IMA Journal of Numerical Analysis
(2005) Page 1 of 27doi: 10.1093/imanum/dri017
A Meshfree Partition of Unity Method for Diffusion Equations onComplex Domains
M. EIGEL
∗
, E. GEORGE†, M. KIRKILIONIS‡
Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL, UK.
[Received on xxx]
We present a numerical method for solving partial differential equations on domains with distinctivecomplicated geometrical properties. These will be called
complex domains
. Such domains occur in manyreal world applications, for example in geology or engineering. We are however in particular interested inapplications stemming from the life sciences, especially cell biology. In this area such complex domainsas retrieved from microscopy images at different scales are the norm and not the exception. Therefore,geometry is expected to directly inﬂuence the physiological function of different systems, for examplesignalling pathways. New numerical methods able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restrictions in the past. The approximation approach presented here for such problems is based on a promisingmeshfree Galerkin method, the
partition of unity method
(PUM). We introduce the main approximationfeatures, and then focus on the construction of appropriate covers as the basis of discretisations. Asa main result we present an extended version of cover construction ensuring fast convergence rates inthe solution process. Parametric patches are introduced as a possible way of approximating complicatedboundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergencebehaviour of the PUM is demonstrated in several numerical examples.
Keywords
: complex domains, meshfree partition of unity method, generalised ﬁnite elements, cover generation
1. Introduction
Simulation of technical and natural phenomena has become a primary tool of scientiﬁc understanding byeither pre or reconstructing the interacting mechanisms of a given problem. The need to understand agrowing number of increasingly complicated problems has generated different sophisticated numericalmethods to approximate solutions of the associated equations. Many such problems typically involvingtransport and reactions, can in general be adequately described by partial differential equations (PDE),although there are many exceptions where stochastic effects and discrete entities do not allow exclusive concentration on spatial continuum approximations. Common and at the same time traditionalapproaches to numerically solving PDE are ﬁnite differences (FD) and ﬁnite element methods (FEM).Both depend on a ﬁxed structure of discretisation nodes and a deﬁned connectivity to neighbour nodes,given by a mesh.Research to generalise these methods and alleviate or entirely remove the dependency on a mesh has
∗
eigel@maths.warwick.ac.uk
†
E.George@warwick.ac.uk
‡
mak@maths.warwick.ac.uk
IMA Journal of Numerical Analysis c
Institute of Mathematics and its Applications 2005; all rights reserved.
2 of 27
M. EIGEL, E. GEORGE, M. KIRKILIONIS
gained momentum since the 1990s with the development of moving least squares (MLS) methods andthe partition of unity method (PUM). Comprehensive overviews of meshfree methods (MM) can befound in [9, 5, 17, 16]. The most important goals for MM is an improved ﬂexibility in the constructionof local approximations and better handling of complicated geometries and reﬁnements. In Sec. 2we describe the partition of unity meshfree method (PUM) which is used throughout the paper. Itis implemented in a computational code, the
Generic Discretisation Framework
(
GDF
), written bythe authors. A key aspect of the PUM, the construction of a cover of the domain, is explained indetail in Sec. 3. As a major extension to existing algorithms and central to this paper we discuss covercreation strategies in settings with complex geometries and holes. This is followed in Sec. 4 with anoverview of the actual discretisation process for a generic linear scalar PDE. Special attention is paidto the accurate treatment of boundary patches where numerical integration can degrade the convergenceorder. Finally, numerical examples demonstrate the capabilities of the method in Sec. 5. We conﬁrmthe anticipated convergence features, which in particular are shown for a geometrically complex setupincluding several holes in the domain. The outlook section, Sec. 6, summarises the main ideas of themanuscript and discusses the necessary future extensions for simulating systems of coupled nonlineartransport equations on complex and, in particular, nested domains.
