THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR
LINEAR INCOMPRESSIBLE FLUID FLOW: A REVIEW
BERNARDO COCKBURN, GUIDO KANSCHAT, AND DOMINIK SCHÖTZAU
Abstract. In this paper, we review the development of the so-called local
discontinuous Galerkin method for linear incompressible fluid flow. This is
a stable, high-order accurate and locally conservative finite element method
whose approximate solution is discontinuous across inter-element boundaries;
this property renders the method ideally suited for hp-adaptivity. In the context of the Oseen problem, we present the method and discuss its stability and
convergence properties. We also display numerical experiments that show that
the method behaves well for a wide range of Reynolds numbers.
Computer and Fluids, Vol. 34, 2005, pp. 491–506
1. Introduction
In this paper, we review the development of the local discontinuous Galerkin
(LDG) method for linear incompressible fluid flow. To do that, we consider the
Oseen equations, namely,
(1)
−ν∆u + (β · ∇)u + γ u + ∇p = f
in Ω,
∇·u=0
in Ω,
u=g
on Γ,
where u is the velocity, p the pressure, f ∈ L2 (Ω)d a prescribed external body
force, ν the kinematic viscosity, β a convective velocity field and γ a given scalar
function. As usual, we take Ω to be a bounded domain of Rd , d = 2, 3, with
boundary RΓ, and the Dirichlet datum g ∈ H 1/2 (Γ)d to satisfy the compatibility
condition Γ g · n ds = 0, where n denotes the outward normal unit vector to Γ.
There are three properties that render the LDG method suitable for application
to incompressible fluid flow problems. The first is that the method is a stable,
high-order accurate and locally conservative method, even in convection-dominated
regimes. The second is that the LDG method can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders;
this renders it ideal for use with hp-adaptive algorithms. Finally, since LDG methods are stabilized mixed methods, there is a significant flexibility in the choice of
polynomial spaces for the velocity and the pressure. For example, it is possible to
2000 Mathematics Subject Classification. 65N30.
Key words and phrases. Finite elements, discontinuous Galerkin methods, Oseen equations.
The first author was supported in part by the National Science Foundation (Grant DMS0107609) and by the University of Minnesota Supercomputing Institute.
c
1997
American Mathematical Society
1
2
B. Cockburn, G. Kanschat and D. Schötzau
use polynomials of degree k for all the components of the velocities and for the
pressure.
To give a basic background about the LDG method for incompressible flows, let
us put it in historical perspective. A fairly complete account of the development of
discontinuous Galerkin (DG) methods can be found in [16]. Here, we are going to
restrict ourselves to what is new or relevant to the topic under consideration.
The LDG method is an extension of the so-called Runge-Kutta discontinuous
Galerkin (RKDG) method for non-linear hyperbolic systems developed in the 90’s
in [12, 17, 18, 19, 21]; see the short introductory monograph [9] and the recent
review [22]. The RKDG method was extended to the Navier-Stokes equations of
compressible fluid flow in [3] and then to general convection-diffusion systems in
[20], thus giving rise to the LDG method; see also, [10].
The LDG method for purely elliptic problems was studied in [7], where general
triangulations were considered, in [13], where super-convergence on Cartesian grids
was proven, and in [33], where its hp-version was analyzed. How to couple the LDG
method with the standard conforming finite element method was shown in [32] and
how to couple it with the mixed method of Raviart-Thomas in [11].
In [7], it was shown that the LDG method is in fact a stabilized mixed method
whose stabilization is achieved by the jumps of the approximate solution across the
boundaries of the elements. It was also shown that, as opposed to classical or stabilized mixed methods, the flux variable can be eliminated element-by-element from
the equations by taking advantage of the discontinuous nature of the approximation.
This allows the LDG method to be compared with the standard conforming method,
the so-called interior penalty methods (developed in the late 70’s and early 80’s),
and other recently introduced DG methods. In [1], a unifying theoretical framework
to study virtually all the known DG methods for elliptic problems was proposed; it
was complemented later by a study of the condition number of the stiffness matrix
and a computational comparison of the main DG methods in [6]. First multi-level
preconditioners for DG methods for elliptic and convection-diffusion problems were
devised, analyzed and numerically tested in [24, 25, 26, 28].
