DISCONTINUOUS GALERKIN METHODS FOR
INCOMPRESSIBLE ELASTIC MATERIALS
BERNARDO COCKBURN, DOMINIK SCHÖTZAU, AND JING WANG
Abstract. In this paper, we introduce and analyze a local discontinuous
Galerkin method for linear elasticity. The distinctive feature of the method is
that it enforces strongly the equation that links the pressure to the divergence
of the displacement. As a consequence, when the material is exactly incompressible, the displacement is also exactly incompressible. This is achieved
without having to deal with the almost impossible task of constructing finite dimensional subspaces of incompressible displacements. Instead, a simple
post-processing is used which takes advantage of the special structure of the
method. We provide an error analysis of the method that holds uniformly
with respect to the Poisson ratio. In particular, we show that displacement
converges in L2 with order k + 1 when polynomials of degree k > 0 are used.
We also display numerical experiments confirming that the theoretical orders
of convergence are actually achieved and that there is no degradation in its
approximating properties when the material becomes incompressible.
November 19, 2004
1. Introduction
In this paper, we introduce, analyze and numerically test a new local discontinuous Galerkin (LDG) for the equations of linear elasticity. The main feature
of this method is that it can handle both compressible and exactly incompressible
materials. In the latter case, it provides an exactly incompressible approximation
to the displacement. There is no other numerical method for these equations with
these properties.
We study the method under consideration as applied to the equations of linear
three-dimensional isotropic elasticity,
(1.1)
(1.2)
(1.3)
ij = (ui,j + uj,i )/2
−ij,j + p,i = fi /(2 µ)
ε p + ui,i = 0
in Ω,
in Ω,
in Ω,
where we have used the standard notation of tensor analysis, with the boundary
conditions
(1.4)
(1.5)
ui = uDi
( ij − p δ ij ) nj = gi /(2 µ)
on ΓD ,
on ΓN .
2000 Mathematics Subject Classification. 65N30.
Key words and phrases. local discontinuous Galerkin methods, linear elasticity, incompressible
materials.
The first author was supported in part by the National Science Foundation (Grant DMS0411254) and by the University of Minnesota Supercomputing Institute. The second author was
supported in part by the Natural Sciences and Engineering Council of Canada (NSERC).
1
2
Here, ni denotes the i-th component of the the unit outward normal to the boundary
of Ω, and
(1.6)
µ=
E
2 (1 + ν)
and
ε=
(1 − 2 ν)(1 + ν)
,
Eν
where E is the Young modulus and ν ∈ (0, 1/2] is the Poisson ratio. For simplicity,
we assume that the area of the Dirichlet boundary ΓD and that of the Neumann
boundary ΓN are both different from zero. The Galerkin least-squares method
introduced and studied in [11] was based on the above set of equations; it provided
approximations for the strain , the displacement ~u and the pressure p.
Note also that when the Poisson ratio is equal to 1/2, the material becomes
incompressible and the equations become identical to those of the Stokes system
for incompressible fluid flow; in this case, p becomes the hydrostatic pressure and
~u the velocity of the fluid. For this very reason, the LDG method introduced in
this paper is an extension of the LDG method introduced in [8] for the steady state
Navier-Stokes equations. Indeed, such a method was the first to provide exactly
divergence-free approximations of the velocity in two space dimensions by using
polynomial approximations of degree one or higher. To achieve this, the task of
having to construct finite dimensional subspaces of exactly incompressible velocities is completely avoided. Instead, the local structure of the method was exploited
to construct a simple and efficient element-by-element post-processing with which
the exactly divergence-free approximation is computed from the discontinuous approximation. (In [3], a similar idea was used in the framework of flow in porous
media to obtain an approximation to the velocity field in H(div, Ω).) In this paper,
we extend the above procedure to LDG methods in several directions. First, we
consider the more complicated equations of elastic materials in three space dimensions; in [8], the two-dimensional case was treated. Second, we approximate all
the components of all the variables by polynomials of degree at most k; in [8], the
pressure and velocity components were approximated with polynomials of degree
k − 1 and k, respectively. Finally, we present an error analysis in which there is no
need to prove the classical inf − sup condition for the discrete spaces.