2. The Partition of Unity Method
2.1
Introduction
The numerical method used in this paper is based on the framework of the partition of unity method(PUM), initially introduced by Melenk, Babuˇska [2, 18]. With regard to the construction of local func
tion spaces, the class of generalised ﬁnite element methods (GFEM) is closely related and can be seen asa subclass of the PUM. Application of PUM and GFEM to real world problems were described in a series of papers by Griebel and Schweitzer [11, 15, 10, 13, 14, 12], the monograph [20] and in Strouboulis
[25, 26, 23, 24]. The PUM exhibits some unique features which renders it especially attractive for someproblems classical FEM cannot deal with efﬁciently:
•
Approximation spaces of arbitrary smoothness can be created. Moreover, special functions suitedfor approximating the sought solution are incorporated and adapted easily (cf. [18] and also [4]).
•
No mesh has to be created for the discretisation. Meshfree methods usually are based on a set of nodes
{
x
i
}
N i
=
1
for which corresponding patches
{
ω
i
}
N i
=
1
are deﬁned, usually with the property thatthe domain
Ω
is covered by the union of the patches. The nodes can be redistributed or the nodedensity changed without geometrical constraints as is the case with triangulations (cf. [8, 13]).
•
Adaptivity by
hp
reﬁnement can be carried out without any complications induced by the domainor function spaces (see [8, 11]).As usual we assume an appropriate Hilbert space
H
(
Ω
)
and the variational form of the PDE writtenwith the help of a continuous bilinear form
a
and a linear form
l
, i.e.
a
:
H
×
H
→
R
and
l
∈
H
′
,
together with appropriate boundary condition. The problem we are concerned with can then be statedas follows:Find
u
∈
H
s.t.
a
(
u
,
v
) =
l
(
v
)
∀
v
∈
H
.
(2.1)The basic method of discretisation in the PUM framework is then given by the following steps:
A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains
3 of 27
•
Given
N
pointsinadomain
Ω
onwhichalinearscalarPDEisdeﬁned, opensetscalled
patches
areassociated with each point to form a cover
Ω
N
of the domain. (
Ω
N
:
=
{
ω
i
}
N i
=
1
, with
Ω
⊂
i
ω
i
).
•
A partition of unity of consistency order 0 and arbitrary smoothness)
{
ϕ
i
}
N i
=
1
subordinate to thecover is constructed.
•
The local function space on patch
ω
i
, 1
i
N
, is given by span
ψ
k i
p
i
k
=
1
, with
{
ψ
k i
}
p
i
k
=
1
beinga set of approximating functions chosen for each patch. The global approximation space, alsocalled the trial or the
PUM space
, is deﬁned by
V
PU
:
=
span
{
ϕ
i
(
x
)
ψ
k i
(
x
)
}
i
,
k
. Replacing
H
by theﬁnite dimensional subspace
V
PU
, a global approximation
u
PU
to the unknown solution
u
of thePDE is deﬁned as a (weighted) sum of local approximation functions on the patches:
u
PU
(
x
) =
N
∑
i
=
1
ϕ
i
(
x
)
ψ
i
(
x
) =
N
∑
i
=
1
ϕ
i
(
x
)
p
i
∑
k
=
1
ξ
k i
ψ
k i
(
x
)
.
In case of Neumann problems, as investigated in this article, the PUM approximation space isconforming, i.e.
V
PU
⊂
H
. Dirichlet problems were treated in, for example, [10].
•
The unknown coefﬁcients
ξ
k i
are determined by substituting the above approximation into thePDE and using the method of weighted residuals to derive an algebraic system of equations
A
ξ
=
b
.
(2.2)Dirichlet boundary conditions are not discussed in this document as they are rarely encounteredin problems arising in cell biology. They are more complicated to deal with than Neumann boundaryconditions in meshfree methods as they have to be imposed (weakly) on boundary patches. This isdue to the fact that the trial functions are not fulﬁlling the so called
Kronecker delta property
. Differentapproaches have been suggested to handle essential boundary conditions. The
Nitsche method
, whichhas been used in the context of mortar ﬁnite elements for many years, and which also is used in somediscontinuous Galerkin methods, is one of the preferred approaches, see [10] for details and [3, 6] for
discussions of other methods.2.2
Partition of Unity Method
Key to the PUM is the notion of a
(
M
,
C
∞
,
C
∇
)
partition of unity. According to Babuˇska and Melenk [18] we deﬁne:
Deﬁnition 2.1 (Partition of Unity)
Let
Ω
⊂
R
d
be an open set, and let
Ω
N
:
=
{
ω
i
}
N i
=
1
be an open coverof
Ω
satisfying a pointwise overlap condition:
∃
M
∈
N
:
∀
x
∈
Ω
card
{
i

x
∈
Ω
i
}
M
.