Given all the above described work on DG methods for second-order elliptic
equations, the main difficulty for an LDG discretization of the Oseen problem was
how to deal with the incompressibility condition on the velocity. This problem
was solved in [15] for the Stokes equations. Later, a unified framework for the
analysis of general DG methods for the Stokes system was proposed in [35] and
then applied to obtain slightly sub-optimal error bounds for the p-version of the DG
methods studied therein. More recently, this study was further refined to obtain
the exponential convergence of the hp-version of the LDG method for the case
of polygonal domains in [36]. In [14], the LDG method for the Oseen equations
was introduced, analyzed and numerically tested. It was shown that if we take
polynomials of degree k for each component of the velocity u and polynomials of
degree k − 1 for the pressure p, the L2 -norm of the error in the velocity is of order
k + 1, and the L2 -norms of the errors in p and in the velocity gradient σ = ν∇u are
of order k. These optimal orders of convergence remain invariant if the pressure p
is approximated by polynomials of degree k. The numerical tests in [14] showed
that the method is very robust for a wide range of Reynolds numbers. The content
of this paper is based on the material of the papers mentioned in this paragraph.
The LDG method for incompressible flows
3
To give the reader a better idea of the LDG method, let us compare it with
other DG methods for incompressible fluid flow. The first of those methods was
proposed in [2], where the incompressibility condition was enforced pointwise inside
each element. To discretize the remaining equations, the above mentioned interior
penalty method was used. In [29], this method was extended to the incompressible
Navier-Stokes equations. It uses continuous approximations for the pressure and
different meshes for the pressure and the velocity. Unlike this method, the LDG
method uses discontinuous pressures and the same mesh for both the pressure and
the velocity.
Another DG method is the one introduced in [4] for the Euler and the incompressible Navier-Stokes equations. This method, applied to a purely diffusive problem,
produces a stiffness matrix with an increased sparsity than the stiffness matrix of
the LDG method. On the other hand, unlike the LDG method, that method is unstable for polynomials of degree 1, produces a stiffness matrix that is not symmetric,
even for self-adjoint elliptic problems, and gives sub-optimal orders of convergence
in the L2 -norm.
Perhaps the closest methods to the LDG method under consideration are the
mixed DG method proposed in [27] for linear elasticity and Stokes flow and its
non-symmetric variant discussed in [37]; both methods use completely discontinuous velocity and pressure spaces. The forms for the incompressibility constraint are
identical to the one in the LDG method, whereas the forms related to the Laplacian are again based on interior penalty approaches. Note, however, that for the
(symmetric) interior penalty approach the stability of the method is achieved only
when the stabilization parameter is properly chosen. In contrast, the LDG method
is always stable regardless of the size of their penalization parameter.
The paper is organized as follows. In section 2, we introduce the LDG method by
using the typical element-by-element flux formulation and then rewrite the method
in compact form by using the classical mixed setting. We then discuss several key
properties of the method, namely, its local conservativity, its ability to eliminate
the auxiliary variable element-by-element, the stabilizing role of the jumps across
the inter-element boundaries, and the relation between the jumps and the local
residuals. In section 3, we briefly state the theoretical convergence results for the
LDG method. In section 4, we present a couple numerical results showing that the
method behaves well for a large range of values of the Reynolds number. Finally,
in section 5, we end with some concluding remarks.
2. The local discontinuous Galerkin method
In this section, we introduce the LDG discretization of the Oseen equations (1);
we follow [14].
P
We use the standard notation (∇u)ij = ∂j ui and (∇ · σ)i = dj=1 ∂j σij . We
also denote by u ⊗ v the matrix whose ij-th component is ui vj and write
σ : τ :=
d
X
i,j=1
σij τij ,
v · σ · n :=
d
X
i,j=1
vi σij nj = σ : (v ⊗ n).
4
B. Cockburn, G. Kanschat and D. Schötzau
Then, we introduce the “velocity gradient” σ = ν∇u and rewrite the Oseen problem
as the following system of first order equations:
σ = ν∇u
in Ω,
−∇ · σ + (β · ∇)u + γ u + ∇p = f
∇·u=0
in Ω,
in Ω,
u=g
on Γ.
2.1. The flux formulation of the LDG method. Multiplying the above equations by smooth test functions τ , v, and q, respectively, and integrating by parts
over an arbitrary subset K ⊂ Ω with outward normal unit vector nK , we obtain
Z
Z
Z
(2)
u · ∇ · τ dx + ν
u · τ · nK ds,
σ : τ dx = −ν
K
∂K
K
Z
Z
(3)
σ : ∇v − p ∇ · v dx −
σ : (v ⊗ nK ) − p v · nK ds
K
∂K
Z
Z
Z
f · v dx,
β · nK u · v ds =
γ u · v − u · ∇ · (v ⊗ β) dx +
+
K
∂K
Z
ZK
(4)
u · nK q ds = 0.
u · ∇q dx +
−
K
∂K
Note that the above equations are well defined for functions (σ, u, p) and (τ , v, q)
in Σ × V × Q where
: σij K ∈ H 1 (K), ∀K ∈ T , 1 ≤ i, j ≤ d},
V:={v ∈ L2 (Ω)d : vi K ∈ H 1 (K), ∀K ∈ T , 1 ≤ i ≤ d},
Z
Q:={q ∈ L2 (Ω) :
q dx = 0, q K ∈ H 1 (K), ∀K ∈ T },
Σ :={σ ∈ L2 (Ω)d
2
Ω
with T being a triangulation of Ω into elements {K}.
