A multi-mesh finite element method for Lagrange elements of arbitrary degree

Abstract

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid-solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier-Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.

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A MULTI-MESH FINITE ELEMENT METHOD FOR LAGRANGE
ELEMENTS OF ARBITRARY DEGREE
AXEL VOIGT∗AND THOMAS WITKOWSKI†
Abstract. We consider within a finite element approach the usage of different adaptively refined
meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different so-
lution behaviours of these variables, the meshes can be independently adapted. The resulting linear
systems are usually much smaller, when compared to the usage of a single mesh, and the overall
computational runtime can be more than halved in such cases. Our multi-mesh method works for
Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The ap-
proach is well defined, and can be implemented in existing adaptive finite element codes with minimal
effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid-solid
phase-transitions. A further application comes from fluid dynamics where we demonstrate the ap-
plicability of the approach for solving the incompressible Navier-Stokes equations with Lagrange
finite elements of the same order for velocity and pressure. The approach thus provides an easy to
implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same
order are required.
1. Introduction. Nowadays, adaptive h-methods for mesh refinement are a
standard technique in finite element codes. They are used to resolve a mesh due
to the local behaviour of the solution. When solving PDEs with multiple variables,
e.g. velocity and pressure in the Navier-Stokes equations, the mesh has to be adapted
to the behaviour of all components of the solution. If these behaviours are different,
the use of a single mesh may lead to an inefficient numerical method. In this work we
propose a multi-mesh finite element method that makes it possible to resolve the local
nature of different components independently of each other. This method works for
Lagrange elements of arbitrary degree in any dimension. Furthermore, the method
works “on the top” of standard adaptive finite element methods. Hence, only small
changes are required if our method has to be implemented in existing finite element
codes. We have implemented the multi-mesh method in our finite element software
AMDiS (adaptive multidimensional simulations), see [1], for Lagrange finite elements
up to fourth degree for 1D, 2D and 3D.
To our best knowledge, Schmidt [2] was the first who has considered the use of
multiple meshes in this context. Li et al. have introduced a very similar technique
and used it to simulate dendritic growth [3, 4, 5]. Although introducing a multi-mesh
technique, in none of these publications the method is formally derived. Furthermore
implementation issues are not discussed and detailed runtime results, which compare
the overall runtime between the single-mesh and the multi-mesh method are miss-
ing. In contrast, in this work we will formally show how multiple meshes are used
in the context of assembling matrices and vectors in the assembly step of the finite
element methods and will discuss issues related to error estimates for each compo-
nent. Furthermore, we will compare the runtimes of both methods and show that
the multi-mesh method is superior to the single-mesh method, when one component
of the PDE can locally be resolved on a coarser mesh. Solin et al. [6, 7] have also
introduced a multi-mesh method, but for hp-FEM. Their method is based on trans-
forming quadrature points which is harder to implement in existing finite element
codes. Furthermore, also in these works detailed runtime studies are missing. We
∗Institut f¨ ur Wissenschaftliches Rechnen, TU Dresden, Zellescher Weg 12-14, 01062 Dresden,
Germany (axel.voigt@tu-dresden.de)
†Institut f¨ ur Wissenschaftliches Rechnen, TU Dresden, Zellescher Weg 12-14, 01062 Dresden,
Germany (thomas.witkowski@tu-dresden.de)
1
arXiv:1005.4808v1 [math.NA] 26 May 2010

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0
1
2
3
0
(a)(b)(c)
Fig. 2.1. (a) A two-dimensional macro mesh. (b) Some refinements of it. (c) Binary tree of
macro element 0
should further mention other approaches which are commonly used to deal with dif-
ferent meshes for different components of coupled systems. Especially in the case of
multi-physics applications a need exists to couple independent simulations code. A
standard tool which can be used to couple various finite element codes is MpCCI
(mesh-based parallel code coupling interface) [8]. In this approach an interpolation
between the different solutions from one mesh to the other is performed which for
different resolutions of the involved meshes will lead to a loss in information and is
thus not the method of choice for the problems to be discussed in this work.
The paper is structured as follows: In the next section we give a brief overview on
adaptive meshes, especially in the context of AMDiS, and introduce the terminology
used throughout this paper. Section 3 introduces the so called virtual mesh assem-
bling, which is the basis of our multi-mesh method. It is shown, how the coupling
meshes are build in a virtual way and how the corresponding coupling operators are
assembled on them. In Section 4, we present several numerical experiments in 2D and
3D that show the advantages of the multi-mesh method. The last section summarizes
our results.
2. Adaptive meshes. The usage of an adaptive mesh, together with error esti-
mators and a refinement strategy, is a standard finite element technique to compute
solution of PDEs with a given accuracy with the lowest possible computational effort.
For a general overview on this topic see for example [9], and references therein. In
this section, we describe how adaptive meshes and the associated algorithms are im-
plemented in our finite element software AMDiS. The data structures that are used to
store and manipulate adaptive meshes, are the basis for a fast and efficient multi-mesh
method, as it is presented in the next section.
A mesh in AMDiS consist of simplicial elements, which are lines in 1D, triangles
in 2D and tetrahedral in 3D. If an element has to be refined within the adaption
loop, it will be bisected into two elements of the same dimension. The refinement
algorithm, that is implemented in AMDiS, is described in [10] in more detail. The
two new elements are called children of the parent element. The coarsest elements
are called macro elements. Accordingly, the macro mesh is the union of all macro
elements. The refinement history of a macro element is represented by a binary tree.
If a node in this tree is a leaf, the corresponding element is not refined. Otherwise, the
children of the node represent the children of the element. To avoid hanging nodes
(vertices of an element which are not vertices of the neighbour element), it may be
necessary to first refine a neighbour of an element before refining the element itself.
2

