applications include tumor tissues [34,35], image processing [6]and
multi-phase fluid systems (see, e.g. the review [26]). The dynamics of this equation
comprises a first stage in which a rapid separation process takes place, leading to
the creation of interfaces, followed by a second stage where aggregation and devel-
opment of bulk phases separated by thin diffuse interfaces take place. These two
phenomena (separation and aggregation) are characterized by different temporal and
spatial scales which, together with the non-linear and fourth-order nature of CH,
makes efficiently solving this equation an interesting computational challenge.
The interface is represented as a layer of small thickness of order εand the auxil-
iary function u(the phase-field function) is used to localize the (pure) phases u=0
and u=1. The chemical potential μis the variational derivative of the Helmholtz
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free energy E(u) with respect to u,μ=∂uE(u),where
E(u) =ε2
2|∇u|2+F (u)dx.(3)
Then, the dynamics of CH correspond to the physical energy dissipation of ut=
−∇ · J (u),whereJ (u) =−M(u)∇μ=−M (u)∇∂uE(u). For existence of solution
for CH equation with a degenerate mobility of the type (2), besides bounds for the
phase over time, we can refer to [16]. A review and many variants can be read in [30].
Numerical approximation of CH is a research topic of great interest due to the wide
spectrum of its applications, as well as a source of interesting mathematical chal-
lenges due to the computational issues referred above. Time approximations include
both linear schemes [21] and non-linear ones, where a Newton method is usually
employed. Although some authors (see, e.g. [15]) consider the fourth-order equation
obtained by substitution in the equation (1a) of the expression of μgivenin(1b),
the spatial discretization is usually based on the mixed phase field/chemical poten-
tial formulation presented in (1). These equations can be approximated by classical
numerical methods, including (a) finite differences (see, e.g. [12,20] for constant
mobility M(u) ≡1and[25] for degenerate mobility similar to (2), see also [32]for
applications in crystallography); (b) finite volumes (see, e.g. [4,13] for degenerate
mobility schemes); (c) and, above all, finite elements (see the precursory papers [15]
(constant mobility) and [5] (degenerate mobility)).
In recent years, an increasing number of advances has been published explor-
ing the use of discontinuous Galerkin (DG) methods both for constant [2,24]and
degenerate mobility [18,28,33,36]. Although DG methods lead in general to more
complex algorithms and to bigger amounts of degrees of freedom, they exhibit some
benefits of which one can take advantage also in CH equations, for instance doing
mesh refining and adaptivity [2]. In this paper, we are interested in the follow-
ing relevant point: the possibility of designing conservative schemes preserving the
maximum-principle 0 ≤u(t , x) ≤1 at the discrete level, even for the CCH variant
of CH equations, namely where a convective term models the convection or transport
of the phase-field.
The idea of introducing a convection term in the CH equations modeling a phase-
field driven by a flow, arriving at the CCH problem (1), arouses great interest,
specially in the case where the phase field is coupled with fluid equations.
To design adequate numerical approximations of these CCH equations is an
extremely challenging problem because it adds the hyperbolic nature of convective
terms to the inherent difficulties of the CH equation. There are not many numeri-
cal methods in the literature on this topic, although some interesting contributions
can be found. The first of them [3] is worth mentioning for the application of high-
resolution spectral Fourier schemes. We also underline the papers [7,27], based on
finite volume approximations, and also [11,22], where finite difference techniques
are applied for Navier-Stokes CH equations.
Some authors have recently worked in DG methods for spatial discretization
of the CCH problem. Using DG schemes in this context is natural because they
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are well-suited in convection-dominated problems, even when the P´
eclet number is
substantially large. For instance, the work of [24] is focused on construction and
convergence analysis of a DG method for the CCH equations with constant mobil-
ity, applying an interior penalty technique to the second-order terms and an upwind
operator for discretization of the convection term. Authors of [18] consider CCH
with degenerate mobility applying also an interior penalty to second-order terms in
the mixed form (1) and a more elaborated upwinding technique, based on a sigmoid
function, to the convective term.
These previous works show that DG methods are well suited for the approximation
of CCH equations, obtaining optimal convergence order and maintaining most of
the properties of the continuous model. But they have room for improvement in one
specific question: getting a maximum principle for the phase in the discrete case.
Although there are some works in which the maximum principle for the CH model
is preserved at the discrete level using finite volumes [4] and flux limiting for DG
[19], to the best knowledge of the authors, no scheme has been published in which
an upwind DG technique is used to obtain a discrete maximum principle property.
Our main contribution in this paper is made in this regard: the development of
numerical schemes which guarantee punctual estimates of the phase at the discrete
level, in addition to maintaining the rest of continuous properties of (1). The main
idea is to introduce an upwind DG discretization associated to the transport of the
phase by the convective velocity v, but also associated to the degenerate mobility
M(u). The main result in this work is Theorem 3.11, where we show that, for a piece-
wise constant approximation of the phase, our DG scheme preserves the maximum
principle, that is the discrete phase is also bounded in [0,1].
Our scheme is specifically designed for the CH equation with non-singular (poly-
nomial) chemical potential and degenerate mobility. For the CH equation with
the logarithmic Flory-Huggins potential, a strict maximum principle is satisfied
without the need of degenerate mobility. In this case, designing maximum principle-
preserving numerical schemes is a different issue. See, for instance, the paper [10]
where a finite difference numerical is proposed in this regard.
The paper is organized as follows: In Section 2, we fix notation and review DG
techniques for discretization of conservative laws. We consider the linear convec-
tion given by a velocity field and show that the usual linear upwind DG numerical
scheme with P0approximation preserves positivity in general, and the maximum
principle in the divergence-free case. In Section 3, we consider the CCH problem
(1). Using a truncated potential and a convex-splitting time discretization, we obtain
a scheme satisfying an energy law, which is decreasing for the CH case (v=0).
Then, we introduce our fully discrete scheme (29) providing a maximum principle
for the discrete phase variable. Finally, in Section 4, we present several numerical
tests, comparing our DG scheme with two different space discretizations found in the
literature: classical continuous P1finite elements and the SIP+upwind sigmoid DG
approximation proposed in [18]. We show error order tests and also we present qual-
itative comparisons where the maximum principle of our scheme is confirmed (and
it is not conserved by the other ones).
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2 DG discretization of conservative laws
Throughout this section, we are going to study the problem of approximating linear
and non-linear conservative laws preserving the positivity of the continuous models.
To this purpose, we introduce an upwinding DG scheme which follows the ideas of
the paper [23] based on an upwinding finite volume method.
2.1 Notation
Firstly, we are going to set the notation that will be used from now on. Let ⊂Rd
be a bounded polygonal domain. Then, we consider a shape-regular triangular mesh
Th={K}K∈Thof size hover , and we denote as Ehthe set of the edges of Th(faces
if d=3) distinguishing between the interior edges Ei
hand the boundary edges Eb
h.
Then, Eh=Ei
h∪Eb
h.
AccordingtoFig.1, we will consider for an interior edge e∈Ei
hshared by the
elements Kand L,i.e.e=∂K ∩∂L, that the associated unit normal vector neis
exterior to Kpointing to L. Moreover, for the boundary edges e∈Eb
h, the unit normal
vector newill be pointing outwards of the domain .
Therefore, we can define the average {{·}} and the jump [[· ]] of a scalar function v
on an edge e∈Ehas follows:
{{v}} := vK+vL
2if e=∂K ∩∂L∈Ei
h
vKif e=∂K∈Eb
h
,
[[ v]] := vK−vLif e=∂K ∩∂L∈Ei
h
vKif e=∂K∈Eb
h
.