We take the LDG approximation to the exact solution (σ, u, p), (σ h , uh , ph ), in
the finite element space Σh × Vh × Qh ⊂ Σ × V × Q, where
: σij K ∈ S(K), ∀K ∈ T , 1 ≤ i, j ≤ d},
Vh :={v ∈ L2 (Ω)d : vi K ∈ V(K), ∀K ∈ T , 1 ≤ i ≤ d},
Z
2
q dx = 0, q K ∈ Q(K), ∀K ∈ T }.
Qh :={q ∈ L (Ω) :
Σh :={σ ∈ L2 (Ω)d
2
Ω
For the sake of simplicity of the presentation, we take the following choice of local
spaces:
S(K) = Q(K) = P ` (K),
V(K) = P k (K),
where P k (K) is the set of polynomials of degree at most k defined on K. Moreover,
we consider only the cases where k ≥ 1 and
`=k
(“equal-order” LDG),
or
` = k − 1 (“mixed-order” LDG).
The LDG method for incompressible flows
5
The approximate solution is then defined by requesting that for each K ∈ T ,
Z
Z
Z
b σh · τ · nK ds,
σ h : τ dx = −ν
(5)
uh · ∇ · τ dx + ν
u
K
∂K
ZK
Z
(6)
σ h : ∇v − ph ∇ · v dx −
σ
b h : (v ⊗ nK ) − pbh v · nK ds
K
∂K
Z
Z
Z
c
b h · v ds =
+
γ uh · v − uh · ∇ · (v ⊗ β) dx +
β · nK u
f · v dx,
∂K
K
ZK
Z
b ph · nK q ds = 0,
uh · ∇q dx +
(7) −
u
K
∂K
2
for all test functions (τ , v, q) ∈ S(K)d × V(K)d × Q(K).
Note how each of the above equations are enforced locally, that is, element by
b σh , σ
element, thanks to the appearance of the so called numerical fluxes u
bh , pbh ,
p
c
b h and u
b h . These fluxes are nothing but discrete approximations to traces on the
u
boundary of the elements. They couple the degrees of freedom between elements
and must be carefully defined since they dramatically influence the stability and
accuracy of the method.
2.2. General properties of the numerical fluxes. To properly describe the
numerical fluxes, we need to introduce some notation associated with traces. Let
K + and K − be two adjacent elements of T ; let x be an arbitrary point of the set
e = ∂K + ∩ ∂K − , which is assumed to have a non-zero (d − 1)-dimensional measure,
and let n+ and n− be the corresponding outward unit normal vectors at that point.
Let (σ, u, p) be a smooth function inside each element K ± , and let us denote by
(σ ± , u± , p± ) the traces of (σ, u, p) on e from the interior of K ± . Then, we define
the mean values {{·}} and jumps [[·]] at x ∈ e as
{{p}} := (p+ + p− )/2,
[[[[[p]]]] := p+ n+ + p− n− ,
{{u}} := (u+ + u− )/2,
[[u]] := u+ ⊗ n+ + u− ⊗ n− ,
{{σ}} := (σ + + σ − )/2,
[[[[[σ]]]] := σ + n+ + σ − n− .
The numerical fluxes that most DG methods are based on are linear combinations of traces on both sides of the set e. We say that the numerical flux is
consistent if it coincides with the variable it approximates when all the functions
are continuous. We say that the numerical flux is conservative when its definition
on e is independent of the order of the elements K + and K − . Methods with this
type of numerical fluxes are called locally conservative and are highly appreciated
by the community of CFD practitioners. A thorough discussion of these concepts
in the framework of DG methods for purely elliptic problems can be found in [1].
2.3. The jumps and the local residuals. Before we define the numerical fluxes,
let us point out an important property of the DG methods, namely, that the local
residuals and the jumps are strongly related.
On each element K ∈ Th , we define the local residual of each of the Oseen
equations as follows:
R = σ h − ν∇uh ,
R = −∇ · σ h + (β · ∇) uh + γ uh + ∇ph − f ,
R = −∇ · uh .
6
B. Cockburn, G. Kanschat and D. Schötzau
In terms of these quantities, the equations defining the approximate solution become
Z
Z
R : τ dx =
J K : τ ds,
Z∂K
ZK
JK · v ds,
R · v dx =
ZK
Z∂K
R q dx =
JK q ds,
K
for all (τ , v, q) ∈ S(K)
d2
∂K
d
× V(K) × Q(K),
J K = ν (b
uσh − uh ) ⊗ nK ,
JK = (b
σ h − σ h ) nK − (b
ph − ph ) nK − β · nK (b
uch − uh ) ,
JK = (b
uph − uh ) · nK .
