Mathematical Models and Methods in Applied Sciences
cfWorld Scientific Publishing Company
STABILIZED HP-DGFEM FOR INCOMPRESSIBLE FLOW
D. SCHÖTZAU1
C. SCHWAB2
A. TOSELLI2
1
Department of Mathematics, University of Basel,
Rheinsprung 21, CH-4051 Basel, Switzerland
2 Seminar for Applied Mathematics, ETH Zürich,
ETH Zentrum, CH-8092 Zürich, Switzerland
Math. Models Methods Appl. Sci., Vol. 13, pp. 1413-1436, 2003
We consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of
the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized
with a term penalizing the pressure jumps. For this approach it is shown that Q k − Qk
and Qk − Qk−1 elements satisfy a generalized inf-sup condition on geometric edge and
boundary layer meshes that are refined anisotropically and non quasi-uniformly towards
faces, edges, and corners. The discrete inf-sup constant is proven to be independent of the
aspect ratios of the anisotropic elements and to decrease as k −1/2 with the approximation
order. We also show that the generalized inf-sup condition leads to a global stability
result in a suitable energy norm.
Keywords: hp-FEM, discontinuous Galerkin methods, Stokes problem, anisotropic refinement
1. Introduction
Over the last few years, several discontinuous Galerkin (DG) methods for incompressible flow problems and for certain saddle-point problems with incompressibility constraints have been proposed in the literature. Here we only mention the
piecewise solenoidal discontinuous Galerkin methods6,20 , the local discontinuous
Galerkin (LDG) methods11,10 , and the interior penalty methods18,29,17 . The methods above all rely on discrete velocity spaces consisting of piecewise polynomial
functions with no continuity constraints between the elements in the underlying triangulation. Interelemental communication is achieved through so-called numerical
fluxes, as in the original discontinuous Galerkin methods for non-linear hyperbolic
systems.12,9,13 The main motivations for using DG methods in fluid flow problems
lie in their robustness in convection-dominated regimes, their conservation properties, and their great flexibility in the mesh-design. Based on completely discontinuous finite element spaces, DG methods easily handle elements of various types
and shapes, non-matching grids and even local spaces of different orders; they are
therefore ideal for hp-adaptivity.
Even if transport phenomena may be dominant in incompressible flow problems,
mixed DG methods still require suitable velocity-pressure pairs in order to ensure
stability and convergence of the underlying Stokes discretization. It was shown
recently that discontinuous Pk − Pk−1 and Qk − Qk−1 pairs are inf-sup stable with
1
2
Stabilized hp-DGFEM for incompressible flow
respect to the mesh-size, as opposed to their conforming counterparts.18,29 These
elements are optimal from an approximation point of view. A slightly different
approach was proposed for the LDG methods.11,10 There, the introduction of a
pressure stabilization term was proven to also render the convenient equal-order
Pk − Pk and Qk − Qk elements stable, uniformly in the mesh-size.
The study of mixed hp-discontinuous Galerkin methods was recently initiated
and it was shown that several discontinuous velocity-pressure pairs possess better
stability properties than their conforming versions.29 In particular, the numerical
results reported in the experiments of Ref. 29 for two-dimensional uniform meshes
show that discontinuous Qk − Qk−1 elements are also uniformly stable with respect
to the polynomial degree k. For this pair, the best available bound of the inf-sup
constant in terms of k was then shown to decrease as k −1 , on shape-regular tensorproduct meshes in two and three dimensions, possibly with hanging nodes.24 This
bound ensures the same p-version convergence rate for the velocity and the pressure
as that of conforming Qk − Qk−2 elements in three dimensions. However, the latter
elements are mismatched with respect to h-approximation.
In laminar regimes, solutions of incompressible flow problems in polyhedral domains have corner and edge singularities. In addition, strong boundary layers may
arise at faces, edges, and corners. In the hp-version of the finite element method,
these solution components can be approximated at exponential rate of convergence
provided that the meshes are geometrically and anisotropically graded towards faces,
edges, and corners.2,5,21,27,28 These anisotropically refined meshes raise serious stability issues in mixed approximations as the inf-sup constants might in general be
very sensitive to the aspect ratios of the elements. It was recently shown for two- and
three-dimensional conforming approximations employing Qk − Qk−2 elements that,
on corner, edge, and boundary-layer tensor-product meshes, the inf-sup constant
for the Stokes problem is in fact independent of the aspect ratios of the anisotropic
elements in the meshes.22,23,1,30 . Recently, discontinuous Qk − Qk−1 elements were
studied on geometric edge meshes designed to resolve corner and edge singularities
in the absence of boundary layers.25 By suitably defining the discontinuity stabilization parameters in the DG bilinear forms on anisotropic elements, it was proven
that this velocity-pressure pair is divergence stable, with an inf-sup constant that
is independent of the aspect ratios of the anisotropic elements and that decreases
as k −3/2 in the approximation order.
In this paper, we analyze stabilized hp-discontinuous Galerkin methods on geometric meshes in three dimensions. We show that the introduction of the pressure
stabilization term originally proposed for the LDG discretization11 leads to a generalized inf-sup constant for Qk − Qk−1 and Qk − Qk elements that decreases only
as k −1/2 in the polynomial degree, and is independent of possibly large aspect ratios of the mesh. As opposed to the work of Ref. 25 that only considers geometric
edge meshes, the results here also hold for geometric boundary layer meshes that
are additionally geometrically refined towards the faces. As a consequence of the
generalized inf-sup condition, we obtain a global stability result in a suitable energy
norm and derive p-version error bounds that are better than those available in the
recent work of Ref. 24, by of half an order of k in the velocity and a full order
of k in the pressure, respectively. We emphasize that, in our analysis, we use a
similar unifying setting as that proposed in the analysis of Ref. 24. Thus, although
we only consider the so-called interior penalty discontinuous Galerkin method, our
results hold true verbatim for the analogues of the methods analyzed there, and, in
particular, extend the LDG methods11,10 to the hp-context.
The outline of the paper is as follows. In Section 2, we introduce stabilized mixed
hp-DGFEM for the Stokes problem. Two classes of geometric meshes are defined
in Section 3. Continuity and coercivity properties of the DG forms on these meshes
are established in Section 4. Our main result is the generalized inf-sup condition
Stabilized hp-DGFEM for incompressible flow
3
that we present and prove in Section 5. A global stability result for the proposed
DG discretizations is then derived in Section 6, together with hp-error bounds on
shape-regular elements.
2. Stabilized mixed hp-DGFEM for the Stokes problem
In this section, we introduce stabilized mixed hp-discontinuous Galerkin methods
using the pressure stabilization form that was originally proposed for the LDG
discretization.11
2.1. The Stokes problem
Let Ω be a bounded polyhedron in R3 , and let n be the outward normal unit
vector to its boundary R∂Ω. Given a source term f ∈ L2 (Ω)3 and a Dirichlet datum
g ∈ H 1/2 (∂Ω)3 with ∂Ω g · n ds = 0, the Stokes problem consists in finding a
velocity field u and a pressure p such that
−ν∆u + ∇p = f
in Ω,
∇·u=0
in Ω,
u=g
(2.1)
on ∂Ω.