We will denote by Pdisc
k(Th)and Pcont
k(Th)the spaces of discontinuous and
continuous functions, respectively, whose restriction to the elements Kof Thare
polynomials of degree k≥0.
Moreover, we will take an equispaced partition 0 =t0<t
1<···<t
N=Tof
the time domain [0,T]with t =tm+1−tmthe time step.
Fig. 1 Orientation of the unit
normal vector ne
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Finally, we set the following notation for the positive and negative parts of a scalar
function v:
v⊕:= |v|+v
2=max {v, 0},v
:= |v|−v
2=−min {v, 0},v=v⊕−v.
2.2 Conservative laws
We consider the following non-linear conservative problem for v:
vt+∇·F(v) =0en×(0,T), (4)
where the flux F(·)is a vectorial continuous function.
Let Sh=Pdisc
k(Th). Multiplying by any v∈Shand using the Green Formula in
each element K∈Th:
∂tv v −
K∈ThK
F(v) ·∇v+
K∈Th∂K F(v) ·nKv=0,
where nKis the normal vector to ∂K pointing outwards K.Ifvis a strong solution
of (4), such that vis continuous over e∈Eh,weget:
∂tv v −
K∈ThK
F(v) ·∇v+
e∈EheF(v) ·ne[[ v]] =0. (5)
But, if we look for functions v∈Shwhich are discontinuous over e∈Eh, then the
term F(v) ·nein the last integral of (5) is not well defined and we need to approach it.
Hence, the concept of numerical flux is used, taking expressions like (vK,v
L,ne)
such that: eF(v) ·ne[[ v]] ≈e
(vK,v
L,ne)[[ v]] .
From now on, we will consider fluxes of the form
F(v) =M(v)β,
with M=M(v) ∈Rand β=β(x, t ) ∈Rd. Since in this paper we are interested in
studying conservative problems defined over isolated domains, we impose from now
on that the transport vector βsatisfies the slip condition
β·n=0on∂.(6)
Remark 2.1 We will have to take into consideration the sign of M(v),i.e.ifM(v)
is non-increasing or non-decreasing, in order to work out a well-suited upwinding
scheme. This is due to
∇·F(v) =∇·(M(v)β)=M(v)β·∇v+M(v)∇·β.(7)
where
•The first term M(v)β·∇vis a transport of vin the direction of M(v)β.
•The second term M(v)∇·βcan be seen as a reaction term with coefficient ∇·β.
It may be specially interesting the case β=−∇v, where the whole term ∇·
(M(v)β)=−∇·(M(v)∇v) is a non-linear diffusion term.
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2.3 Linear convection and positivity
Taking M(v) =v, we arrive at the linear conservative problem:
vt+∇·(v β)=0in×(0,T), (8a)
v(0)=v0in , (8b)
where β:→Rdis continuous.
Remark 2.2 The problem (8) is well-posed without imposing boundary conditions
due to the slip condition (6) (it can be derived writing an integral expression of solu-
tion in terms of the characteristic curves, see for example [31]). In particular, this
integral expression implies the positivity of the solution, namely, if v0≥0in,then
the solution vof (8) satisfies v≥0in×(0,T).
For the linear problem (8a), we introduce the upwind numerical flux
(vK,v
L,ne):= (β·ne)⊕vK−(β·ne)vL,(9)
which leads to the following discrete scheme for (8a): Given vm∈Sh,findvm+1∈
Shsolving
δtvm+1v+aupw
h(β;vm+1
⊕,v) =0,∀v∈Sh,(10)
where
aupw
h(β;v, v) := −
K∈ThK
v(β·∇v)
+
e∈Ei
h,e=K∩Le
((β·ne)⊕vK−(β·ne)vL)[[v]] (11)
and δtvm+1=(vm+1−vm)/t denotes a discrete time derivative.
Notice that we have truncated vby its positive part v⊕, taking into account that
the solution of the continuous model (8a) is positive.
Remark 2.3 The numerical flux (vK,v
L,ne)giving in (9) can be rewritten as
follows:
(vK,v
L,ne)=(β·ne){{v}} +1
2|β·ne|[[ v]] ,
where (β·ne){{v}} is a centered-flux term and 1
2|β·ne|[[ v]] is the upwind term.
Now, we will prove that if we set Sh=Pdisc
0(Th), the scheme (10) preserves the
positivity of the continuous model. In this case, since K∈ThK(β·∇v)v =0, the
upwind term (11) reduces to
aupw
h(β;v, v) =
e∈Ei
h,e=K∩Le
((β·ne)⊕vK−(β·ne)vL)[[ v]] . (12)
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Theorem 2.4 (DG scheme (10) preserves positivity) If Sh=Pdisc
0(Th), then the
scheme (10)preserves positivity, that is, for any vm∈Shwith vm≥0in , then any
solution vm+1of (10)satisfies vm+1≥0in .
Proof Taking the following test function
v=(vm+1
K∗)in K∗
0 out of K∗, v ∈Sh,
where K∗is an element of Thsuch that vm+1
K∗=minK∈Thvm+1
K, the scheme (10)
becomes K∗δtvm+1
K∗(vm+1
K∗)=−aupw
h(β;vm+1
⊕,(v
m+1
K∗)). (13)
By applying for all L∈Th,vm+1
L≥vm+1
K∗hence in particular (vm+1
L)⊕≥
(vm+1
K∗)⊕(the positive part is a non-decreasing function), we deduce
aupw
h(β;vm+1
⊕,(v
m+1
K∗))=
=
e∈Ei
h,e=K∗∩Le(β·ne)⊕(vm+1
K∗)⊕−(β·ne)(vm+1
L)⊕(vm+1
K∗)
≤
e∈Ei
h,e=K∗∩Le(β·ne)⊕(vm+1
K∗)⊕−(β·ne)(vm+1
K∗)⊕(vm+1
K∗).
Since (vm+1
K∗)⊕(vm+1
K∗)=0then
aupw
h(β;vm+1
⊕,(v
m+1
K∗))≤0.
Therefore, from (13),
|K∗|δtvm+1
K∗(vm+1
K∗)≥0.
On the other hand,
0≤|K∗|δtvm+1
K∗(vm+1
K∗)=−|K∗|
t (vm+1
K∗)2
+vm
K∗(vm+1
K∗)≤0;
hence, (vm+1
K∗)=0. Thanks to the choice of K∗, we can assure that vm+1≥0.
Theorem 2.5 (Existence of DG scheme (10)) Assume that Sh=Pdisc
0(Th). For any
vm∈Sh, there is at least one solution vm+1∈Shof the scheme (10).
Proof Consider the following well-known theorem:
Theorem 2.6 (Leray-Schauder fixed point theorem) Let Xbe a Banach space
and let T:X−→ Xbe a continuous and compact operator. If the set
{x∈X:x=αT(x) for some 0≤α≤1}
is bounded (with respect to α), then Thas at least one fixed point.
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Given a function w∈Pdisc
0(Th)with w≥0, we are going to define the operator
T:Pdisc
0(Th)−→ Pdisc
0(Th)such that T(v) =vwhere vis the unique solution of the
linear scheme:
v∈Pdisc
0(Th),
v−w
t v=−aupw
h(β;v⊕,v), ∀v∈Pdisc
0(Th). (14)
The idea is to use the Leray-Schauder fixed point Theorem 2.6 to the operator T.