Note that this implies that the DG method under consideration actually forces
the local residuals R, R and R to be L2 -orthogonal to all the functions in the
2
space S(K)d × V(K)d × Q(K) that vanish on ∂K. We thus see that the projection
of the local residuals into the L2 -orthogonal complement of that space is a linear
functional of the functions J K , JK and JK . Note that these functions are linear
combinations of the jumps of the approximate solution if we assume that all the
numerical fluxes are consistent. Indeed, in such a case, when all the jumps of the
approximation are zero we see that the functions J K , JK and JK are equal to zero.
In summary, the L2 -projections of the local residuals are liftings of the jumps into
2
the L2 -orthogonal complement of functions in S(K)d × V(K)d × Q(K) that vanish
on ∂K.
2.4. The numerical fluxes of LDG method. Next, we give the numerical fluxes
that define the LDG method for the Oseen equations; cf. [15, 14].
b c in (6), we take the stanThe convective numerical flux. For the convective flux u
dard upwind flux introduced in [31, 34], namely,
(8)
b c (x) = lim u (x − β(x)) .
u
↓0
The diffusive numerical fluxes. If a face e lies inside the domain Ω, we take
(9)
σ
b = {{σ}} − C11 [[u]] − [[[[[σ]]]] ⊗ C12 ,
and, if e lies on the boundary, we take
(10)
σ
b = σ − C11 (u − g) ⊗ n,
b σ = {{u}} + [[u]] · C12 ,
u
b σ = g.
u
As will be shown later, the role of the parameter C11 is to enhance the stability
of the method; see also [7]. A proper choice of the parameter C12 can sometimes
enhance the accuracy of the method; see [13].
The numerical fluxes we just displayed are direct generalizations of the numerical
fluxes for the LDG method for the second-order elliptic problems; see [20], [7] and
[13]. The diffusive numerical fluxes corresponding to other DG methods can be
found in [1].
The LDG method for incompressible flows
7
The numerical fluxes related to the incompressibility constraint. The numerical
b p and pb, are defined by
fluxes associated with the incompressibility constraint, u
using an analogous recipe. If the face e lies on the interior of Ω, we take
b p = {{u}} + D11[[[[[p]]]] + D12 tr [[u]],
u
(11)
pb = {{p}} − D12 · [[[[[p]]]]],
where D11 , D12 depend on x ∈ e. Here, tr [[u]] denotes the trace of [[u]]. Again, the
parameter D11 is a stabilizing parameter and the parameter D12 could be used to
enhance the accuracy of the method.
On the boundary, we set
b p = g,
u
(12)
pb = p+ .
This completes the definition of the LDG method.
2.5. The LDG method as a stabilized mixed method. Next, we recast the
LDG method in a classical mixed setting in order to show that it is a stabilized
mixed method; we also show how the parameters C11 and D11 are related to the
stabilization of the method. More precisely, after eliminating the auxiliary variable σ h , we show that the approximation (uh , ph ) ∈ Vh × Qh given by the LDG
method satisfies
(13)
Ah (uh , v) + Oh (uh , v) + Bh (v, ph ) = Fh (v) + Gh (v),
(14)
−Bh (uh , q)
+ Ch (ph , q) = Hh (q),
for all (v, q) ∈ Vh ×Qh . Here, the forms Ah , Oh and Bh are forms that discretize the
Laplacian, the convective term and the incompressibility constraint, respectively.
The form Ch is a pressure stabilization form. The corresponding right-hand sides
are given by linear forms Fh , Gh and Hh .
For simplicity, we take, from now on:
C11 = κ > 0,
C12 = 0,
D11 = δ > 0,
D12 = 0.
Using the first equation (5) to eliminate σ h . To eliminate the auxiliary variable σ h ,
we introduce the lifting operator L : Vh → Σh and the datum G ∈ Σh by
Z
Z
Z
Z
L(v) : τ dx =
[[v]] : {{τ }} ds
G : τ dx =
(g ⊗ n) : τ ds
Ω
E
Ω
ED
for all τ ∈ Σh . Here, E denotes the union of all edges (d = 2) or faces (d = 3) of
elements in T , and ED the union of all boundary edges or faces.
With this notation, it is not difficult to see that the equation defining σ h in terms
of uh , equation (5) can be rewritten as
(15)
σ h = ν ∇h uh − L(uh ) + G ,
with ∇h denoting the elementwise gradient. Note that, σ h can be computed from
uh in an element-by-element fashion. Using this identity, it is easy to eliminate σ h
from the equations.