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In this way, a single refinement can cause some propagation refinements of elements in
the neighbourhood. In Figure 2.1, a triangular macro mesh consisting of four elements
(a), some refinements of this macro triangulation (b) and the corresponding binary
tree for macro element 0 (c) are shown.
All topological and geometrical information are only stored for the macro ele-
ments. To get these information for other elements in the mesh, an algorithm called
mesh traverse is used. This algorithm traverses all binary trees, corresponding to the
macro elements, and recursively computes the requested information, e.g. coordinates
of vertices, for the requested child elements from the information of its parent element.
The mesh traverse algorithm than calls a function that computes on the element data.
An algorithm, that makes use of the mesh traverse, can specify the element level on
which it wants to process. The level of an element is defined to be the depth of its
node representation in the binary tree. In most cases, the leaf level, i.e., the union of
all leaves in all binary trees, is used for mesh traverse.
The additional effort to compute the element data for the leaf mesh multiple
times is usually less than 1% of the computational time of the function that process
on the mesh. Instead, a large amount of memory can be saved. In AMDiS, the
information data for one element requires around 200 bytes per element. Many of
our simulations make use of around one million elements per processor. Storing all
element information explicitly for these simulations would require more than 200
Mbyte of extra memory.
2.1. Error estimation and adaptive strategies. AMDiS provides different
methods for error estimation and mesh refinement. Furthermore, due to an abstract
interface, is is easily possible to implement other methods, that are more adapted to
a specific PDE. In the following, we describe the most common ones.
In AMDiS, the standard estimator for the spatial error is the residual error esti-
mator as for example described by Verf¨ urth [11] for general non-linear elliptic PDEs.
This local element wise estimator defines for each element T of a mesh an indicator,
depending on the finite element solution uh, by
ηT(uh) = CoRT(uh) + C1
?
E⊂T
JE(uh),(2.1)
where RT is the element residual on element T and JEis the jump residual, defined
on an edges E. C0and C1are user defined constants that can be used to adjust the
estimator. The global error estimation of a finite element solution is than the sum of
the local estimates:
??
where T is a partitions of the domain into simplices. The user has to specify an error
tolerance tol. If η(uh) > tol, the mesh must be refined in some way in order to reduce
the error. Several strategies are implemented in AMDiS:
• the maximum strategy, as described by Verf¨ urth [12],
• the equidistribution strategy, as described by Eriksson and Johnson [13],
• the guaranteed error reduction strategy, as described by D¨ orfler [14].
In all of the numerical experiments in Section 4, we make use of the equidistribution
strategy, which we found out to be easy to use and which provided good results. The
basis for this strategy is the idea that the error is assumed to be equidistributed over
η(uh) =
T∈T
ηT(uh)p
?1/p
,p ≥ 1,(2.2)
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all elements. Hence, if the partition consists of n elements, the element error estimates
should fulfill
ηT(uh) ≈
tol
n1/p≡ ηeq(uh). (2.3)
The equidistribution strategy now introduces two parameters, θR∈ (0,1) and θC∈
(0,1). An element is refined, if ηT(uh) > θRηeq(uh), and the element is coarsen, if
ηT(uh) ≤ θCηeq(uh). The parameter θRis usually chosen to be close to 1, and θCto
be close to 0, respectively.
If the equation consists of more than one variable, one can define multiple error
estimators and mesh adaption strategies. This is also the case, if only one mesh is used
to resolve all variables. The error estimators work independently of each other, and
the user can provide the constants C0and C1for each component. Correspondingly,
independent mesh adaption strategies may be defined for each component. In the case
of the single-mesh method, an element is refined, if at least one strategy has marked
the element to be refined. An element is coarsen, if all strategies have marked it to be
coarsen. In the multi-mesh method, these conditions are omitted, because the meshes
are adapted independently of each other.
3. Virtual mesh assembling. The basis of our multi-mesh method is the so
called virtual mesh assembling. Systems of PDEs usually involve coupling terms. If
each component of the system is assembled on a different mesh, special care has to
be devoted to these coupling terms. In the next section, we shortly describe this
situation. The section 3.2 than introduces the dual mesh traverse. This algorithm
create a virtual union of two meshes without creating it explicitly. To the last, we
show how to compute integrals, that appear within the assemble procedure, on these
virtual meshes.
3.1. Coupling terms in systems of PDEs. To illustrate the techniques pre-
sented in this section, we consider the homogeneous biharmonic equation as a simple
example for general systems of PDEs. This equation reads:
∆2u = 0 in Ω andu =∂u
∂n= 0 on ∂Ω,(3.1)
with u ∈ C4(Ω) ∩ C1(¯Ω). Using operator splitting, the biharmonic equation can be
rewritten as a system of two second order elliptic PDEs:
−∆u + v = 0
∆v = 0
(3.2)
The standard mixed variational formulation of this system is: Find (u,v) ∈ H1
H1(Ω) such that
?
?
To discretize these equations, we assume that T0
of the domain Ω into simplices. Than, V0
and V1
0(Ω) ×
Ω
∇u∇φdx +
?
Ω
vφdx = 0
∀φ ∈ H1(Ω)
Ω
∇v∇ψdx = 0
∀ψ ∈ H1
0(Ω)
(3.3)
hand T1
hare different partitions
h= {vh ∈ H1: vh|T ∈ Pn∀T ∈ T0
h} are finite element spaces of globally
h}
h= {vh ∈ H1
0: vh|T ∈ Pn∀T ∈ T1
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continuous, piecewise polynomial functions of an arbitrary but fixed degree. We thus
obtain: Find (uh,vh) ∈ V0
?
?
Let define {φi| 1 ≤ i ≤ n} and {ψi| 1 ≤ i ≤ m} to be the nodal basis of V0
and V1
h, respectively. Hence, uh and vh can be written by the linear combinations
uh=?n
(3.4) rewrites to