First of all, Tis well defined.
Secondly, we will check that Tis continuous. Let {vj}j∈N⊂Pdisc
0(Th)be a
sequence such that limj→∞vj=v. Taking into account that all norms are equiv-
alent in Pdisc
0(Th)since it is a finite-dimensional space, the convergence vj→vis
equivalent to the convergence elementwise (vj)K→vKfor every K∈Th(this may
be seen, for instance, by using the norm ·
L∞()). Taking limits when j→∞in
the scheme (14) (with v:= vjand v:= T(vj)), and using the notion of convergence
elementwise, we get that
lim
j→∞ T(vj)=T(v) =Tlim
j→∞vj;
hence, Tis continuous.
In particular, as Tis a continuous operator defined over Pdisc
0(Th), which is finite-
dimensional, Tis also compact.
Finally, let us prove that the set
B={v∈Pdisc
0(Th):v=αT(v) for some 0 ≤α≤1}
is bounded. The case α=0 is trivial so we will assume that α∈(0,1].Ifv∈B,
then vis the solution of the scheme
v∈Pdisc
0(Th),
v−αw
t v=−αa
upw
h(β;v⊕,v), ∀v∈Pdisc
0(Th). (15)
Now, testing (15)byv=1, we get that
v=α
w,
and, since w≥0, and since it can be proved that v≥0 using the same arguments
than in Theorem 2.4, we get that
vL1() ≤wL1().
Hence, since Pdisc
0(Th)is a finite-dimensional space where the norms are equiva-
lent,wehaveprovedthatBis bounded. Thus, using the Leray-Schauder fixed point
Theorem 2.6, there is a solution of the scheme (10).
Let us focus on the the following linear scheme (without truncating vby its pos-
itive part, v⊕): Given vm∈Pdisc
0(Th),findvm+1∈Pdisc
0(Th)such that, for every
v∈Pdisc
0(Th):
δtvm+1v+aupw
h(β;vm+1,v) =0. (16)
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It is well-known (see, e.g. [14] and references therein) that (16) has a unique solution.
Since we have shown positivity of the non-linear truncated scheme (10), then any
solution of (10) is the unique solution of (16). This argument implies uniqueness of
(10) and positivity of (16).
Corollary 2.7 (Linear DG scheme (16) preserves positivity) For any vm∈Pdisc
0(Th)
with vm≥0in , the unique solution vm+1of the linear scheme (16)is positive, i.e.
vm+1≥0in .
2.4 Linear convection with incompressible velocity and maximum principle
Now, let us focus on the particular case of the conservation problem (8)where
β:→Rdis continuous and incompressible,i.e.∇·β=0in. In this case,
the solution of the problem (8), satisfies the following maximum principle (it can be
proved, for instance, using the characteristics method as in Remark 2.2):
min
v0≤v≤max
v0in ×(0,T).
We are going to show that the solution of the linear scheme (16) (without trun-
cating v) preserves this maximum principle. The proof is based on the fact that, as
consequence of the divergence theorem, one has
∂K
β·nK=0,∀K∈Th. (17)
Proposition 2.8 (Linear DG (16) preserves the maximum principle) For any vm∈
Pdisc
0(Th), the solution vm+1∈Pdisc
0(Th)of the scheme (16)satisfies the maximum
principle, that is
min
vm≤vm+1≤max
vmin .
Proof First, we will prove that vm+1≥minvm. Let us denote vmi n =minvm.
Taking the following test function
v=(vm+1
K∗−vmin )in K∗
0 out of K∗,
where K∗is an element of Thsuch that the value vm+1
K∗=minK∈Thvm+1
K, the scheme
(16) becomes
|K∗|δtvm+1
K∗(vm+1
K∗−vmin )=−aupw
h(β;vm+1,(v
m+1
K∗−vmin )).
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Now, as we have chosen K∗we can assure that vm+1
L≥vm+1
K∗for all L∈Th. Hence,
using (17), one has
aupw
h(β;vm+1,(v
m+1
K∗−vmin ))=
=
e∈Ei
h,e=K∗∩Le(β·ne)⊕vm+1
K∗−(β·ne)vm+1
L(vm+1
K∗−vmin )
≤
e∈Ei
h,e=K∗∩Le(β·ne)⊕vm+1
K∗−(β·ne)vm+1
K∗(vm+1
K∗−vmin )
=vm+1
K∗(vm+1
K∗−vmin )
e∈Ei
h,e=K∗∩Le
(β·ne)=0.
Therefore,
|K∗|δtvm+1
K∗(vm+1
K∗−vmin )≥0.
Moreover,
0≤|K∗|(δtvm+1
K∗)(vm+1
K∗−vmin )
=|K∗|
t (vm+1
K∗−vmin )+(vmin −vm
K∗)(vm+1
K∗−vmin )
=|K∗|
t −(vm+1
K∗−vmin )2
+(vmin −vm
K∗)(vm+1
K∗−vmin )≤0,
then we have proved that (vm+1
K∗−vmin )=0. Hence, from the choice of K∗,we
can assure vm+1≥minvm.
Now, we will prove that vm+1≤maxvm. Let us denote vmax =maxvm.
Taking the following test function
v=(vm+1
K∗−vmax )⊕in K∗
0 out of K∗,
where K∗is an element of Thsuch that the value vm+1
K∗=maxK∈Thvm+1
Kand using
similar arguments to those above, we arrive at
|K∗|δtvm+1
K∗(vm+1
K∗−vmax )⊕≤0.
Moreover,
0≤|K∗|
t (vm+1
K∗−vmax )2
⊕+(vmax −vm
K∗)(vm+1
K∗−vmax )⊕
=|K∗|
t (vm+1
K∗−vmax )+(vmax −vm
K∗)(vm+1
K∗−vmax )⊕
=|K∗|δtvm+1
K∗(vm+1
K∗−vmax )⊕≤0,
then we have proved that (vm+1
K∗−vmax )⊕=0. From the choice of K∗, we can assure
that vm+1≤maxvm.
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3 Cahn-Hilliard with degenerate mobility and incompressible
convection
At this point, given v:×(0,T) −→ Rda continuous incompressible velocity field
satisfying the slip condition (6), we are in position to consider the CCH problem (1).
Remark 3.1 Any smooth enough solution (u, μ) of the CCH model (1) satisfies the
maximum principle 0 ≤u≤1in×(0,T)whenever 0 ≤u0≤1in. The proof
of this statement is a straightforward consequence of Remark 2.2.
•To prove that u≥0, it is enough to notice that uis the solution of (8) with
β:= −(1−u)∇μ+vandtouseRemark2.2.
•To check that u≤1 we make the change of variables w:= 1−uand, using that
∇·v=0, notice that wis the solution of (8) with β:= (1−w)(∇μ+v). Then,
we just use Remark 2.2.
Owing to this maximum principle, the following C2(R)truncated potential is
considered
F (u) := 1
4⎧
⎨
⎩
u2u<0,
u2(1−u)2u∈[0,1],
(u −1)2u>1.
(18)
This truncated potential will allow us to define a linear time discrete convex-splitting
scheme satisfying an energy law (where the energy is decreasing in the case v=0)
(see Section 3.1).