8
B. Cockburn, G. Kanschat and D. Schötzau
Rewriting the second equation (6). If we insert the expression of σ h and the definitions of the numerical fluxes into the equation (6), we readily get
Ah (uh , v) + Oh (uh , v) + Bh (v, ph ) = Fh (v) + Gh (v),
where
Z
Z
ν ∇h u − L(u) : ∇h v − L(v) dx + ν
κ [[u]] : [[v]] ds,
Ω
E
Z
Z
X
X
b c · v ds,
β · nK u
γ u · v − u · ∇ · (v ⊗ β) dx +
Oh (u, v) :=
Ah (u, v) :=
K
K∈Th
Bh (u, p) := −
and
Z
p ∇h · u dx +
Ω
Fh (v) :=
Z
f · v dx − ν
Ω
Gh (v) := −
Z
Z
Z
K
∂K\Γ−
{{p}}[[u]] ds,
E
G : ∇h v − L(v) dx + ν
Ω
β · n g · v ds.
Z
κ g · v ds,
ED
Γ−
Of course, Γ− = {x ∈ Γ : β(x) · n < 0} denotes the inflow part of the boundary Γ.
The reader might want to see [1] for details concerning the elimination of the
variable σ h in the equation (6). For the remaining computations, see [14].
We note that exactly the same form Bh is also used in the mixed DG approaches
of [27, 37, 35].
Rewriting the third equation (7). It is now a simple exercise to see that the equation
(7) can be rewritten as
−Bh (uh , q) + Ch (ph , q) = Hh (q),
where
Ch (p, q) :=
and
Z
Hh (q) := −
δ [[[[[p]]]] · [[[[[q]]]] ds,
EI
Z
q g · n ds.
ED
Here, we write EI to denote the union of all interior edges or faces of elements in T .
2.6. Stabilization mechanisms. Here, we discuss the crucial properties of each
of the forms of the equations (13) and (14). They are all related with the fact that
the jumps of the approximate solution help to enhance the stability properties of
the numerical method.
The stabilization properties of the forms Ah , Oh and Ch . Note that, if in the equations (13) and (14), we take (v, q) = (uh , ph ) and add the equations, we get
Ah (uh , uh ) + Oh (uh , uh ) + Ch (ph , ph ) = Fh (uh ) + Gh (uh ) + Hh (ph ).
It is thus clear that the stability of the scheme strongly depends on the stability
properties of the bilinear forms of the left-hand side. Let us show how for each of
these forms, there is a stabilization term associated with the jumps.
Let us begin with the form Ah . Obviously,
Z
Z
2
Ah (v, v) =
ν ∇h v − L(v) dx + ν
κ |[[v]]|2 ds.
Ω
E
The LDG method for incompressible flows
9
We immediately see that if the parameter κ is positive, the above quantity is also
positive. In this case, the second term of the right-hand side clearly helps to enhance
the stability of the method. This extra stabilization is reflected in the ellipticity of
the bilinear form Ah we describe next.
If we introduce the norm (in the space Vh )
Z
X
2
2
kvk1,h = ν
|v|1,K + ν
κ |[[v]]|2 ds,
E
K∈Th
and consider the edgewise (or facewise) meshsize function h ∈ L∞ (E) given by
(
1
(h + hK 0 ) x in the interior of ∂K ∩ ∂K 0 ,
(16)
h(x) := 2 K
hK
x in the interior of ∂K ∩ ∂Ω,
with hK denoting the diameter of element K ∈ T . As usual, we also define the
meshsize by h = maxK∈T hK . We have the following result.
Proposition 2.1. Let κ be given in the form
κ = κ0 h−1 ,
with the local meshsize function h in (16) and a parameter κ0 independent of the
meshsize. Then, for any κ0 > 0, there exists a constant α > 0 independent of the
meshsize such that
Ah (v, v) ≥ αkvk21,h
for all v ∈ Vh .
For a proof, we refer to [1] or [33]. An equivalent coercivity result involving also
the discrete velocity gradients was used in [15] and [14]. For the similar symmetric
interior penalty forms Ah used in the DG approach of [27] the parameter κ0 has to
be chosen large enough.
Now, let us consider the form Oh . It is well known that we have
Z
Z
1
1
1
|β ·n| |[[v]]|2 ds+
|β ·n||v|2 ds,
v ∈ Vh ,
(17) Oh (v, v) ≥ kγ02 vk20,Ω +
2 EI
2 Γ
provided that
1
γ(x) − ∇ · β(x) =: γ0 (x) ≥ 0,
x ∈ Ω.
2
We see once more the stabilizing effect of the jumps. This time, the stabilization
is due to the convective velocity β and appears thanks to the use of the upwinding
numerical flux.
Finally, since we have
Z
Ch (p, p) :=
δ |[[[[[p]]]]]|2 ds,
EI
we see, again, that the jumps (this time in the pressure) have also a stabilizing
effect, provided, of course, that the parameter δ is positive.