Thanks to the continuous inf-sup condition7,16
R
− Ω q ∇ · v dx
inf
sup
≥ CΩ > 0,
06=q∈L2 (Ω)/R 06=v∈H 1 (Ω)3
|v|1 kqk0
0
(2.2)
with a constant CΩ depending only on Ω, the Stokes problem (2.1) has a unique
solution (u, p) with
u ∈ V := H 1 (Ω)3 ,
p ∈ Q := L20 (Ω) = L2 (Ω)/R.
Here, we denote by k · ks,D and | · |s,D the norm and seminorm of the Sobolev space
H s (D), s ≥ 0, on a domain D in Rd , d = 1, 2, 3. The same notation is used to
denote norms for vector fields. In case D = Ω, we drop the subscript.
2.2. Meshes and trace operators
Throughout, we consider triangulations T on Ω that consist of affine hexahedral
elements {K}. More precisely, each element K ∈ T is obtained from the reference
b = (−1, 1)3 by an affine mapping. In general, we allow for irregular meshes,
cube Q
i.e., meshes with hanging nodes (see, e.g., Sect. 4.4.1 of Ref. 26), but suppose that
the intersection between neighboring elements is either a common vertex, a common
edge, a common face, or an entire face of one of the two elements. An interior face
of T is the (non-empty) two-dimensional interior of ∂K + ∩∂K − , where K + and K −
are two adjacent elements of T . Similarly, a boundary face of T is the (non-empty)
two-dimensional interior of ∂K ∩∂Ω which consists of entire faces of ∂K. We denote
by FI the union of all interior faces of T , by FB the union of all boundary faces,
and set F = FI ∪ FB .
For an element K ∈ T , we denote its diameter by hK and the radius of the
largest circle that can be inscribed into K by ρK . A mesh T is called shape-regular
if
hK ≤ cρK ,
∀K ∈ T ,
(2.3)
for a shape-regularity constant c > 0 that is independent of the elements. As will
be discussed below, our meshes are not necessarily shape-regular.
4
Stabilized hp-DGFEM for incompressible flow
We next define some trace operators. Let f ⊂ FI be an interior face shared by
two elements K + and K − and v, q, and τ be vector-, scalar- and matrix-valued
functions, respectively, that are smooth inside each element K ± . We denote by v± ,
q ± and τ ± the traces of v, q and τ on f from the interior of K ± and define the
mean values {{·}} and normal jumps [[·]] at x ∈ f as
{{v}} := (v+ + v− )/2,
[[v]] := v+ · nK + + v− · nK − ,
{{q}} := (q + + q − )/2,
[[[[q]]
[[ ]]]] := q + nK + + q − nK − ,
{{τ }} := (τ + + τ − )/2,
[[[[τ]]
[[ ]]]] := τ + nK + + τ − nK − .
Here, we denote by nK the outward normal unit vector to the boundary ∂K of an
element K. We also define the matrix-valued jump of the velocity v given by
[[v]] := v+ ⊗ nK + + v− ⊗ nK − ,
where, for two vectors a and b, [a ⊗ b]ij = ai bj . On a boundary face f ⊂ FB given
by f = ∂K ∩ ∂Ω, we set {{v}} := v, {{q}} := q, {{τ }} := τ , as well as [[v]] := v · n,
[[v]] := v ⊗ n, [[[[q]]
[[ ]]]] := qn and [[[[τ]]
[[ ]]]] := τ · n.
2.3. Finite element spaces
Given a mesh T on Ω and an approximation order k ≥ 0, we introduce the finite
element spaces Vhk (T ) and Qkh (T ):
Vhk (T ) := { v ∈ L2 (Ω)3 : v|K ∈ Qk (K)3 , K ∈ T },
Qkh (T ) := { q ∈ L20 (Ω) : q|K ∈ Qk (K), K ∈ T },
(2.4)
where Qk (K) is the space of polynomials of maximum degree k in each variable on
element K.
2.4. Mixed discontinuous Galerkin approximations
We approximate the velocities and pressures in the spaces Vh and Qh given by
Vh := Vhk (T ),
Qh := Q`h (T ),
(2.5)
with k ≥ 1 and ` = k or ` = k − 1. We refer to these velocity-pressure pairs as
(discontinuous) equal-order Qk −Qk elements and mixed-order Qk −Qk−1 elements,
respectively.
We consider the following stabilized mixed DG methods: find (uh , ph ) ∈ Vh ×Qh
such that
(
Ah (uh , v) + Bh (v, ph ) = Fh (v)
(2.6)
−Bh (uh , q) + Ch (ph , q) = Gh (q)
for all (v, q) ∈ Vh × Qh . The forms Ah , Bh , and Ch are given by
Z
Z
{{ν∇h v}} : [[u]] + {{ν∇h u}} : [[v]] ds
Ah (u, v) =
ν∇h u : ∇h v dx −
Ω
F
Z
+ν
δ [[u]] : [[v]] ds,
Z F
Z
Bh (v, q) = −
q ∇h · v dx + {{q}}[[v]] ds,
Ω
E
Z
−1
γ [[[[p]]
[[ ]][[
]][[q]]
[[ ]]]] ds.
Ch (p, q) = ν
FI
Stabilized hp-DGFEM for incompressible flow
5
Here, ∇h is the discrete gradient, taken elementwise. The functions δ ∈ L∞ (F) and
γ ∈ L∞ (FI ) are the so-called discontinuity and pressure stabilization functions, respectively, for which we will make a precise choice below. Finally, the corresponding
right-hand sides Fh and Gh are
Z
Z
Z
f · v dx −
(g ⊗ n) : {{ν∇h v}} ds + ν
Fh (v) =
δ g · v ds,
FB
FB
Ω
Z
Gh (q) = −
q g · n ds.
FB
Remark 2.1 It follows from the stability results below that problem (2.6) has a
unique solution (uh , ph ) ∈ Vh × Qh .
Remark 2.2 The form Ah (·, ·) discretizing the Laplacian is the so-called interior
penalty (IP) form. Several other choices are possible for A h (·, ·). All the results of
this paper also hold verbatim for the forms considered in the work of Ref. 24. The
form Bh (·, ·) is related to the incompressibility constraint; it is used in several related
mixed DG approaches.11,18,29,24,17 Finally, Ch (·, ·) is the pressure stabilization form
that was originally introduced in the local discontinuous Galerkin methods. 11,10
2.5. Perturbed mixed formulation
eh and B
eh ,
For the purpose of the analysis, we introduce perturbed forms A
following the ideas of Ref. 4 and of Ref. 24. To this end, we define the space
V(h) := V + Vh , and introduce the lifting operators L : V(h) → Σh and M :
V(h) → Qh by
Z
Z
[[v]] : {{τ }} ds,
∀τ ∈ Σh ,
L(v) : τ dx =
ZF
ZΩ
[[v]]{{q}} ds,
∀q ∈ Qh ,
M(v)q dx =
Ω
E
where we use the auxiliary space Σh = { τ ∈ L2 (Ω)3×3 : τ |K ∈ Qk (K)3×3 , K ∈ T }.