Assume 0 ≤u0≤1in. The weak formulation of problem (1) consists of
finding (u, μ) such that, u(t) ∈H1(),μ(t ) ∈H1() with ∂tu(t ) ∈H1(),
M(u(t))∇μ(t ) ∈L2() a.e. t∈(0,T), and satisfying the following variational
problem a.e. t∈(0,T)for every μ, u∈H1():
∂tu(t), μ=−(M(u(t ))∇μ(t) −u(t)v(t), ∇μ)(L2())d,(19a)
(μ(t ), u)L2() =ε2(∇u(t), ∇u)(L2())d+(F (u(t )), u)L2(),(19b)
with the initial condition u(0)=u0in .
Remark 3.2 By taking μ=1in(19a), any solution uof (19) conserves the mass,
because
d
dt
u(x, t )dx =0.
Remark 3.3 By taking μ=μ(t) and u=∂tu(t ) in (19), and adding the resulting
expressions, one has that any solution (u, μ) of (19) satisfies the following energy
law
d
dt E(u(t )) +
M(u(x, t ))|∇μ(x, t )|2dx =
u(x, t ) v(x, t ) ·∇μ(x, t )dx, (20)
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where E:H1() −→ Ris the Helmholtz free energy
E(u) := ε2
2|∇u|2+F (u)dx. (21)
Indeed, by applying the chain rule we get
d
dt E(u(t )) =δE
δu (u(t)), ∂tu(t)=
μ(x, t )∂tu(x, t )dx
=−
M(u(x, t )) |∇μ(x, t )|2dx +
u(x, t ) v(x, t ) ·∇μ(x, t )dx.
In particular, in the CH case (v=0), the energy E(u(t)) is dissipative. Contrarily,
to the best knowledge of the authors, for the CCH problem with v= 0, there is no
evidence of the existence of a dissipative energy.
3.1 Convex splitting time discretization
Now, we are ready to focus on a convex splitting time discretization (of Eyre’s type
[17])of(19). Specially, we decompose the truncated potential (18) as follows:
F (u) =Fi(u) +Fe(u),
where
Fi(u) := 3
8u2,F
e(u) := 1
4⎧
⎪
⎨
⎪
⎩
−1
2u2u<0,
u4−2u3−1
2u2u∈[0,1],
1−2u−1
2u2u>1.
It can be easily proved that Fi(u) is a convex operator, which will be treated implicitly
whereas Fe(u) is a concave operator that will be treated explicitly. Then, we consider
the following convex Ei(u(t)) and concave Ee(u(t)) energy terms:
Ei(u) := ε2
2|∇u(x )|2dx +
Fi(u(x))d x,
Ee(u) :=
Fe(u(x))d x,
such that the free energy (21) is split as E(u) =Ei(u) +Ee(u).
Finally, we define the following time discretization of (19): find um+1∈H1()
and μm+1∈H1() such that, for every μ, u∈H1():
(δtum+1,μ)L2() =−(M(um+1)⊕∇μm+1−um+1v(tm+1), ∇μ)(L2())d,(22a)
(μm+1,u)L2() =ε2(∇um+1,∇u)(L2())d+(f (um+1,u
m), u)L2(),(22b)
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where we denote the convex-implicit and concave-explicit linear approximation of
the potential as follows
f(u
m+1,u
m):= F
i(um+1)+F
e(um)
=3
4um+1+1
4⎧
⎨
⎩
−um,u
m∈(−∞,0),
4(um)3−6(um)2−um,u
m∈[0,1],
−(um+2),u
m∈(1,+∞).
(23)
Notice that the positive part of the mobilitity has been taken in (22a), regarding the
Remark 4, in order to prevent possible overshoots of the solution um+1beyond the
interval [0,1].
3.1.1 Discrete energy law
By adding (22a)and(22b)forμ=μm+1and u=δtum+1in (22), we get:
M(um+1)⊕|∇μm+1|2+ε2(∇um+1δt∇um+1)(L2())d
+(f (um+1,u
m), δtum+1)L2() =
um+1v(·,t
m+1)·∇μm+1. (24)
Taking into account that
ε2(∇um+1,δ
t∇um+1)(L2())d=ε2
2δt∇um+12
+kε2
2δt∇um+12
,
and by adding and substracting δtF(u
m+1), we get the following equality
δtE(um+1)+
M(um+1)⊕|∇μm+1|2+kε2
2|δt∇um+1|2
+(f (um+1,u
m), δtum+1)L2() −δt
F(u
m+1)=
um+1v(·,t
m+1)·∇μm+1,
where E(u) is defined in (21).
Then, using the Taylor theorem, we get
Fi(um)=Fi(um+1)+F
i(um+1)(um−um+1)+F
i(um+ξ)
2(um−um+1)2,
Fe(um+1)=Fe(um)+F
e(um)(um+1−um)+F
e(um+η)
2(um+1−um)2,
for certain ξ,η ∈(0,1)with um+ξ=ξum+1+(1−ξ)um,um+η=ηum+1+(1−η)um.
Hence, adding these expressions and taking into consideration that F (u) =Fi(u) +
Fe(u) for every u∈R, we arrive at
F(u
m+1)−F(u
m)=
=f(u
m+1,u
m)(um+1−um)−F
i(um+ξ)−F
e(um+η)
2(um+1−um)2.
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Furthermore, as
F
i(um+ξ)−F
e(um+η)
2k=
=⎧
⎨
⎩
1
2k,u
m+η∈(−∞,0)∪(1,+∞),
3−12(um+η)2+12(um+η)+1
8k,u
m+η∈[0,1],
=1
2k≥0,u
m+η∈(−∞,0)∪(1,+∞),
1
2k1−3um+η(um+η−1)≥0,u
m+η∈[0,1],
we have
F
i(um+ξ)−F
e(um+η)
2k≥0,
and finally
(f (um+1,u
m), δtum+1)L2() −
δtF(u
m+1)≥0.
Therefore, we arrive at the following result:
Theorem 3.4 Any solution of the scheme (22)satisfies the following discrete energy
law
δtE(um+1)+
M(um+1)⊕|∇μm+1|2+kε2
2|δt∇um+1|2
≤
um+1v(·,t
m+1)·∇μm+1. (25)
In particular, if v=0the time-discrete scheme is unconditionally energy stable,
because E(um+1)≤E(um).
3.2 Fully discrete scheme
At this point, we are going to introduce the key idea for the spatial approximation:
to treat (1a) as a conservative problem where there are two different fluxes (linear
and non-linear). In this sense, we propose an upwind DG scheme approximating the
non-linear flux F(u) =M(u)∇μproperly.
To this aim, we take for values v∈Rthe increasing and decreasing part of M(v)⊕,
denoted respectively by M↑(v) and M↓(v), as follows:
M↑(v) =v
0
(∂s(M(s)⊕))⊕ds =min(v,1)
0
M(s)⊕ds =min(v ,1)
0
(1−2s)⊕ds,
M↓(v) =−
v
0
(∂s(M(s)⊕))ds =−min(v,1)
0
M(s)ds
=−
min(v,1)
0
(1−2s)ds.
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Therefore,
M↑(v) =M(v)⊕if v≤1
2
M1
2if v>1
2
,M
↓(v) =0ifv≤1
2
M(v)⊕−M1
2if v>1
2
. (26)
Notice that M↑(v) +M↓(v) =M(v)⊕. We define the following generalized
upwind bilinear form to be applied for the non-linear flux F(u) =M(u)βwhere
now βcan be discontinuous over Ei
h(in fact we will take β=∇μ):
aupw
h(β;M(v)⊕,v) := −
(β·∇v)M(v)⊕
+
e∈Ei
h,e=K∩Le({{β}} ·ne)⊕(M ↑(vK)+M↓(vL))
−({{β}} ·ne)(M ↑(vL)+M↓(vK))[[ v]] . (27)
Remark 3.5 We refer to (27) as a generalized bilinear form since it generalizes
the definition of (11) considering the case where βmay be discontinuous. If βis
continuous both definitions are equivalent.