The generalized inf − sup condition. It is well known that stabilized mixed methods,
like the LDG method, can circumvent the so-called inf − sup condition. This is
captured in the fact that the classical inf − sup condition can be replaced by a
weaker condition that takes into account the stabilizing effects of the bilinear form
Ch (·, ·). Next, we state such a result.
10
B. Cockburn, G. Kanschat and D. Schötzau
Proposition 2.2. (Generalized inf − sup condition) Let δ be given in the form
δ = δ0 h,
with the local meshsize function h in (16) and a parameter δ0 independent of the
meshsize. Then, for any δ0 > 0, there exist constants γ1 > 0 and γ2 > 0 independent of the meshsize such that
sup
06=v∈Vh
1
Bh (v, q)
≥ γ1 kqk0,Ω − γ2 Ch (q, q) 2 ,
kvk1,h
∀q ∈ Qh \ {0}.
This results holds true for equal- and mixed-order elements.
The proof of this estimate can be found in [15, Section 3.4], although there the
result was not stated in this form. For mixed-order elements, the estimate above
can be proved with a constant γ2 = 0, giving rise to a standard inf − sup condition;
see [27]. A more refined analysis was then given in [37] and [35] on quadrilateral and
hexahedral meshes, covering also hanging nodes and extensions to the hp-version.
Thus, the pressure stabilization form Ch is not necessary for mixed-order elements
where ` = k − 1.
3. The theoretical results
In this section, we present the main error estimates for the LDG method.
We begin by stating our assumptions on the exact solution, the meshes, the local
finite element spaces and the parameters in the definition of the numerical fluxes.
In what follows, we assume again that
(18)
1
γ(x) − ∇ · β(x) =: γ0 (x) ≥ 0,
2
x ∈ Ω,
as this condition guarantees the existence and uniqueness of a solution (u, p) ∈
Hg1 (Ω)d × L20 (Ω) where, as usual,
Z
1
d
1
d
2
2
Hg (Ω) := {u ∈ H (Ω) : u|Γ = g},
L0 (Ω) := {p ∈ L (Ω) :
p dx = 0};
Ω
see, for example, [5, 23]. We also take β and γ such that
(19)
β ∈ L∞ (Ω)d ,
γ ∈ L∞ (Ω),
γ − ∇ · β ∈ L∞ (Ω),
and assume the following standard smoothness properties for the exact solution
(20)
u ∈ H s+1 (Ω)d ,
p ∈ H s (Ω),
with integer s ≥ 1. We are going to use the following norm,
(21)
1
1
||| (u, p) |||s = ν 2 kuks+1,Ω + ν − 2 kpks,Ω ,
for integer s.
The error estimates we present next are stated in terms of the constant ζ in the
continuous inf-sup condition for the divergence operator [5, 23]:
R
Ω q ∇ · v dx
≥ ζ = ζ(Ω) > 0,
(22)
inf
sup
2
q∈L0 (Ω) v∈H 1 (Ω)d kqk0,Ω kvk1,Ω
0
The LDG method for incompressible flows
11
and in terms of the following two other dimensionless parameters:
h k β kL∞ (Ω)d h2 k γ − ∇ · β kL∞(Ω)
(23)
µh = max
,
,
ν
ν
h CPoinc k γ − ∇ · β kL∞ (Ω)
Mh =
(24)
,
ν
where CPoinc is the Poincaré constant that we use here to dimensionally balance
the term Mh . The number ν −1 hk β kL∞ (Ω)d is the cell Peclet number.
We assume that the triangulations T can have hanging nodes and elements of
different shapes that are affinely equivalent to one of several reference elements in
an arbitrary but fixed set, see [8, Section 2.3]. The triangulations are also shaperegular , and have elements with an arbitrary but fixed number of neighbors.
Finally, for technical reasons, we assume that the boundary of the domain Ω is
of class C `+2 for some ` ≥ 0; see [14].
We have the following result.
Theorem 3.1. Under the assumptions of this section, we have that the error
(eu , ep ) between the exact solution (u, p) and the LDG approximation (uh , ph ) satisfies the following bounds:
1
1
ν 2 keu k0,Ω + [ ζ hmin{1,k} ] ν − 2 kep k0,Ω ≤ C hmin{1,k}+min{k,s} ||| (u, p) |||s ,
where the dimensionless constant C is a continuous functions of µh , Mh , k and the
mesh regularity constants.
We point out that the estimates above are valid (and sharp) for both mixedand equal-order elements, but, from the approximation point of view, are slightly
suboptimal for the latter pairing. However, the numerical tests in [15] indicate that
in practice the use of equal-order elements is not less efficient than mixed-order
elements.
A more complete set of theoretical results can be found in [14].