Let us briefly describe our main result. We show how to obtain an approximate
displacement, ~u h , that satisfies the equation
h
ε p h + ~ui,i
= 0,
pointwise inside each element (ph is the approximate pressure), and that, if the
mesh does not have any hanging node, has continuous normal components across
inter-element boundaries. Moreover, we show that it converges optimally and uniformly in the Poisson ratio in the L2 -norm. As a consequence, when the material is
incompressible, the approximation of the displacement ~u h is an optimally convergent, exactly incompressible approximation. No other numerical method for linear
incompressible elasticity has these properties. In [12], two DG methods based on
the interior penalty approach (see the discussion about DG methods in [2]) were
obtained whose convergence properties are also invariant as the Poisson ratio ν
tends to zero.
The paper is organized as follows. In Section 2, we describe the LDG method and
state and discuss our main results: Proposition 2.4, which contains two remarkable
properties of the post-processed displacement, and Theorem 2.5, which contains a
priori error estimates for the LDG method under consideration. Section 3 is devoted
Discontinuous Galerkin methods for incompressible elastic materials
3
to the proof of Theorem 2.5 and Section 4 to numerical results that verify that the
theoretical orders of convergence are sharp and that the approximation properties
of the method are independent of the value of the Poisson ratio. We end in Section
5 with a few concluding remarks.
2. Main results
In this section, we devise the LDG method and show it is well defined. Then, we
describe the post-processing leading to a new approximation to the displacement.
Finally, we state and briefly discuss our main result, Theorem 2.5.
2.1. Weak formulation and numerical traces. The LDG method seeks an approximation (h , ~u h , ph ) to the exact solution (, ~u, p) in the space Σh × V~ h × Qh ,
where
(2.1)
Σh = {Symmetric S ∈ L2 (Ω)3×3 : S|K ∈ Pk (K)3×3 for all K ∈ T h },
(2.2)
~
V
(2.3)
Qh = {q ∈ L2 (Ω) : q|K ∈ Pk (K) for all K ∈ T h }.
h
= {~v ∈ L2 (Ω)3 : ~v |K ∈ Pk (K)3 for all K ∈ T h },
Here, T h denotes a triangulation of Ω made of tetrahedra K. To define such an
approximation, we need a weak formulation we describe next.
To obtain it, we formally multiply the equations solved by the exact solution by
smooth test functions (S, ~v , q) and integrate over the element K. We obtain
(ij , S ij )K = −(ui , S ij,j )K + hui , S ij nj i∂K
(ij , vi,j )K − hij , nj vi i∂K − (p, vi,i )K + hp, ni vi i∂K =
1
(fi , vi )K ,
2µ
(ε p, q)K − (ui , q,i )K + hui , ni qi∂K = 0,
where
(p, ψ)K :=
Z
p ψ dx,
K
hp, ψi∂K :=
Z
p ψ dγ,
∂K
and ni is the i-th component of the outward unit normal to the boundary of K.
Notice that to obtain the first equation, we have strongly exploited the fact that
the test function S is symmetric.
Now, if we add the above equations, we get
(ij , S ij )Ωh = −(ui , S ij,j )Ωh + hui , S ij nj i∂Ωh ,
(ij , vi,j )Ωh − hij , nj vi i∂Ωh − (p, vi,i )Ωh + hp, ni vi i∂Ωh =
1
(fi , vi )Ωh ,
2µ
(ε p, q)Ωh − (ui , q,i )Ωh + hui , ni qi∂Ωh = 0,
where
Ωh = {interior of K : K ∈ Th }
These are the equations we desired.
and
∂Ωh = {∂K : K ∈ Th }.