Then, we propose the following fully discrete DG+Eyre scheme (named DG-
UPW) for the model (1):
Find um+1∈Pdisc
0(Th), with μm+1,w
m+1∈Pcont
1(Th), solving
(δtum+1,u)L2()
+aupw
h(−∇μm+1;M(um+1)⊕,u) +aupw
h(v(tm+1);um+1,u) =0,(28a)
(μm+1,μ)L2() =ε2(∇wm+1,∇μ)L2() +(f (um+1,u
m), μ)L2(),(28b)
(wm+1,w)L2() =(um+1,w)L2(),(28c)
for all u∈Pdisc
0(Th)and μ, w∈Pdisc
1(Th). Following the notation of the Section 2.3,
aupw
h(v;u, u) =
e∈Ei
h,e=K∩Le
((v·ne)⊕uK−(v·ne)uL)[[ u]] .
In this scheme, we have introduced a truncation of the function M(u) taking its
positive part M(u)⊕, which is consistent as the solution of the continuous model (1)
satisfies 0 ≤u≤1.
Notice that we have introduced a new continuous variable w∈Pcont
1(Th)in (28c).
It can be seen as a regularization of the variable u∈Pcont
0(Th), which is used in the
diffusion term in (28b) (which corresponds to the philic term in the energy of the
model (21)). In fact, both variables wm+1and um+1are approximations of u(tm+1).
Remark 3.6 We are using the same notation in the fully discrete scheme (28)than
the one we have used in the time-discrete scheme (3.1) given in the Section 3.1.1,
satisfying an energy law.
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Nevertheless, in this case, we are changing the meaning of the equations since we
are treating (28a)astheu-equation and (28b)astheμ-equation, contrary to computa-
tions done to reach the energy law. This has been done for the purpose of preserving
the maximum principle in the (28a) and adequately approximating the laplacian term
of the (28b).
Remark 3.7 The boundary condition ∇wm+1·n=0on∂ ×(0,T) is imposed
implicitly by the term (∇wm+1,∇μ)L2() in (28b).
Remark 3.8 Since f(·,u
m)is linear, we have the following equality of the potential
term of (28b):
(f (wm+1,u
m), μ)L2() =(f (um+1,u
m), μ)L2().
Remark 3.9 The scheme (28) is non-linear; hence, we will have to use an iterative
procedure, the Newton’s method, to approach its solution.
Proposition 3.10 The scheme (28)conserves the mass of both um+1and wm+1
variables:
um+1=
umand
wm+1=
wm.
Proof Just need to take u=1in(28a)andw=1in(28c).
Theorem 3.11 (DG (28) preserves the maximum principle) For any um∈Pdisc
0(Th)
with 0≤um≤1in , then any solution um+1of (28)satisfies 0≤um+1≤1in .
Proof Firstly, we prove that um+1≥0. Taking the following Pdisc
0(Th)test function
u=(um+1
K∗)in K∗
0 out of K∗,
where K∗is an element of Thsuch that um+1
K∗=minK∈Thum+1
K,(28a) becomes
|K∗|δtum+1
K∗(um+1
K∗)=
=−aupw
h−∇μm+1;M(um+1)⊕, u−aupw
hv(tm+1);um+1, u. (29)
Now, since um+1
L≥um+1
K∗for all L∈Th, we can assure that
M↑(um+1
L)≥M↑(um+1
K∗)and M↓(um+1
L)≤M↓(um+1
K∗).
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Then, we can bound as follows:
aupw
h(−∇μm+1;M(um+1)⊕,u) =
=
e∈Ei
h,e=K∗∩Le(−∇μm+1·ne)⊕(M ↑(um+1
K∗)+M↓(um+1
L))
−(−∇μm+1·ne)(M↑(um+1
L)+M↓(um+1
K∗))(um+1
K∗)
≤
e∈Ei
h,e=K∗∩Le(−∇μm+1·ne)⊕(M ↑(um+1
K∗)+M↓(um+1
K∗))
−(−∇μm+1·ne)(M↑(um+1
K∗)+M↓(um+1
K∗))(um+1
K∗)
=
e∈Ei
h,e=K∗∩Le
(−∇μm+1·ne)M(um+1
K∗)⊕(um+1
K∗)=0.
On the other hand, applying the incompressibility of vand proceeding as in
Section 2.4, one has that
aupw
h(v(tm+1);um+1,u) ≤0.
Therefore, from (29)
|K∗|δtum+1
K∗(um+1
K∗)≥0.
Consequently, it is satisfied that
0≤|K∗|(δtum+1
K∗)(um+1
K∗)=−|K∗|
t (um+1
K∗)2
+um
K∗(um+1
K∗)≤0;
hence, since um
K∗≥0, we prove that (um+1
K∗)=0. Hence, um+1≥0.
Secondly, we prove that um+1≤1. Taking the following test function
u=(um+1
K∗−1)⊕in K∗
0 out of K∗,
where K∗is an element of Thsuch that um+1
K∗=maxK∈Thum+1
Kand using similar
arguments than above, we arrive at
|K∗|δtum+1
K∗(um+1
K∗−1)⊕≤0.
Besides, it is satisfied that
0≥|K∗|δtum+1
K∗(um+1
K∗−1)⊕=|K∗|
t (um+1
K∗−1)+(1−um
K∗)(um+1
K∗−1)⊕
=|K∗|
t (um+1
K∗−1)2
⊕+(1−um
K∗)(um+1
K∗−1)⊕≥0;
hence, we deduce that (um+1
K∗−1)⊕=0 and, therefore, um+1≤1.
The following result is a direct consequence of Theorem 3.11.
Corollary 3.12 If we use mass-lumping to compute wm+1in (28c),then0≤
wm+1≤1in for m≥0.
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Theorem 3.13 There is at least one solution of the scheme (28).
Proof Given a function z∈Pdisc
0(Th)with 0 ≤z≤1, we define the map
T:Pdisc
0(Th)×Pcont
1(Th)×Pcont
1(Th)−→ Pdisc
0(Th)×Pcont
1(Th)×Pcont
1(Th)
such that T(
u, μ, w) =(u, μ, w) ∈Pdisc
0(Th)×Pcont
1(Th)×Pcont
1(Th)is the unique
solution, for every u∈Pdisc
0(Th),μ, w∈Pcont
1(Th), of the linear (and decoupled)
scheme:
1
t (u−z, u)L2() +aupw
h(v;u, u) =−aupw
h(−∇μ;M(
u)⊕,u), (30a)
(μ, μ)L2() =ε2(∇w, ∇μ)L2() +(f (u, z), μ)L2(),(30b)
(w, w)L2() =(u, w)L2(),(30c)
To check that Tis well defined, one may use the following steps. First, it is easy
to prove that there is a unique solution uof (30a) which implies that there is a unique
solution wof (30c) using, for instance, the Lax-Milgram theorem. Then, it is straight-
forward to see that the solution μof (30b) is unique, which implies its existence as
Pcont
1(Th)is a finite-dimensional space.