4. Numerical results
Numerical results reported in [15] and [14] show that the theoretical orders of
convergence are actually realized in practice. In this section, we briefly discuss
two numerical experiments for the LDG method. The first is the classical Stokes
flow (β = 0 and γ = 0) over a backward facing step. In the second experiment,
we assess the performance of the LDG method as the Reynolds number varies.
To this end, we consider the so-called Kovasznay solution, [30], which for a given
Reynolds number Re is a two-dimensional analytical solution of the incompressible
Navier-Stokes equations.
Stokes flow. Figure 1 shows the approximation produced by the LDG method over a
backward facing step using equal-order elements with tensor product polynomials of
order one. In the velocity arrows, the length is proportional to the absolute velocity;
the lines are pressure iso-lines. Boundary conditions are as follows: parabolic inflow
profile at the left end of the channel, natural boundary conditions ν∇u · n − pn = 0
at the right end, and no-slip conditions at the walls. We note that incorporation
of natural boundary conditions is straightforward in the LDG framework. It can
be seen that the LDG method produces a reasonable approximation. For detailed
convergence results, the reader is again referred to [15].
12
B. Cockburn, G. Kanschat and D. Schötzau
Figure 1. Stokes flow over a backward facing step.
Kovasznay flow. We solve the Oseen equations with β = u, where u is the velocity
given by Kovasznay’s exact solution. Accordingly, we take the Dirichlet boundary
condition g = u. In our case, we have γ = 0. The vector plot on the left-hand side
of Figure 2 shows that this flow is not trivial.
Figure 2. Kovasznay velocity field for Re = 10.
Quadrilateral meshes generated by consecutive refinement of the original computational square were used. In each refinement, each grid cell is divided into four
similar cells by connecting the edge midpoints. Thus, grid level L corresponds to a
mesh width hL = 21−L .
For bi-quadratic elements, we display in Figure 3 the dependence of the L2 -norms
of the errors eu and ep on the Reynolds number. There, the norms are scaled with
the appropriate powers of ν taken from the estimates in Theorem 3.1. Note how
robust is the behavior of the LDG method when the Reynolds number varies from
1 to 1000.
5. Concluding remarks
In this paper, we have reviewed the development of the LDG method for linear
incompressible flows. The method works very well and has been shown to be
The LDG method for incompressible flows
13
1e+01
1e+00
ν1/2||eu||L2
1e-01
1e-02
1e-03
Re = 1
Re = 2
Re = 5
Re = 10
Re = 20
Re = 50
Re = 100
Re = 200
Re = 500
Re = 1000
1e-04
1e-05
1e-06
1e-07
2
3
4
5
6
7
6
7
Refinement level
1e+02
1e+01
1e+00
ν-1/2||ep||L2
1e-01
1e-02
Re = 1
Re = 2
Re = 5
Re = 10
Re = 20
Re = 50
Re = 100
Re = 200
Re = 500
Re = 1000
1e-03
1e-04
1e-05
1e-06
1e-07
2
3
4
5
Refinement level
Figure 3. Scaled L2 -norms of the errors eu and ep with k = 2 for
different Reynolds numbers for the Kovasznay flow.
robust with respect to the Reynolds number. Extensions of the LDG method to
the incompressible Navier-Stokes equations are under way.
References
[1] D.N. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Unified analysis of discontinuous
Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001), 1749–1779.
[2] G.A. Baker, W.N. Jureidini, and O.A. Karakashian, Piecewise solenoidal vector fields and
the Stokes problem, SIAM J. Numer. Anal. 27 (1990), 1466–1485.
[3] F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the
numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys. 131 (1997),
267–279.
[4] C.E. Baumann and J.T. Oden, A discontinuous hp-finite element method for the Euler and
the Navier-Stokes equations, Int. J. Numer. Meth. Fluids (Special Issue: Tenth International
Conference on Finite Elements in Fluids, Tucson, Arizona ) 31 (1999), 79–95.
[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Verlag, 1991.
14
B. Cockburn, G. Kanschat and D. Schötzau
[6] P. Castillo, Performance of discontinuous Galerkin methods for elliptic PDE’s, SIAM J.
Math. Anal. 24 (2002), 524–5547.
[7] P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the
local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000),
1676–1706.
[8] P. Ciarlet, The finite element method for elliptic problems, North-Holland, Armsterdam,
1978.
[9] B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, HighOrder Methods for Computational Physics (T. Barth and H. Deconink, eds.), Lecture Notes
in Computational Science and Engineering, vol. 9, Springer Verlag, 1999, pp. 69–224.
[10] B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method
for convection-diffusion equations in multidimensions, The Proceedings of the Conference on
the Mathematics of Finite Elements and Applications: MAFELAP X (J.R. Whiteman, ed.),
Elsevier, 2000, pp. 225–238.