We then introduce the following perturbed forms on V(h) × V(h) and V(h) × Q:
Z
Z
eh (u, v) =
ν ∇h u : ∇h v − L(u) : ∇h v − L(v) : ∇h u dx + ν
δ[[u]] : [[v]] ds,
A
Ω
F
Z
eh (v, q) = −
B
q [∇h · v − M(v)] dx.
Ω
(2.7)
eh = Ah on Vh × Vh , and B
eh = Bh on Vh × Qh , respectively. Thus, we
We have A
may rewrite the method (2.6) as: find (uh , ph ) ∈ Vh × Qh such that
(
eh (uh , v) + B
eh (v, ph ) = Fh (v)
A
(2.8)
eh (uh , q) + Ch (ph , q) = Gh (q)
−B
for all (v, q) ∈ Vh × Qh .
3. Geometric edge and boundary layer meshes
In this section, we define two classes of geometric meshes, namely geometric
edges meshes that are employed in the presence of corner and edge singularities (as,
6
Stabilized hp-DGFEM for incompressible flow
e.g., in Stokes flow or nearly incompressible elasticity), and geometric boundary layer
meshes that are used when, in addition to corner/edge singularities, boundary layers
are present as well. Both meshes are characterized by a geometric grading factor
σ ∈ (0, 1) and the number of layers n, the thinnest layer having width proportional
to σ n .2,5,21,27,28,30
3.1. Geometric edge meshes
n,σ
A geometric edge mesh Tedge
is constructed by considering an initial shaperegular macro-triangulation Tm = {M } of Ω, with no hanging nodes, possibly
consisting of just one element. The macro-elements M in the interior of Ω are
then refined isotropically and regularly (not discussed further) while the macroelements M on the boundary of Ω are refined geometrically and anisotropically
towards edges and corners. This geometric refinement is obtained by affinely mapb onto the
ping corresponding reference triangulations (referred to as patches) on Q
b → M . This process is illusmacro-elements M using the elemental maps FM : Q
b = I 3 , I = (−1, 1),
trated in Figure 1. For edge meshes, the following patches on Q
are used for the geometric refinement towards the boundary of Ω:
b is given by
Edge patches: An edge patch Teedge on Q
Teedge := {Kxy × I | Kxy ∈ Txy },
where Txy is an irregular corner mesh, geometrically refined towards a vertex of
Sb = (−1, 1)2 with grading factor σ and n refinement levels; see Figure 1 (level 2,
left).
b we first consider
Corner patches: In order to build a corner patch Tcedge on Q,
b
an initial, irregular, corner mesh Tc,m , geometrically refined towards a vertex of Q,
with grading factor σ and n refinement levels; see the mesh in bold lines in Figure
1 (level 2, right). The elements of this mesh are then irregularly refined towards
the three edges adjacent to the vertex in order to obtain the mesh Tcedge ; see also
Figure 3.
For simplicity, we always assume that the only hanging nodes in geometric edge
n,σ
meshes Tedge
are those in the closure of edge and corner patches.
Level 1
Level 2
n,σ
Figure 1: Hierarchic structure of a geometric edge mesh Tedge
. The macro-elements
M on the boundary of Ω (level 1) are further refined as edge and corner patches
(level 2). The geometric grading factor is here σ = 0.5.
Stabilized hp-DGFEM for incompressible flow
7
3.2. Geometric boundary layer meshes
As for edge meshes, the construction of a geometric boundary layer mesh T bln,σ
starts from an initial shape-regular macro-triangulation Tm = {M } of Ω, with no
hanging nodes, possibly consisting of just one element. The macro-elements M on
the boundary of Ω are now also refined geometrically towards faces; see Figure 2.
b = I 3 are used:
More precisely, the following face, edge, and corner patches on Q
bl
b
Face patches: A face patch Tf on Q is given by an anisotropic triangulation
of the form
Tfbl := {Kx × I × I | Kx ∈ Tx },
where Tx is a mesh of I, geometrically refined towards one of the vertices, say x = 1,
with grading factor σ ∈ (0, 1) and total number of layers n; see Figure 2 (level 2,
left).
b is given by
Edge patches: An edge patch Tebl on Q
Tebl := {Kxy × I | Kxy ∈ Texy },
where Texy is a triangulation of Sb = I 2 obtained by first considering an irregular
corner mesh Txy as in a patch Teedge of an edge mesh, geometrically refined towards
b say (x, y) = (1, 1), with grading factor σ and n refinement levels (see
a vertex of S,
Figure 1 below, level 2, left). The elements of the mesh Txy are then anisotropically
refined towards the two edges x = 1 and y = 1, in order to obtain a regular mesh
Texy . We refer to Figure 2 (level 2, center) for an example.
b we first consider
Corner patches: In order to build a corner patch Tcbl on Q,
the same initial, irregular corner mesh Tc,m , geometrically refined towards a vertex
b with grading factor σ and n refinement levels; see the mesh in bold lines
of Q,
in Figure 2 (level 2, right). The elements of Tc,m are then anisotropically refined
towards the three faces x = 1, y = 1, and z = 1 in order to obtain a regular mesh
Tcbl ; see also Figure 3.
For simplicity, we always assume that the three types of patches above are
combined in such a way that geometric boundary layer meshes Tbln,σ do not contain
hanging nodes.
Level 1
Level 2
Figure 2: Hierarchic structure of a geometric boundary layer mesh Tbln,σ . The
macro-elements M on the boundary of Ω (level 1) are further refined as face, edge
and corner patches (level 2). The geometric grading factor is here σ = 0.5.
8
Stabilized hp-DGFEM for incompressible flow
Remark 3.1 We note that the underlying mesh Tc,m is the same for the corner
patches Tcedge and Tcbl in edge and boundary layer meshes, respectively. However,
Tcedge is irregular and contains hanging nodes. Figure 3 shows the difference between
corner patches for boundary layer and edge meshes.
Figure 3: Geometrically refined corner patches Tcbl and Tcedge for boundary layer
(left) and edge (right) meshes. The geometric grading factor is σ = 0.5.
The geometric edge and boundary layer meshes defined above satisfy the following property; see Sect. 3 of Ref. 25.
n,σ
Property 3.2 Let T be a geometric edge mesh Tedge
or a geometric boundary layer
n,σ
mesh Tbl , with a grading factor σ ∈ (0, 1) and n levels of refinement. Then, any
K ∈ T can be written as K = FK (Kxyz ), where Kxyz is of the form
Kxyz = Ix × Iy × Iz = (x1 , x2 ) × (y1 , y2 ) × (z1 , z2 ),
and FK is an affine mapping, the Jacobian of which satisfies
| det(JK )| ≤ C,
−1
| det(JK
)| ≤ C,
kDFK k ≤ C,
−1
kDFK
k ≤ C,
with constants only depending on the angles of K but not on its dimensions.
We note that the constants in Property 3.2 only depend on the shape-regularity
constant in (2.3) of the underlying macro-element mesh Tm . The dimensions of Kxyz
on the other hand may depend on the geometric grading factor and the number of
refinements.
For an element K of a geometric edge mesh, we define, according to Property 3.2,
hK
x = hx = x2 − x1 ,
hK
y = hx = y 2 − y 1 ,
hK
z = hx = z 2 − z 1 .