It can be proved, using the notion of convergence elementwise, as it was done in
Theorem 2.5 and taking into consideration that ∇μ∈Pdisc
0(Th)d, that the operator
Tis continuous, and, therefore, it is compact since Pdisc
0(Th)and Pcont
1(Th)have finite
dimension.
Finally, let us prove that the set
B={(u, μ, w) ∈Pdisc
0(Th)×Pcont
1(Th)×Pcont
1(Th):
(u, μ, w) =αT (u, μ, w) for some 0 ≤α≤1}
is bounded (independent of α). The case α=0 is trivial so we will assume that
α∈(0,1].
If (u, μ, w) ∈B,thenu∈Pdisc
0(Th)is the solution, for every u∈Pdisc
0(Th),of
1
t (u −αz, u)L2() +aupw
h(v;u, u) =−αa
upw
h(−∇μ;M (u)⊕,u). (31)
Now, testing (31)byu=1, we get that
u=α
z,
and, since 0 ≤z≤1, and since it can be proved that 0 ≤u≤1usingthesame
arguments than in Theorem 3.11, we get that
uL1() ≤zL1() .
Moreover, w∈Pcont
1(Th)is the solution of the equation
(w, w)L2() =(u, w)L2(),∀w∈Pcont
1(Th). (32)
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Testing with w=wand using that u≤1 and that the norms are equivalent in
Pcont
1(Th), we obtain
w2
L2() =(u, w)L2() ≤wL1() ≤||1/2wL2() ;
hence, wL2() ≤||1/2holds.
Finally, we will check that μis bounded. Regarding that, μ∈Pcont
1(Th)is the
solution of
(μ, μ)L2() =ε2(∇w, ∇μ)L2() +(f (u, z), μ)L2(),∀μ∈Pcont
1(Th), (33)
by testing by μ=μwe get that
μ2
L2() ≤ε2∇w(L2()) ∇μ(L2()) +f (u, z)L2() μL2()
≤ε2wH1() μH1() +f (u, z)L2() μL2() .
The norms are equivalent in the finite-dimensional space Pcont
1(Th); therefore, there
is Ccont ≥0 such that
μL2() ≤ε2Ccont wL2() +f (u, z)L2() .
Hence, as 0 ≤u, z ≤1, we know that f (u, z)L2() is bounded, and therefore
μL2() is bounded.
Since Pdisc
0(Th)and Pcont
1(Th)are finite-dimensional spaces where all the norms
are equivalent, we have proved that Bis bounded.
Thus, using the Leray-Schauder fixed point Theorem 2.6, there is a solution
(u, μ, w) of the Scheme (28).
Corollary 3.14 There is at least one solution of the following (non-truncated)
scheme:
Find um+1∈Pdisc
0(Th)with 0≤um+1≤1and μm+1,w
m+1∈Pcont
1(Th)with
0≤wm+1≤1, solving
(δtum+1,u)L2()
+aupw
h(−∇μm+1;M(um+1), u) +aupw
h(v(tm+1);um+1,u) =0,(34a)
μm+1, μL2() =ε2∇wm+1,∇μL2() +f(u
m+1,u
m), μL2() ,(34b)
wm+1, wL2() =um+1, wL2() . (34c)
for all u∈Pdisc
0(Th)and μ, w∈Pcont
1(Th). Here, we have considered M(um+1)
instead of M(um+1)⊕.
Proof By Theorems 3.11 and 3.13 we know that there is a solution of the scheme
(28) such that 0 ≤um≤1infor every m≥0. Hence, M(um)=M(um)⊕for
every m≥0, and therefore the solution of (28) is also a solution of (34), which
moreover satisfies the discrete maximum principle.
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Fig. 2 Left: initial condition in the case v=0. Right: initial condition in the case v=100(y, −x)
4 Numerical experiments
We now present several numerical tests in which we explore the behaviour of the
new upwind DG scheme presented in this work (DG-UPW)(28) and compare it
with two other space semidiscretizations found in the literature: firstly a classical
FEM discretization (FEM) and secondly the DG scheme proposed in [18], based on
an SIP + (sigmoid upwind) technique, that we call (DG-SIP).WeuseP1piecewise
polynomials for both schemes unless otherwise specified.
For the DG-UPW scheme, we use mass lumping to compute wm+1in (28c)so
that wm+1∈[0,1]by the Corollary 3.12. This wm+1, which is a regularization of
the primal variable um+1, is considered as the main phase-field variable, which is
used when showing the results of the numerical experiments. Moreover, we consider
Pe =1 unless another value is specified.
4.1 Qualitative tests and comparisons
Our first numerical tests are devoted to qualitative experiments about our DG-UPW
scheme in rectangular and circular domains with different kinds of velocity fields.
We also inspect the discrete energy and the maximum principle property, confirming
Fig. 3 Aggregation of circular regions at T=0.001 without convection, 3D view (height represents the
value of the phase variable on each point of the squared domain)
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Fig. 4 Aggregation of circular phases. On the left, maximum (top) and minimum (bottom) of the phase
field variable over time without convection (v=0). On the right, energy over time
that the latter one holds for our scheme but not for the two aforementioned ones,
FEM and DG-SIP.
4.1.1 Agreggation of circular regions without convection
First, we consider the Cahn–Hilliard equation without convection (v=0) in the
squared domain =(0,1)2and the following initial condition (two small circles of
radius 0.2, see Fig. 2,left):
u0=1
2⎛
⎝tanh 0.2 −(x −x1)2+(y −y1)2
√2ε+1⎞
⎠
+1
2⎛
⎝tanh 0.2 −(x −x2)2+(y −y2)2
√2ε+1⎞
⎠,(35)
with centers (x1,y
1)=(0.3,0.5)and (x2,y
2)=(0.7,0.5). We take a structured
mesh with h≈2.8284 ·10−2and run time iterations with t =10−6. Each itera-
tion consists of solving a non-linear system for computing (un+1,μ
n+1,w
n+1),for
which we use Newton’s method iterations, programmed on the FEniCS finite ele-
ment library [1,29]. For linear systems, we used a MPI parallel solver (GMRES) in
the computing cluster of the Universidad de C´
adiz.
In Fig. 3, we show a 3D view of the phase field function at the time step T=
0.001, when the aggregation process has started. It is interesting to notice that for our
upwind DG scheme (28) there are no spurious oscillations meanwhile for the FEM
and DG-SIP schemes we obtain several numerical issues (vertical fluctuations in the
3D graphics).
Moreover, in Fig. 4, we can clearly observe how the maximum principle is pre-
served by DG-UPW scheme (and not by the two other ones). Regarding the energy,
we obtain a non-increasing behaviour as expected from the continuous model. An
analytical proof of this property in the discrete case is left as future work.
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Fig. 5 Aggregation of circular regions over time with a strong convection (v=100(y, −x))
4.1.2 Agreggation of circular regions with convection
Second, we define as the unit ball in R2and, again, the initial condition (35)(two
small circles of radius 0.2), with centers (x1,y
1)=(−0.2,0)and (x2,y
2)=(0.2,0)
(see Fig. 2(right)). Moreover, for testing the effect of convection in our scheme, we
take ε=0.001 and v=100(y, −x),sothatv·n=0on∂. We take an unstructured
mesh with h≈4·10−2and run time iterations with t =10−3. Figure 5shows
the values of the phase field function at different time steps. We can observe that,
despite of our election of a highly significant convection term, the results of scheme
DG-UPW are qualitatively correct. Nevertheless, we can observe how the spurious
oscillations become more important and the solution begins to have an unexpected
behaviour when using both the FEM and DG-SIP schemes.