Moreover let
{
ϕ
i
}
be a Lipschitz partition of unity subordinate to the cover
Ω
N
satisfyingsupp
(
ϕ
i
)
⊂
Ω
N
∀
i
,
∑
i
ϕ
i
≡
1 on
Ω
,
ϕ
i
L
2
(
R
d
)
C
∞
,
∇
ϕ
i
L
∞
(
R
d
)
C
∇
diam
(
ω
i
)
,
4 of 27
M. EIGEL, E. GEORGE, M. KIRKILIONIS
with two positive constants
C
∞
and
C
∇
. Then
{
ϕ
i
}
N i
=
1
is called a
(
M
,
C
∞
,
C
∇
)
partition of unity
subordinate to the cover
{
ω
i
}
. The PU is said to be of degree
m
∈
N
0
if
ϕ
i
∈
C
m
(
R
d
)
for all
i
.There are several ways for constructing a partition of unity satisfying this deﬁnition. We use thewell known
Shepard functions
which are described later in this paper. In order to provide good (global)approximation results, the numerical method has to fulﬁl two main criteria:
Local Approximation:
On each patch
ω
i
a local approximation space
V
i
=
span
{
ψ
k i
}
p
i
k
=
1
of arbitrarydegree
p
i
is deﬁned. Functions in
V
i
have to be able to approximate the solution in
ω
i
accurately.Polynomial bases are most common but may not provide the best possible approximation forspeciﬁc problems, see [2, 18, 3] for further examples.
Global Continuity:
The global approximation space
V
PU
is created by blending together the set of local spaces
{
V
i
}
N i
=
1
by a partition of unity
{
ϕ
i
}
N i
=
1
on
Ω
while keeping the local approximationproperties.
Deﬁnition 2.2 (PUM space)
Let
{
ω
i
}
N i
=
1
be an open cover of
Ω
⊂
R
d
, let
{
ϕ
i
}
N i
=
1
be a
(
M
,
C
∞
,
C
∇
)
partition of unity subordinate to
{
ω
i
}
and let
V
i
⊂
H
1
(
Ω
∩
ω
i
)
be given. Then the space
V
PU
:
=
N
∑
i
=
1
ϕ
i
V
i
=
N
∑
i
=
1
ϕ
i
v

v
∈
V
i
(2.4)
=
span
{
ϕ
i
ψ
k i

ψ
k i
∈
V
i
,
i
=
1
...
N
,
k
=
1
...
p
i
}
is called a
PUM space
of degree
m
∈
N
if
V
PU
⊂
C
m
(
Ω
)
.The PUM space is used as approximation space for the discretisation of the problem equation. Ageneral approximation result is provided by the following theorem.T
HEOREM
2.1 (A
PPROXIMATION
P
ROPERTY
[18]) Let
Ω
⊂
R
d
be given,
{
ω
i
}
,
{
ϕ
i
}
and
{
V
i
}
asin Deﬁnitions 2.2 and 2.1. Let
u
∈
H
be the function to be approximated and assume that the localapproximation spaces
V
i
possesses the following approximation properties: On each patch
Ω
∩
ω
i
thefunction
u
can be approximated by a function
v
i
∈
V
i
such that
u
−
v
i
L
2
(
Ω
∩
ω
i
)
ε
1
,
i
and
∇
(
u
−
v
i
)
L
2
(
Ω
∩
ω
i
)
ε
2
,
i
hold. Then the function
u
PU
:
=
∑
i
ϕ
i
v
i
∈
V
PU
⊂
H
satisﬁes
u
−
u
PU
L
2
(
Ω
)
√
MC
∞
∑
i
ε
21
,
i
1
/
2
,
(2.5a)
∇
(
u
−
u
PU
)
L
2
(
Ω
)
√
2
M
∑
i
C
∇
diam
(
ω
i
)
2
ε
21
,
i
+
C
2
∞
ε
22
,
i
1
/
2
.