3.3. Stabilization on geometric meshes
In this section, we define the discontinuity and pressure stabilization functions
δ ∈ L∞ (F) and γ ∈ L∞ (FI ) on geometric meshes.
To this end, let f be an entire face of an element K of a geometric mesh T on Ω.
According to Property 3.2, K can be obtained by a stretched parallelepiped Kxyz
by an affine mapping FK that only changes the angles. Suppose that the face f is
the image of, e.g., the face {x = x1 }. We set hf = hx . For a face perpendicular to
the y- or z-direction, we choose hf = hy or hf = hz , respectively.
Stabilized hp-DGFEM for incompressible flow
9
Let then K and K 0 be two elements with entire faces f and f 0 that share an
interior face f = f ∩ f 0 in FI . We have
chf ≤ hf 0 ≤ c−1 hf ,
(3.1)
with a constant c > 0 that only depends on the geometric grading factor σ and
the constant in (2.3) for the underlying macro-mesh Tm . We define the function
h ∈ L∞ (F) by
(
min{hf , hf 0 } x ∈ f ∩ f 0 ⊂ FI ,
(3.2)
h(x) :=
hf
x ∈ f ⊂ FB .
We then set
and
δ(x) = δ0 h−1 k 2 ,
x ∈ F,
γ(x) = γ0 min{hf , hf 0 } max{1, `}−1,
(3.3)
x ∈ FI ,
(3.4)
with δ0 > 0 and γ0 > 0 independent of h and k.
Remark 3.3 For isotropically refined and shape-regular meshes, the definitions in
(3.2) and (3.3) are equivalent to the usual definition of δ.24 Similarly, the definition of γ in (3.4) generalizes the definition of Ref. 11 to the hp-version context on
geometric meshes.
4. Continuity and coercivity on geometric meshes
eh and B
eh as well as the coercivity of
On geometric meshes, the continuity of A
Ah can be established as in Sect. 4 of Ref. 25.
To this end, we equip V(h) = V + Vh with the broken norm
Z
X
2
2
kvkh :=
|v|1,K +
δ|[[v]]|2 ds,
v ∈ V(h).
K∈T
F
We have the following result.
n,σ
Theorem 4.1 Let T be a geometric edge mesh Tedge
or a geometric boundary layer
n,σ
mesh Tbl , with a grading factor σ ∈ (0, 1) and n levels of refinement. Let the
eh and
stabilization functions δ be defined as in (3.2) and (3.3). Then, the forms A
e
Bh in (2.7) are continuous:
eh (v, w)| ≤ να1 kvkh kwkh
|A
eh (v, q)| ≤ α2 kvkh kqk0
|B
∀ v, w ∈ V(h),
∀ u ∈ V(h), q ∈ Q,
with continuity constants α1 and α2 that depend on δ0 and the constants in Property 3.2, but are independent of ν, k, n, and the aspect ratio of T . Furthermore,
there exists a constant δmin > 0 that depends on the constants in Property 3.2, but
is independent of ν, k, n, and the aspect ratio of T , such that, for any δ0 ≥ δmin ,
Ah (v, v) ≥ νβkvk2h
∀ v ∈ Vh ,
for a coercivity constant β > 0 depending on δ0 and the constants in Property 3.2,
but independent of ν, k, n, and the aspect ratio of T .
10
Stabilized hp-DGFEM for incompressible flow
Remark 4.2 The results in Theorem 4.1 are based on anisotropic stability estimates for the lifting operators L and M that can be found in Sect. 4 of Ref. 25.
These operators are identical for all the DG forms considered in the framework of
Ref. 24 and, thus, the results in Theorem 4.1 also hold for all the mixed DG methods
considered there. We note that the restriction on δ0 is typical for the interior penalty
form Ah and can be avoided if Ah were chosen to be, e.g., the local discontinuous
Galerkin form, the nonsymmetric interior penalty form, or the second Bassi-Rebay
form.24
Next, we address the continuity of Fh and Gh .
n,σ
Theorem 4.3 Let T be a geometric edge mesh Tedge
or a geometric boundary layer
n,σ
mesh Tbl , with a grading factor σ ∈ (0, 1) and n levels of refinement. Let the
stabilization functions δ be defined as in (3.2) and (3.3). Then we have
1
∀ v ∈ Vh ,
|Fh (v)| ≤ C kf k0 + νkδ 2 gk0,∂Ω kvkh
1
|Gh (q)| ≤ C kδ 2 gk0,∂Ω kqk0
∀ q ∈ Qh ,
with continuity constants that depend on δ0 , Ω, and the constants in Property 3.2,
but are independent of ν, k, n, and the aspect ratio of T .
Proof : We first note that we have the Poincaré inequality
kvk0 ≤ Ckvk1,h
∀ v ∈ V(h),
(4.1)
with a constant depending on δ0 , Ω, and the constants in Property 3.2. The bound
(4.1) follows by proceeding as in the original proof in Lemma 2.1 of Ref. 3, taking
into account elliptic regularity theory for polyhedral domains and by using the
anisotropic trace inequality
kϕk0,f ≤ Ch−1
f kϕk3/2+ε,K ,
ε > 0,
for an element K ∈ T and an entire face f of ∂K, with a constant depending on
the constants in Property 3.2.
R
Let now v ∈ Vh . From (4.1), we obtain | Ω f · v dx| ≤ Ckf k0 kvkh . Further,
applying the discrete trace inequality from Lemma 3.3 of Ref. 25 as in the proof
of Theorem 4.1 of Ref. 25,
Z
1
(g ⊗ n) : {{ν∇h v}} ds| ≤ Cνkδ 2 gk0,∂Ω kvkh ,
|
EB
with a constant depending on δ0 , and
3.2. Finally, the
R the constants in Property
1
Cauchy-Schwarz inequality yields |ν EB δg · v ds| ≤ νkδ 2 gk0,∂Ω kvkh . This proves
the assertion for Fh . Similarly, for q ∈ Qh ,
Z
Z
1
1
2
|Gh (q)| ≤ |
q g · n ds| ≤ kδ gk0,∂Ω
δ −1 |q|2 ds 2 .
EB
EB
Using again the techniques in Lemma 3.3 and Theorem 4.1 of Ref. 25, we have
Z
δ −1 |q|2 ds ≤ Ckqk20 ,
EB
with a constant depending on δ0 , and the constants in Property 3.2. This completes
the proof.
2
Stabilized hp-DGFEM for incompressible flow
11
Remark 4.4 The same continuity properties hold for all the functionals F h and
Gh in the mixed DG methods analyzed in the setting of Ref. 24.
5. Generalized inf-sup condition on geometric meshes
Our main result establishes a generalized inf-sup condition on geometric meshes.
To this end, we introduce the following seminorm on Qh
Z
2
|q|FI :=
γ |[[[[q]]
[[ ]]|
]] 2 ds,
FI
with γ the pressure stabilization function defined in (3.4).