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Fig. 6 Aggregation of circular phases with strong convection (v=100(y, −x)). On the left, maximum
and minimum of the phase field variable over time. On the right, we plot
um+1−um
L∞()
umL∞() to observe the
dynamics of the approximations
Concerning the maximum principle, in Fig. 6, we can see how this property is
preserved by the DG-UPW scheme (28), while the phase field variable reaches non-
physical values, very far from [0,1], when using the other aforementioned schemes.
Moreover, it is interesting to observe the approximation of the steady state of the
schemes, represented by the quantity
um+1−um
L∞()
umL∞() , which tends to 0 in the case
DG-UPW. This fact indicates that the solution converges to a stationary state, while
for the other schemes it remains in an oscillatory state.
The computational time spent to obtain the results (computed sequentially) with
each of these schemes is 2:32min using DG-UPW, 1:10min using FE and 3:24min
using DG-SIP.
For the fairness of comparisons, we redo the tests using both FE and DG-SIP
reducing the step size of the mesh, on the one hand, and using higher order polyno-
mials, on the other hand. First, if we reduce the mesh size to h/2≈2·10−2and we
Fig. 7 Aggregation of circular phases with strong convection (v=100(y, −x)) using the DG-SIP scheme.
On the left, the result obtained with h/2≈2·10−2and Pdisc
1(Th). On the right, the result obtained with
h≈4·10−2and Pdisc
2(Th)
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Fig. 8 Random initial
perturbation for the spinoidal
decomposition test
use P1polynomials, the FEM scheme does not converge (the linear solver, GMRES,
fails to converge) while the DG-SIP scheme gives us the results shown in Fig. 7
(left), requiring 35:20min to complete the computations sequentially. Second, if we
keep the mesh size h≈4·10−2and we use P2polynomials, the FEM scheme does
not converge either (Newton’s method does not converge) while the DG-SIP scheme
gives us the results shown in Fig. 7(right), requiring 11min to complete the compu-
tations sequentially. Therefore, the FEM scheme does not even converge if we try to
improve the results above and, while the DG-SIP scheme does converge, the results
still show spurious oscillations and require a much longer computational time to be
completed than those shown in Fig. 5(right) using our DG-UPW scheme.
4.1.3 Spinoidal decomposition driven by Stokes cavity flow
We show the results from a spinoidal decomposition test, in which the initial con-
dition is a small uniformly distributed random perturbation around 0.5, u0(x) ∈
[0.49,0.51]for x∈asshowninFig.8.
As convection vector v, we take the flow resulting from solving a cavity test for
the Stokes equations in the domain =[0,2]×[0,1]with Dirichlet boundary
conditions given by a parabolic profile
v(x, y ) =(x (2−x),0), ∀(x , y) ∈top ={(x, 1)∈R2:x∈[0,2]}.
We used this parabolic profile in order to avoid the discontinuities of the Stokes
velocity for a standard boundary condition v=1ontop, which produces a non-
vanishing divergence in the corners where the discontinuities arise. In fact, we have
checked that, in this particular case where v=1ontop, the scheme DG-UPW does
not preserve the upper bound um≤1, although it does preserve the lower bound
0≤um(recall that, for compressible velocity, positivity is the only property of the
solution, see Remark 2.2).
We set ε=0.005, t =0.001, h≈0.07 and, in this case, we take Pe =10 in
order to emphasize the convection effect. We can observe in Figs. 9and 10 how the
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Fig. 9 Spinoidal decomposition over time with convection vector obtained from a cavity test
maximum principle is preserved by our DG-UPW scheme (28) while for the other
schemes the solution takes values out of the interval [0,1](to notice that, we must
take into account the scale of the values shown on the right-hand side of each picture).
Furthermore, in Fig. 10, we can also notice that the approximation obtained using
the DG-UPW scheme converges to a stationary state, while it remains in an non-
physical oscillatory state when we use the other schemes.
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Table 1 Errors and convergence orders at T=0.001 without convection (v=0)
Scheme Norm h≈2.8284 ·10−2h/2≈1.4142 ·10−2h/3≈9.428 ·10−3h/4≈7.071 ·10−3
Error Error Order Error Order Error Order
DG-UPW ·L28.5268 ·10−33.0933 ·10−31.46 1.7645 ·10−31.38 1.2134 ·10−31.30
·H18.0000 ·10−14.0199 ·10−10.99 2.6081 ·10−11.07 1.8849 ·10−11.13
FEM ·L25.3224 ·10−31.5679 ·10−31.76 6.9944 ·10−41.99 4.0191 ·10−41.93
·H18.9963 ·10−14.1080 ·10−11.13 2.5252 ·10−11.2 1.7799 ·10−11.22
DG-SIP ·L24.6466 ·10−31.3023 ·10−31.84 5.8945 ·10−41.96 3.2710 ·10−42.05
·H11.1784 5.8331 ·10−11.01 3.6254 ·10−11.17 2.6024 ·10−11.15
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Table 2 Errors and convergence orders at T=0.001 with convection (v=(y, −x))
Scheme Norm h≈4·10−2h/2≈2·10−2h/3≈1.3333 ·10−2h/4≈1·10−2
Error Error Order Error Order Error Order
DG-UPW ·L21.7288 ·10−26.9446 ·10−31.32 3.3102 ·10−31.83 2.0578 ·10−31.65
·H11.4549 6.0305 ·10−11.27 3.0204 ·10−11.71 2.0315 ·10−11.38
FEM ·L26.8347 ·10−32.1213 ·10−31.69 9.7749 ·10−41.91 5.3883 ·10−42.07
·H18.3104 ·10−13.8060 ·10−11.13 2.1887 ·10−11.36 1.4991 ·10−11.32
DG-SIP ·L26.5242 ·10−31.9557 ·10−31.74 8.9471 ·10−41.93 5.0257 ·10−42.00
·H11.1980 6.1624 ·10−10.96 3.8451 ·10−11.16 2.7439 ·10−11.17
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Fig. 10 Spinoidal decomposition with convection vector obtained from a cavity test. On the left, maximum
and minimum of the phase field variable over time. On the right, we plot
um+1−um
L∞()
umL∞() to observe the
dynamics of the approximations
4.2 Error order test
We now introduce the results of a numerical test in which we study the convergence
order of our numerical scheme DG-UPW, verifying experimentally that the expected
order is obtained. For the sake of completeness, we also compare the convergence
orders of the two others aforementioned space semidiscretizations: FEM, DG-SIP.
We consider again the same initial conditions than in Section 4.1.1 with ε=0.01
(see Fig. 2).
First, in the non-convective case, errors and convergence order are compared with
respect to an approximate solution which is computed using the FEM scheme in a
very fine mesh of size h=1.414 ·10−3and a time step t =10−6(which is taken
as the “exact solution”). In this case, we have used conforming structured meshes for
the space discretization. The results for the DG-UPW scheme, which are shown in
the first row of Table 1, confirm order 1 (in fact, slightly over 1) in norm ·L2.These
results match our expectations for the P0approximation of um+1, with the upwind
discretization of the non-linear second-order term. It is interesting to emphasize that,
unexpectedly, the scheme produces kind results in ·
H1, reaching order 1. On the
other hand, the FEM and DG-SIP schemes reach order 2 in L2and order 1 in H1
norms, as expected (see also Table 1).