(2.5b)For instance, if
u
∈
H
2
(
Ω
)
, assuming a patchwise error bound depending on the patch size
h
i
andlinear polynomial local spaces such that
ε
1
,
i
(
h
i
,
p
i
)
C
i
h
2
i
p
−
1
i
u
H
2
(
Ω
∩
ω
i
)
,
A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains
5 of 27then Theorem 2.1 yields
u
−
u
PU
L
2
(
Ω
)
MC
∞
max
i
{
C
i
h
2
i
p
−
1
i
}
u
H
2
(
Ω
∩
ω
i
)
.
This shows that the method can be used as a
h
,
p
 or
hp
version by adjusting density of nodes orthe consistency order of the local spaces which makes the method very versatile. While this a prioriapproximation result is shown for polynomial spaces, it is obvious from the construction of the localspaces that the approximation can be improved if a base of the solution space of the problem is known,see [18]. Reﬁnement strategies (
h
reﬁnement by decreasing patch sizes or
p
reﬁnement by increasinglocal approximation quality) can be carried out independently on each patch. Moreover, the burden tofulﬁl strict structuring criteria (e.g. avoiding hanging nodes and overlapping meshes) when dealing withreﬁnement and remeshing of triangulations is entirely omitted. The partition of unity ensures smoothness of the global function space while local features of the employed function spaces are preserved.When using an arbitrary cover of patches it can happen that the union of local bases is not linearlyindependent on some patches. While this might not be a major problem as long as only a few functionspaces exhibit linear dependence, the solution process may become more difﬁcult. This complication of obtaining singular operator matrices was reported on in [24], where also an iterative solution algorithmwas suggested based on perturbing the algebraic system. A recent investigation of this topic can befound in [27].By the cover construction algorithm described in section 3 we completely abolish this difﬁculty byensuring linearly independent function spaces. In [11] Schweitzer deﬁned the so called
ﬂat top property
which is a sufﬁcient (but not necessary) condition for this. First, for a cover
Ω
N
:
=
{
ω
i
}
N i
=
1
we deﬁnethe neighbour index sets
n
(
x
)
:
=
{
i

x
∈
ω
i
} ∀
x
∈
Ω
,
n
(
i
)
:
=
{
j

ω
i
∩
ω
j
=
/ 0
}
i
=
1
,...,
N
.
Deﬁnition 2.3 (Flat top property)
Let
Φ
:
=
{
ϕ
i
}
be a partition of unity subordinate to the cover
{
ω
i
}
N i
=
1
of domain
Ω
as deﬁned in 2.2,
µ
the Lebesgue measure in
R
d
. Then
Φ
is said to have the
ﬂat top property
, if
µ
(
ω
i
)
˜
C
µ
(
˜
ω
i
)
,
i
=
1
...
N
(2.6)with a constant ˜
C
∈
R
+
independent of the patch and the subpatch ˜
ω
i
⊂
ω
i
deﬁned as˜
ω
i
:
=
{
x
∈
ω
i

n
(
x
) =
{
i
}}
.
(2.7)R
EMARK
2.1 In a partition of unity exhibiting this property all patches
ω
i
have a subset ˜
ω
i
larger thana nullset in the Lebesgue sense with no other patches overlapping.2.3
Partition of unity construction
We now construct a straightforward 0th order PU due to
Shepard
[21]. First,
weight functions w
i
:
R
d
→
[
0
,
1
]
are deﬁned on each patch. These are used to localise functions by requiring supp
(
w
i
)
⊂
ω
i
.