We have the following result.
n,σ
Theorem 5.5 Let T be a geometric edge mesh Tedge
or a geometric boundary layer
n,σ
mesh Tbl , with a grading factor σ ∈ (0, 1) and n levels of refinement. Let the
stabilization functions δ and γ be defined according to (3.2), (3.3), and (3.4). Then,
there exists a constant C > 0 that depends on Ω, δ0 , γ0 , and the constants in
Property 3.2 and (3.1), but is independent of ν, k, `, n, and the aspect ratio of T ,
such that, for any n and k ≥ 1, ` = k or ` = k − 1,
sup
06=v∈Vh
Bh (v, q)
1
|q|FI ≥ Ck − 2 kqk0 1 −
,
kvkh
kqk0
∀q ∈ Qh \ {0}.
Remark 5.6 For h-version DG approximations on shape-regular meshes, the generalized inf-sup condition in Theorem 5.5 was established recently for the LDG discretizations, in a form that also involves the auxiliary stresses present in the LDG
approach.11,10 Similar inf-sup conditions also play an important role in the analysis
of conforming stabilized mixed methods.15,14
The proof of Theorem 5.5 is carried out in the rest of this section. We begin
by collecting several properties of L2 -projections and by deriving bounds for averages and jumps over faces of geometric meshes. We then complete the proof of
Theorem 5.5.
5.1. L2 -projections
For an interval Ix = (x1 , x2 ), let Πx : L2 (Ix ) → Qk (Ix ) denote the onedimensional L2 -projection onto the space Qk (Ix ) of polynomials of degree at most
k on Ix ; given v ∈ L2 (Ix ), this projection is defined by imposing
Z
Z
Πx v ϕ dx =
v ϕ dx,
∀ϕ ∈ Qk (Ix ).
Ix
Ix
The L2 -projection is stable:
kΠx vk0,Ix ≤ kvk0,Ix ,
∀v ∈ L2 (Ix ).
(5.2)
Moreover, applying similar techniques as in Theorem 2.2 of Ref. 8, we have, for
k ≥ 1,
1
∀v ∈ H 1 (Ix ),
(5.3)
|Πx v|1,Ix ≤ Ck 2 |v|1,Ix ,
with a constant C > 0 independent of k, Ix , and v.
We next recall the following approximation result from Lemma 3.5 of Ref. 19.
12
Stabilized hp-DGFEM for incompressible flow
Lemma 5.7 Let Ix = (x1 , x2 ), hx = x2 − x1 and v ∈ H 1 (Ix ). Then, there holds
|v(x1 ) − Πx v(x1 )|2 + |v(x1 ) − Πx v(x2 )|2 ≤ C hx k −1 kv 0 k20,Ix ,
for k ≥ 1 and with a constant C > 0 independent of hx , k, and v.
We will also make use of an approximation result from Lemma 3.9 of Ref. 19
for the two-dimensional L2 -projection Πx ⊗ Πy ; here, the subscripts indicate the
variables the projections Πx and Πy act on.
Lemma 5.8 Let Ix = (x1 , x2 ), Iy = (y1 , y2 ), hx = x2 − x1 and hy = y2 − y1 .
Assume that there exists a constant c > 0 such that chx ≤ hy ≤ c−1 hx . Then, for
v ∈ H 1 (Ix × Iy ) and k ≥ 1, we have
kv − Πx ⊗ Πy vk20,∂(Ix ×Iy ) ≤ C hx k −1 |v|21,Ix ×Iy ,
with a constant C > 0 depending on c, but independent of hx , hy , k, and v.
For an axiparallel element Kxyz = (x1 , x2 ) × (y1 , y2 ) × (z1 , z2 ), the L2 -projection
ΠKxyz : L2 (Kxyz ) → Qk (Kxyz ) is the product operator ΠKxyz = Πx ⊗ Πy ⊗ Πz of
one-dimensional L2 -projections. For v ∈ L2 (Kxyz ), it satisfies
Z
Z
ΠKxyz v ϕ dx =
v ϕ dx,
∀ϕ ∈ Qk (Kxyz ).
Kxyz
Kxyz
For an element K of a geometric edge or boundary layer mesh T , the L2 -projection
ΠK : L2 (K) → Qk (K) is defined by
Z
Z
ΠK v ϕ dx =
v ϕ dx,
∀ϕ ∈ Qk (K).
K
K
Thanks to Property 3.2, we have K = FK (Kxyz ) for an axiparallel element Kxyz =
(x1 , x2 ) × (y1 , y2 ) × (z1 , z2 ). For v ∈ L2 (K), we therefore have
on Kxyz .
(5.4)
ΠK v ◦ FK = ΠKxyz v ◦ FK ,
We have the following stability result.
Lemma 5.9 Let T be a geometric edge or boundary layer mesh. Let K ∈ T and
v ∈ H 1 (K). Then we have for k ≥ 1
1
|ΠK v|1,K ≤ Ck 2 |v|1,K ,
with a constant C > 0 depending on the bounds in Property 3.2, but independent of
k, K, and v.
Proof : Let K = Kxyz according to Property 3.2. The bounds (5.2) and (5.3) imply
that
1
k∂x ΠKxyz vk0,Kxyz
≤ Ck 2 k∂x vk0,Kxyz ,
k∂y ΠKxyz vk0,Kxyz
≤ Ck 2 k∂y vk0,Kxyz ,
k∂z ΠKxyz vk0,Kxyz
≤ Ck 2 k∂z vk0,Kxyz ,
1
1
for any v ∈ H 1 (Kxyz ), with a constant C > 0 independent of k, Kxyz , and v. A
scaling argument and the bounds in Property 3.2 prove the assertion for a general
element K.
2
Stabilized hp-DGFEM for incompressible flow
13
Finally, the L2 -projection Π : L2 (Ω) → {v ∈ L2 (Ω) : v|K ∈ Qk (K), K ∈ T } is
defined elementwise by Πv|K = ΠK v|K , K ∈ T . For vector fields, we use bold-face
notation (such as ΠKxyz , ΠK , and Π) to denote the L2 -projections that are applied
componentwise.
5.2. Auxiliary results
In this section, we derive bounds for the averages and jumps over faces. We
start by considering interior faces.
5.2.1. Interior faces
Let K and K 0 be two elements of a geometric mesh with entire faces f and f 0
that share an interior face f ∩ f 0 in FI . We may assume that f ∩ f 0 is an entire
face of K, that is, f ∩ f 0 = f . By Property 3.2, we have K = FK (Kxyz ) and
0
K 0 = FK 0 (Kxyz
) with, e.g.,
0
Kxyz
= (x2 , x3 ) × (y1 , y3 ) × (z1 , z2 ),
Kxyz = (x1 , x2 ) × (y1 , y2 ) × (z1 , z2 ),
and y2 ≤ y3 . The face f is then given by f = Ff (fyz ), with fyz = {x2 } × (y1 , y2 ) ×
(z1 , z2 ), and Ff (y, z) = FK (x2 , y, z) = FK 0 (x2 , y, z) for y1 ≤ y ≤ y2 , z1 ≤ z ≤ z2 .
0
Similarly, we have f 0 = Ff 0 (fyz
). For a function v ∈ H 1 (K ∪ K 0 )3 , we define
0
vxyz = v|K ◦ FK and vxyz = v|K 0 ◦ FK 0 .