Next, we focus on the case with convection, where we take v=(y, −x) in the unit
ball. The resulting errors and convergence orders computed using the three different
schemes over a conforming unstructured mesh are shown in Table 2. In this case, the
errors are computed with respect to the solution obtained for the FEM scheme in a
mesh of size h=4·10−3.
In this case, it is interesting to notice that the error order in ·L2of the DG-UPW
scheme is improved and it approaches the order 2 of the other schemes, while order
in ·
H1slightly beats the other schemes.
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As a technical comment, notice that, in order to compute the errors, we projected
on a Pcont
1space both the exact and the DG solution obtained when using the DG-SIP.
In the case of the DG-UPW scheme, we have taken was the continuous solution.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
The first author has been supported by UCA FPU contract UCA/REC14VPCT/2020 funded by Univer-
sidad de C´adiz andbyaGraduate Scholarship funded by the University of Tennessee at Chattanooga.
The second and third authors have been supported by Grant PGC2018-098308-B-I00 by MCI N/AEI/
10.13039/501100011033 and by ERDF a way of making Europe.
Data availability Data sharing not applicable to this article as no datasets were generated or analyzed
during the current study.
Declarations
Conflictofinterest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
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References
1. Alnæs, M., Blechta, J., Hake, J., et al.: The FEniCS project version 1.5. Arch. Numer. Softw 3(100),
9–23 (2015)
2. Aristotelous, AC., Karakashian, OA., Wise, SM.: Adaptive, second-order in time, primitive-variable
discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source. IMA J. Numer.
Anal 35(3), 1167–1198 (2015)
3. Badalassi, V., Ceniceros, H., Banerjee, S.: Computation of multiphase systems with phase field
models. J. Comput. Phys 190(2), 371–397 (2003)
4. Bailo, R., Carrillo, JA., Kalliadasis, S., et al.: Unconditional bound-preserving and energy-dissipating
finite-volume schemes for the Cahn-Hilliard equation. arXiv:210505351 (2021)
5. Barrett, JW., Blowey, JF., Garcke, H.: Finite element approximation of the Cahn–Hilliard equation
with degenerate mobility. SIAM J. Numer. Anal 37(1), 286–318 (1999)
6. Bertozzi, AL., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn–Hilliard
equation. IEEE Trans. Image Process 16(1), 285–291 (2007)
7. Boyer, F., Nabet, F.: A DDFV method for a Cahn-Hilliard/Stokes phase field model with dynamic
boundary conditions. ESAIM Math. Model. Numer. Anal 51(5), 1691–1731 (2017)
8. Cahn, JW.: On spinodal decomposition. Acta Metall 9(9), 795–801 (1961)
9. Cahn, JW., Hilliard, JE.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys
28(2), 258–267 (1958)
10. Chen, W., Wang, C., Wang, X., et al.: Positivity-preserving, energy stable numerical schemes for the
Cahn-Hilliard equation with logarithmic potential. J. Comput. Phys. X 3, 100,031 (2019)
11. Chen, Y., Shen, J.: Efficient, adaptive energy stable schemes for the incompressible Cahn–Hilliard
Navier–Stokes phase-field models. J. Comput. Phys 308, 40–56 (2016)
12. Cheng, K., Feng, W., Wang, C., et al.: An energy stable fourth order finite difference scheme for the
Cahn–Hilliard equation. J. Comput. Appl. Math 362, 574–595 (2019)
1618 Numerical Algorithms (2023) 92:1589–1619
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
13. Cueto-Felgueroso, L., Peraire, J.: A time-adaptive finite volume method for the Cahn–Hilliard and
Kuramoto–Sivashinsky equations. J. Comput. Phys 227(24), 9985–10,017 (2008)
14. Di Pietro, DA., Ern, A.: Mathematical aspects of discontinuous Galerkin methods. Springer, New
York (2012)
15. Elliott, CM., French, DA.: A nonconforming finite-element method for the two-dimensional Cahn–
Hilliard equation. SIAM J. Numer. Anal 26(4), 884–903 (1989)
16. Elliott, CM., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math.
Anal 27(2), 404–423 (1996)
17. Eyre, DJ.: An unconditionally stable one-step scheme for gradient systems. Unpublished article (1998)
18. Frank, F., Liu, C., Alpak, FO., et al.: A finite volume / discontinuous Galerkin method for the advec-
tive Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT
imaging. Comput Geosci 22(2), 543–563 (2018)
19. Frank, F., Rupp, A., Kuzmin, D..: Bound-preserving flux limiting schemes for DG discretizations of
conservation laws with applications to the Cahn–Hilliard equation. Comput. Methods Appl. Mech.
Eng 359, 112–665 (2020)
20. Furihata, D.: A stable and conservative finite difference scheme for the Cahn–Hilliard equation.
Numer. Math 87(4), 675–699 (2001)
21. Guill´
en-Gonz´
alez, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model.
JComput. Phys 234, 140–171 (2013)
22. Guo, Z., Lin, P., Lowengrub, J., et al.: Mass conservative and energy stable finite difference methods
for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: primitive variable and projection-
type schemes. ComputMethods Appl. Mech. Eng 326, 144–174 (2017)
23. Ibrahim, M., Saad, M.: On the efficacy of a control volume finite element method for the capture of
patterns for a volume-filling chemotaxis model. Comput. Math. Appl 68(9), 1032–1051 (2014)
24. Kay, D., Styles, V., S¨uli, E.: Discontinuous Galerkin finite element approximation of the Cahn–
Hilliard equation with convection. SIAM J. Numer. Anal 47(4), 2660–2685 (2009)
25. Kim, J.: A numerical method for the Cahn–Hilliard equation with a variable mobility. Commun.
Nonlinear Sci. Numer. Simul 12(8), 1560–1571 (2007)
26. Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Phys 12(03), 613–
661 (2012)
27. Li, R., Gao, Y., Chen, J., et al.: Discontinuous finite volume element method for a coupled Navier-
Stokes-Cahn-Hilliard, phase field model. Adv. Comput. Math 46(2), 1–35 (2020)
28. Liu, H., Yin, P.: Unconditionally energy stable discontinuous Galerkin schemes for the Cahn–Hilliard
equation. JComput. Appl. Math 390, 113,375 (2021)
29. Logg, A., Mardal, KA., Wells, GN., et al.: Automated solution of differential equations by the finite
element method. Springer Berlin, Heidelberg (2012)
30. Miranville, A.: The Cahn–Hilliard equation and some of its variants. AIMS Math 2(3), 479–544
(2017)
31. Pironneau, O.: Finite element methods for fluids. Wiley (1989)
32. Wang, C., Wise, SM.: An energy stable and convergent finite-difference scheme for the modified
phase field crystal equation. SIAM J. Numer. Anal 49(3), 945–969 (2011)
33. Wells, GN., Kuhl, E., Garikipati, K.: A discontinuous Galerkin method for the Cahn–Hilliard equation.
J. Comput. Phys 218(2), 860–877 (2006)
34. Wise, S., Lowengrub, J., Frieboes, H., et al.: Three-dimensional multispecies nonlinear tumor
growth—I. J. Theor. Biol 253(3), 524–543 (2008)
35. Wu, X., Zwieten, GJ., Zee, KG.: Stabilized second-order convex splitting schemes for Cahn–Hilliard
models with application to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed
Eng 30(2), 180–203 (2014)
36. Xia, Y., Xu, Y., Shu, CW.: Local discontinuous Galerkin methods for the Cahn–Hilliard type
equations. J. Comput Phys 227(1), 472–491 (2007)
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