[11]
, Approximation of the velocity by coupling discontinuous Galerkin and mixed finite
element methods for flow problems, Computational Geosciences (Special issue: Locally Conservative Numerical Methods for Flow in Porous Media) 6 (2002), 502–522.
[12] B. Cockburn, S. Hou, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous
Galerkin finite element method for conservation laws IV: The multidimensional case, Math.
Comp. 54 (1990), 545–581.
[13] B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal.
39 (2001), 264–285.
[14] B. Cockburn, G. Kanschat, and D. Schötzau, Local discontinuous Galerkin methods for the
Oseen equations, Math. Comp., to appear.
[15] B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002), no. 1, 319–343.
[16] B. Cockburn, G.E. Karniadakis, and C.-W. Shu, The development of discontinuous Galerkin
methods, Discontinuous Galerkin Methods. Theory, Computation and Applications (B. Cockburn, G.E. Karniadakis, and C.-W. Shu, eds.), Lecture Notes in Computational Science and
Engineering, vol. 11, Springer Verlag, February 2000, pp. 3–50.
[17] B. Cockburn, S.Y. Lin, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous
Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys. 84 (1989), 90–113.
[18] B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite
element method for scalar conservation laws II: General framework, Math. Comp. 52 (1989),
411–435.
, The Runge-Kutta local projection P 1 -discontinuous Galerkin method for scalar con[19]
servation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), 337–361.
[20]
, The local discontinuous Galerkin method for time-dependent convection-diffusion
systems, SIAM J. Numer. Anal. 35 (1998), 2440–2463.
[21]
, The Runge-Kutta discontinuous Galerkin finite element method for conservation
laws V: Multidimensional systems, J. Comput. Phys. 141 (1998), 199–224.
, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems,
[22]
J. Sci. Comput. 16 (2001), 173–261.
[23] V. Girault and P.A. Raviart, Finite element approximations of the Navier-Stokes equations,
Springer-Verlag, New York, 1986.
[24] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer.
Math., to appear.
[25] J. Gopalakrishnan and G. Kanschat, Multi-level preconditioners for the interior penalty
method, 2002, to be published in the proceedings of ENUMATH 01.
[26] J. Gopalakrishnan and G. Kanschat, Application of unified analysis to preconditioning discontinuous Galerkin methods, Proceedings of the Second M.I.T. Conference on Computational
Fluid and Solid Mechanics, June 17 - 20, 2003, to appear.
[27] P. Hansbo and M.G. Larson, Discontinuous finite element methods for incompressible and
nearly incompressible elasticity by use of Nitsche’s method, Comput. Methods Appl. Mech.
Engrg. 191 (2002), 1895–1908.
The LDG method for incompressible flows
15
[28] G. Kanschat, Preconditioning methods for local discontinuous Galerkin discretizations, SIAM
Sci. Comp., to appear.
[29] O. Karakashian and W.N. Jureidini, A nonconfroming finite element method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 35 (1998), 93–120.
[30] L. I. G. Kovasznay, Laminar flow behind a two-dimensional grid, Proc. Camb. Philos. Soc.
44 (1948), 58–62.
[31] P. LeSaint and P.A. Raviart, On a finite element method for solving the neutron transport
equation, Mathematical aspects of finite elements in partial differential equations (C. de Boor,
ed.), Academic Press, 1974, pp. 89–145.
[32] I. Perugia and D. Schötzau, On the coupling of local discontinuous Galerkin and conforming
finite element methods, J. Sci. Comput. 16 (2001), 411–433 (2002).
[33]
, An hp-analysis of the local discontinuous Galerkin method for diffusion problems, J.
Sci. Comput. (Special Issue: Proceedings of the Fifth International Conference on Spectral
and High Order Methods (ICOSAHOM-01) (Uppsala)) 17 (2002), 561–571.
[34] W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech.
Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
[35] D. Schötzau, C. Schwab, and A. Toselli, hp-DGFEM for incompressible flows, SIAM J.
Numer. Anal., to appear.
[36] D. Schötzau and T. Wihler, Exponential convergence of mixed HP-DGFEM for Stokes flow
in polygons, Numer. Math., to appear.
[37] A. Toselli, hp-discontinuous Galerkin approximations for the Stokes problem, Math. Models
Methods Appl. Sci. 12 (2002), 1565–1616.
Bernardo Cockburn, School of Mathematics, University of Minnesota, Vincent Hall,
Minneapolis, MN 55455, USA
E-mail address: cockburn@math.umn.edu
Guido Kanschat, Institut für Angewandte Mathematik, Universität Heidelberg, Im
Neuenheimer Feld 293/294, 69120 Heidelberg, Germany
E-mail address: guido.kanschat@na-net.ornl.gov
Dominik Schötzau, Department of Mathematics, University of Basel, Rheinsprung 21,
4051 Basel, Switzerland
E-mail address: schotzau@math.unibas.ch