We set hx = x2 − x1 , h0x = x3 − x2 , hy = y2 − y1 , h0y = y3 − y1 , and hz = z2 − z1 ,
and may assume that
chx ≤ h0x ≤ c−1 hx ,
(5.5)
according to (3.1). In the case where the elements K and K 0 match regularly (i.e.,
f = f 0 ) the ratios of the mesh-sizes hx , hy and hz can be arbitrary. However,
when K and K 0 match irregularly (i.e., f 6= f 0 ), it is essential to observe that, by
definition of geometric meshes, we also have
chy ≤ h0y ≤ c−1 hy ,
chy ≤ hx ≤ c−1 hy ,
(5.6)
with a constant c > 0 depending solely on the bounds in Property 3.2. The situation
when K and K 0 match irregularly is shown in Figure 4. We point out that the above
configuration covers all interior faces in geometric edge and boundary layer meshes.
We first show the following result.
Lemma 5.10 Let K, K 0 ∈ T share a face f ⊂ FI . Then, for q ∈ Qh and v ∈
H 1 (K ∪ K 0 )3 ,
|
Z
[[[[q]]
[[ ]]]] · {{v − Πv}} ds| ≤ C
f
Z
2
γ |[[[[q]]
[[ ]]|
]] ds
f
12 |v|21,K
+
|v|21,K 0
12
,
with a constant C > 0 that depends on γ0 and the bounds in Property 3.2 and (3.1).
R
Proof : We begin by noting that f [[[[q]]
[[ ]]]] · {{v − Πv}} ds = 12 SK + 12 SK 0 , where
SK
SK 0
=
=
Z
Z
[[[[q]]
[[ ]]]] · (v|K − ΠK v|K ) ds,
f
[[[[q]]
[[ ]]]] · (v|K 0 − ΠK 0 v|K 0 ) ds.
f
14
Stabilized hp-DGFEM for incompressible flow
z2
z1
fyz
y3
K’xyz
y2
Kxyz
y1
x1
x2
x3
0
Figure 4: The axiparallel elements Kxyz and Kxyz
match irregularly. The face fyz
is given by fyz = {x2 } × (y1 , y2 ) × (z1 , z2 ).
Step 1: We start by bounding the term SK . Setting qyz = [[[[q]]
[[ ]]]] ◦ Ff , we obtain
SK
=
=
=
=
Z
Z
Z
[[[[q]]
[[ ]]]] · (v|K − ΠK v|K ) ds
f
qyz · (vxyz − ΠKxyz vxyz ) | det(DFf )| dy dz
fyz
Z
qyz · (vxyz − Πx ⊗ Πy ⊗ Πz vxyz ) | det(DFf )| dy dz
fyz
qyz · (Πy ⊗ Πz vxyz − Πx ⊗ Πy ⊗ Πz vxyz ) | det(DFf )| dy dz.
fyz
Here, we have used identity (5.4), the factorization ΠKxyz = Πx ⊗Πy ⊗Πz into onedimensional L2 -projections, and the fact that each component of qyz is a polynomial
of degree ` = k or ` = k − 1 in y- and z-direction. The Cauchy-Schwarz inequality,
the definition of hf in (3.1), the definition of γ in (3.4), and (5.5) yield
SK
≤ TK
Z
hx max{1, `}
−1
2
|qyz | | det(DFf )| dy dz
fyz
≤ C · TK ·
Z
2
γ|[[[[q]]
[[ ]]|
]] ds
f
21
12
,
with the term TK given by
2
TK
:= max{1, `}h−1
x
Z
| Πy ⊗ Πz vxyz − Πx ⊗ Πy ⊗ Πz vxyz |2 | det(DFf )| dy dz.
fyz
From the stability of the one-dimensional projections Πy and Πz in (5.2) (taking
into account that | det(DFf )| is constant), the approximation result in Lemma 5.7,
Stabilized hp-DGFEM for incompressible flow
15
and the bounds in Property 3.2, we obtain
2
TK
≤
max{1, `}h−1
x | det(DFf )|
Z
| vxyz − Πx vxyz |2 dy dz
fyz
≤ C max{1, `}k −1 | det(DFf )| k∂x vxyz k20,Kxyz
−1
≤ C | det(DFf )| kDFK k | det(DFK
)| |v|21,K ≤ C |v|21,K .
Combining the above estimates shows that
12
Z
2
γ |[[[[q]]
[[ ]]|
]] ds ,
SK ≤ C |v|1,K
(5.7)
f
with a constant C depending on γ0 and the bounds in Property 3.2 and (3.1).
Step 2: Let us now consider the term SK 0 . We note that there is an entire face
f 0 of K 0 , such that f = f ∩ f 0 . If f = f 0 , then SK 0 can be bounded as SK in Step 1.
Thus, we only need to consider the case where f is an irregular face of K 0 , i.e., f
is a proper subset of f 0 as in Figure 4. As in the proof of Step 1, since qyz is a
polynomial in z-direction, we have:
Z
[[[[q]]
[[ ]]]] · (v|K 0 − ΠK 0 v|K 0 ) ds
SK 0 =
f
Z
0
0
0
qyz · (vxyz
− ΠKxyz
vxyz
) | det(DFf )| dy dz
=
Z
=
fyz
fyz
0
0
0
) | det(DFf )| dy dz.
vxyz
− Π0x ⊗ Π0y ⊗ Π0z vxyz
qyz · (Π0z vxyz
0
Here, we denote by Π0x , Π0y , and Π0z the one-dimensional L2 -projections on Kxyz
.
We obtain
Z
21
γ |[[[[q]]
[[ ]]|
]] 2 ds ,
SK 0 ≤ C · T K 0 ·
f
with the term TK 0 given by
Z
2
−1
0
0
TK
| Π0z vxyz
− Π0x ⊗ Π0y ⊗ Π0z vxyz
|2 | det(DFf )| dy dz.
0 := max{1, `}hx
fyz
From the stability (5.2) of Π0z in z-direction, (5.5), (5.6) and Lemma 5.8, we obtain
Z
0
0
2
| vxyz
− Π0x ⊗ Π0y vxyz
|2 dy dz
≤ max{1, `}h−1
|
det(DF
)|
TK
0
f
x
0
fyz
≤
0
C|vxyz
|21,Kxyz
0
≤
|v|21,K 0 .
Combining the bounds above gives
SK 0 ≤ C |v|1,K 0
Z
γ |[[[[q]]
[[ ]]|
]] 2 ds
f
21
,
(5.8)
with a constant C > 0 depending on γ0 and the bounds in Property 3.2 and (3.1).
Combining (5.7) and (5.8) concludes the proof.
2
Next, we estimate the jump of the L2 -projection over the face f .
16
Stabilized hp-DGFEM for incompressible flow
Lemma 5.11 Let K, K 0 ∈ T share a face f ⊂ FI . Suppose that f = f ∩ f 0 , with
f and f 0 entire faces of K and K 0 , respectively. Then, for v ∈ H 1 (K ∪ K 0 )3 ,
Z
|[[Πv]]|2 ds ≤ C min{hf , hf 0 }k −1 |v|21,K + |v|21,K 0 ,
f
with a constant C > 0 that depends only on the bounds in Property 3.2 and (3.1).