T∈T0
?
h× V1
hsuch that
?
∇vh∇ψdx = 0
Ω
∇uh∇φdx +
Ω
vhφdx = 0
∀φ ∈ V0
h(Ω)
Ω
∀ψ ∈ V1
h(Ω).
(3.4)
h
i=1uiφiand vh=?m
i=1viψi, with uiand vithe unknown real coefficients.
Using these relations and braking up the domain in the partitions of Ω, Equation
n
?
j=1
uj
?
h
?
T
∇φj· ∇φi

+
m
?
m
j=1
vj


?
?
T∈T0
h∪T1
h
?
T
ψjφi


= 0
= 0
i = 1...n
j=1
vj

?
T∈T1
h
T
∇ψj· ∇ψi
i = 1...m.
(3.5)
To compute the coupling term, we have to define the union of two different partitions
T0
is either a subelement of an element T1∈ T1
very strict. It is always fulfilled for the standard bisectioning algorithm, if the initial
meshes for all components share the same macro mesh. Then, T0
of the locally finest simplices.
The most common way to compute the integrals in Equation (3.5) is to define
local basis functions. We define φi,jto be the j-th local basis function on an element
Ti∈ T0
Because the global basis functions φiand ψjare defined on different triangulations
of the same domain, it is not straightforward to calculate the coupling term?
either the integral has to be evaluated on an element from the partition T0
an element from T1
functions we have to evaluate
?
for some j and l, and there exists an element Tk∈ T1
develop a multi-mesh method that works on the top of existing finite element software.
These have usually implemented special methods to evaluate local basis functions, or
the multiplication of two basis functions, respectively, on elements very fast. This
involves for example precalculated integral tables or fast quadrature rules. All these
methods cannot directly be applied to the coupling terms, because ψk,lin Equation
(3.6) is not a local basis function of the element Ti. The general idea to overcome
this problem is to define the basis functions ψk,lby a linear combination of local basis
functions of Ti. Thus,?
5
h∪ T1
h. For this, we make a restriction on the partitions: Any element T0∈ T0
h, or vice versa. This restriction is not
h
h∪ T1
his the union
h. ψi,jis defined in the same way for elements in the partition T1
h.
Ωψjφi
in an efficient manner. For evaluating this integral, two different cases may occur:
hor on
h. For what follows, we fix the first case. In terms of local basis
Ti∈T0
h
ψk,lφi,j
(3.6)
h, with Ti⊂ Tk. Our aim is to
Ti∈T0
h
ψk,lφi,j=
?
Ti∈T0
h
?
m
(ck,mφi,m)φi,j,(3.7)