Proof : Equality (5.4) ensures that
Z
Z
2
|[[Πv]]| ds ≤
|ΠK v|K − ΠK 0 v|K 0 |2 ds
f
f
Z
0
0
=
|ΠKxyz vxyz − ΠKxyz
vxyz
|2 | det(DFf )| dy dz.
fyz
We consider two cases separately.
0
Case 1: Let K and K 0 match regularly, i.e., f = f 0 . Since vxyz and vxyz
0
0 0
coincide on the face fyz , we have Πy ⊗ Πz vxyz = Πy ⊗ Πz vxyz on fyz . We thus
obtain from the triangle inequality
Z
|[[Πv]]|2 ds ≤ C · | det(DFf )| · [ TK + TK 0 ],
f
with
TK
TK 0
Z
=
Z
=
|Πy ⊗ Πz vxyz − Πx ⊗ Πy ⊗ Πz vxyz |2 dy dz,
fyz
fyz
0
0
|2 dy dz.
− Π0x ⊗ Π0y ⊗ Π0z vxyz
|Π0y ⊗ Π0z vxyz
Using the stability (5.2) of the projections Πy and Πz in y- and z-directions, as
well as the approximation result in Lemma 5.7, we obtain
TK ≤ Chx k −1 k∂x vxyz k20,Kxyz ≤ Chf k −1 |v|21,K .
An analogous bound for TK 0 and (5.5) prove the assertion in this case.
Case 2: Assume that K and K 0 are non-matching (f 6= f 0 ). We then have that
0
Πz vxyz = Π0z vxyz
on fyz . Thus,
Z
|[[Πv]]|2 ds ≤ C · | det(DFf )| · [ TK + TK 0 ],
f
with
TK
TK 0
=
=
Z
Z
|Πz vxyz − Πx ⊗ Πy ⊗ Πz vxyz |2 dy dz,
fyz
fyz
0
0
|Π0z vxyz
− Π0x ⊗ Π0y ⊗ Π0z vxyz
|2 dy dz.
Stabilized hp-DGFEM for incompressible flow
17
Since the underlying elements are shape-regular in x- and y-directions thanks to
(5.5) and (5.6), we can invoke the stability (5.2) of Πz and the approximation
result in Lemma 5.8. This gives
Z
0
0
|vxyz
− Π0x ⊗ Π0y vxyz
|2 dy dz
TK 0 ≤
≤
Z
fyz
0
fyz
0
0
|vxyz
− Π0x ⊗ Π0y vxyz
|2 dy dz
0
≤ Chf 0 k −1 |v|21,K 0 .
≤ Ch0x k −1 |vxyz
|21,Kxyz
0
An analogous bound for TK and (5.5) prove the assertion in this case.
2
5.2.2. Boundary faces
We conclude by stating an analogous result to Lemma 5.11 for boundary faces
that can be proved with exactly the same techniques. Let K be an element on the
boundary and f an entire face of K in FB .
Lemma 5.12 For v ∈ H01 (K)3 , we have
Z
|[[Πv]]|2 ds ≤ C hf k −1 |v|21,K ,
f
with a constant C > 0 depending on the bounds in Property 3.2.
5.3. Proof of Theorem 5.5
Fix q ∈ Qh . From the continuous inf-sup condition (2.2), there exists a field
w ∈ H01 (Ω)3 such that
Z
−
q∇ · w dx = kqk20 ,
|w|1 ≤ CΩ−1 kqk0 ,
(5.9)
Ω
where CΩ > 0 is the continuous inf-sup constant. We then set v = Πw, with Π the
L2 -projection defined previously. Using [[w]] = 0 on F, (5.9), integration by parts,
and the properties of the L2 -projection, we find
Bh (v, q)
= Bh (w, q) + Bh (Πw − w, q)
Z
Z
= kqk20 +
∇h q · (Πw − w) dx −
[[[[q]]
[[ ]]]] · {{Πw − w}} ds
Ω
FI
Z
= kqk20 +
[[[[q]]
[[ ]]]] · {{w − Πw}} ds.
FI
Applying Lemma 5.10 gives
Z
≤
[[q]]
]]
[[
[
[
]
]
·
{
{w
−
Πw}
}
ds
FI
X Z
[[[[q]]
]]
[
[
]
]
·
{
{w
−
Πw}
}
ds
f ⊂FI
≤ C
f
X
|w|21,K
K∈T
≤ C|w|1 |q|FI .
21 X Z
f ⊂FI
2
γ|[[[[q]]
[[ ]]|
]] ds
f
21
18
Stabilized hp-DGFEM for incompressible flow
Combining the above estimates with (5.9) yields
|q|FI
,
Bh (v, q) ≥ Ckqk20 1 −
kqk0
(5.10)
with a constant C > 0 depending on CΩ , γ0 , and the bounds in Property 3.2 and
(3.1).
We have from Lemma 5.9, Lemma 5.11 and Lemma 5.12, together with the
definition of the discontinuity stabilization function δ,
X
XZ
kvk2h =
|Πw|21,K +
δ |[[Πw]]|2 ds
K∈T
≤ Ck
X
f ⊂F
|w|21,K + Ck
K∈T
f
X
|w|21,K ≤ Ck|w|21 .
K∈T
Thus, invoking (5.9),
1
kvkh ≤ Ck 2 kqk0 .
(5.11)
Combining (5.10) and (5.11) concludes the proof of Theorem 5.5.
6. Global stability and a-priori error estimates
In this section, we show how the stability results in the previous sections can
be used to obtain a global stability result and to derive a-priori error estimates.
The technique we use is closely related to that used in the analysis of conforming
stabilized mixed methods.15,14
6.1. Global stability
Let Wh be the product space Wh = Vh × Qh , endowed with the norm
|||(v, q)|||2DG = νkvk2h + ν −1 k −1 kqk20 + ν −1 |q|2FI .
In Wh we define the forms
eh (u, v) + B
eh (v, p) − B
eh (u, q) + Ch (p, q),
Ah (u, p; v, q) = A
Lh (v, q) = Fh (v) + Gh (q),
and reformulate (2.8) equivalently as: find (uh , ph ) ∈ Wh such that
Ah (uh , ph ; v, q) = Lh (v, q)
(6.1)
for all (v, q) ∈ Wh .
The following stability result holds.
n,σ
Theorem 6.1 Let T be a geometric edge mesh Tedge
or a geometric boundary layer
mesh Tbln,σ , with a grading factor σ ∈ (0, 1) and n levels of refinement. Let the
stabilization functions δ and γ be defined according to (3.2), (3.3), and (3.4). Then,
there exists a constant C > 0 that depends on Ω, δ0 , γ0 , and the constants in
Property 3.2 and (3.1), but is independent of ν, k, `, n, and the aspect ratio of of
T , such that, for any n and k ≥ 1, ` = k or ` = k − 1,
inf
sup
(0,0)6=(u,p)∈Wh (0,0)6=(v,q)∈Wh
Ah (u, p; v, q)
≥ C.
|||(u, p)|||DG |||(v, q)|||DG
Stabilized hp-DGFEM for incompressible flow
19
Proof : Fix (0, 0) 6= (u, p) ∈ Vh × Qh . Thanks to the coercivity of Ah in Theorem 4.1 and the definition of Ch , we have
Ah (u, p; u, p) ≥ νβkuk2h + ν −1 |p|2FI .
(6.2)
Furthermore, Theorem 5.5 guarantees the existence of a velocity w ∈ Vh satisfying
Bh (w, p) ≥ Ckpk20 − C|p|FI kpk0 ,
1
kwkh ≤ Ck 2 kpk0 .
(6.3)
From the definition of Ah , the continuity properties in Theorem 4.1, weighted
Cauchy-Schwarz inequalities, and (6.3), we obtain
Ah (u, p; w, 0) = Ah (u, w) + Bh (w, p)
−1
2
2
2
2
≥ −Cε1 νkuk2h − Cε−1
1 νkwkh + Ckpk0 − Cε2 kpk0 − Cε2 |p|FI
−1
2
2
2
≥ C(1 − ε−1
2 − νε1 k)kpk0 − Cε1 νkukh − Cε2 |p|FI ,
(6.4)
with parameters ε1 , ε2 > 0 at our disposal. We next set (v, q) = (u, p) + ε3 (w, 0),
with ε3 > 0. Then, combining (6.2) and (6.4), yields
−1
2
Ah (u, p; v, q) ≥ Cν(1 − ε1 ε3 )kuk2h + C(ν −1 − ε3 ε2 )|p|2FI + Cε3 (1 − ε−1
2 − νε1 k)kpk0 .
It is now easy to see that one can select ε1 of order O(kν), ε2 of order O(k), and
ε3 of order O(ν −1 k −1 ), respectively, in such a way that
Ah (u, p; v, q) ≥ Cνkuk2h + Cν −1 |p|2FI + Cν −1 k −1 kpk20 = C|||(u, p)|||2DG .
(6.5)
Using the fact that ε3 is of order O(ν −1 k −1 ) and (6.3) give
|||(v, q)|||2DG
≤ Cνkuk2h + Cνε23 kwk2h + ν −1 k −1 kpk20 + ν −1 |p|2FI
≤ Cνkuk2h + Cν −1 k −1 kpk20 + ν −1 k −1 kpk20 + ν −1 |p|2FI
≤ C|||(u, p)|||2DG .
(6.6)
Combining (6.5) and (6.6) completes the proof.
2
6.2. A-priori error estimates
In order to derive a-priori error estimates, we let (u, p) be the exact solution
of the Stokes system (2.1) and assume that p ∈ H 1 (Ωint ) in a domain Ωint ⊂ Ω
containing all the interior faces in FI . Thus, [[[[p]]
[[ ]]]] = 0 on FI . We define Q(h) :=
Qh + H 1 (Ωint ) and W(h) := V(h) × Q(h), equipped with the norm |||(v, q)|||DG .
From the continuity properties in Theorem 4.1, Theorem 4.3 and the CauchySchwarz inequality, it can be seen that
1
|Ah (u, p; v, q)| ≤ Ck 2 |||(u, p)|||DG |||(v, q)|||DG ,
∀(u, p), (v, q) ∈ W(h), (6.7)
and
1
1
|Lh (v, q)| ≤ C ν −1 kf k20 + νkkδ 2 gk20,∂Ω 2 |||(v, q)|||DG ,
∀(v, q) ∈ Wh ,
(6.8)
with constants as in Theorem 4.1 and Theorem 4.3, respectively.
Taking into account (6.7), the global inf-sup condition in Theorem 6.1, and the
eh and B
eh , we obtain straightforwardly the following
non-consistency of the forms A
a-priori bound.
20
Stabilized hp-DGFEM for incompressible flow
Corollary 6.2 Let (u, p) be the exact solution of the Stokes system (2.1), with
p ∈ H 1 (Ωint ), and let (uh , ph ) be its discontinuous Galerkin approximation (2.6) on
n,σ
a geometric edge mesh T = Tedge
or a geometric boundary layer mesh T = Tbln,σ ,
with a grading factor σ ∈ (0, 1) and n levels of refinement. Let the stabilization
functions δ and γ be defined as in (3.2), (3.3) and (3.4), respectively. Then,
1
|||(u − uh , p − ph )|||DG ≤ Ck 2
inf
(v,q)∈Wh
|||(u − v, p − q)|||DG + C Rh (u, p),
with a constant C > 0 that depends on Ω, δ0 , γ0 , and the constants in Property 3.2
and (3.1), but is independent of ν, k, `, n, and the aspect ratio of the anisotropic
elements in T . Here, Rh (u, p) is the residual
Rh (u, p) =
sup
(w,s)∈Wh
|Ah (u, p; w, s) − Lh (w, s)|
.
|||(w, s)|||DG
Let us make precise the abstract error bound above for a smooth solution (u, p) ∈
H s+1 (Ω)3 × H s (Ω), s ≥ 1, on isotropically refined meshes with mesh-size h with
possible hanging nodes and for mixed-order elements where ` = k − 1.
In this case, the residual Rh (u, p) can be bounded (see Proposition 8.1 of Ref. 24)
by
R
R
| F {{ν∇u − T (ν∇u)}} : [[w]] ds| + | F {{p − T (p)}}[[w]] ds|
Rh (u, p) ≤ sup
,
|||(w, s)|||DG
(w,s)∈Wh
where T and T are the L2 -projections onto Σh and Qh , respectively. The CauchySchwarz inequality and standard hp-approximation properties then give
hmin{s,k} 1
− 21
2
Rh (u, p) ≤ C
ν kuks+1 + ν kqks .
1
k s+ 2
Furthermore,
inf
(v,q)∈Wh
|||(u − v, p − q)|||DG ≤ C
hmin{s,k}
1
k s− 2
1
2
ν kuks+1 + ν
− 12
kqks
and thus
|||(u − uh , p − ph )|||DG ≤ C
hmin{s,k} 1
− 12
2 kuk
.
ν
+
ν
kqk
s+1
s
k s−1
(6.9)
This estimate is optimal in the mesh-size h and suboptimal in k by one power of k
in the velocity and by a power k 3/2 in the pressure, respectively.
Similarly to Sect. 8 of Ref. 24, we obtain a slightly better result on conforming
meshes, that is,
hmin{s,k} 12
− 12
|||(u − uh , p − ph )|||DG ≤ C
ν
kuk
+
ν
kqk
.
(6.10)
s+1
s
1
k s− 2
We point out that the a-priori error bounds (6.9) and (6.10) hold verbatim for
equal-order elements.
Stabilized hp-DGFEM for incompressible flow
21
Remark 6.3 We note that the dependence on the polynomial degree k in (6.9) and
(6.10) is better than in the hp-estimates of Ref. 24 for mixed-order Q k − Qk−1
elements without pressure stabilization, by half an order of k in the velocity and a
full order of k in the pressure, respectively.
Acknowledgments
The first author was partially supported by the Swiss National Science Foundation under Project 21-068126.02. The last two authors were partially supported by
the Swiss National Science Foundation under Project 20-63397.00.
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