4
The approximate solution (h , ~u h , ph ) is determined by requiring that, for all
~ h × Qh ,
(S, ~v , q) ∈ Σh × V
(2.4)
(2.5)
(2.6)
(hij , S ij )Ωh = −(uhi , S ij,j )Ωh + hb
uih,T , S ij nj i∂Ωh ,
1
(fi , vi )Ωh ,
(hij − ph δ ij , vi,j )Ωh − hb
ijh − pbh δ ij , nj vi i∂Ωh =
2µ
(ε ph , q)Ωh − (uhi , q,i )Ωh + hb
uih,p , ni qi∂Ωh = 0,
where u
b h,T , b
h− I pbh and u
b h,p are the so-called numerical traces which we define
next.
To do that, we need to introduce some notation. We denote by Eho the set of
inter-element boundaries. We say that the set e belongs to Eho if it is the intersection
of the boundaries of any two elements K + and K − and if it has nonzero area. We
denote the trace of the function ϕ to the boundary of the element K ± by ϕ± . The
average of these traces is denoted by
1
{{ϕ}} := (ϕ+ + ϕ− ).
2
We also define the jump of the function (along the i-th canonical direction) on the
inter-element boundaries as
− −
[[ϕ ni ]] = ϕ+ n+
i + ϕ ni
where n±
i is the i-th component of the outward unit normal to the boundary of
K ± . For convenience, on the border of Ω, we take
[[ϕ ni ]] = ϕ ni .
We are now ready to define the numerical traces. On inter-element boundaries,
we take
(2.7)
(2.8)
(2.9)
u
bih,T = {{uhi }} + c` [[uhi n` ]],
b
ijh − δ ij pbh = {{Tijh }} − δ ij {{ph }} − cj [[hi` n` ]] + δ ij d` [[ph n` ]] − κσ [[uhi nj ]],
u
bh,p
= {{uhi }} + di [[uh` n` ]] + κp [[ph ni ]].
i
On the Dirichlet part of the boundary of Ω, ΓD , we take
(2.10)
(2.11)
(2.12)
u
bih,T = uDi ,
b
ijh − δ ij pbh = Tijh − δ ij ph − κσ (uhi − uDi ) nj ,
u
bih,p = uDi ,
and on the Neumann part, ΓN ,
(2.13)
(2.14)
(2.15)
u
bih,T = uhi ,
b
ijh − δ ij pbh = gi nj /(2µ),
u
bih,p = uhi .
This completes the definition of the DG method.
Let us point out that the choice of the vector-valued functions ~c and ~d can affect
the sparsity of the matrices associated to the method and even its accuracy; see the
discussion in [2] in the framework of the Laplace equation. However, these functions
do not have any impact on the existence and uniqueness of the approximate solution.
Discontinuous Galerkin methods for incompressible elastic materials
5
As we see next, this property only depends on a simple positivity condition on the
real-valued functions κσ and κp .
Proposition 2.1. If ε > 0, the approximate solution ( h , ~u h , ph ) provided by the
method under consideration exists and is unique provided the function κ σ is strictly
positive and κp is nonnegative. If ε = 0, the approximate solution exists and is
unique (up to a constant in the pressure) provided that both functions κσ and κp
are strictly positive.
2.2. Proof of Proposition 2.1. Let us prove this result. To do that, let us
rewrite the method in compact form. It is not difficult to see that the definition of
the approximate solution ( h , ~u h , ph ) can be rewritten as follows
(2.16)
a(h , S) + b(~u h , S) = F (S),
(2.17)
−b(~v , h ) + c(~u h , ~v ) + d(ph , ~v ) = G(~v ),
−d(q, ~u h ) + e(ph , q) = H(q),
(2.18)
~
for all (S, ~v , q) ∈ Σh × V
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
h
× Qh . Here, the bilinear forms are given by
a(, S) = (ij , S ij )Ωh ,
T
b
b(~u, S) = (ui , S ij,j )Ωh − hu
bi , [[S ij nj ]]iEho ∪ΓN
b
b ij iE o ∪ΓD
= −(ui,j , S ij )Ωh + h[[ui nj ]], S
h
c(~u, ~v ) = hκσ [[ui nj ]], [[vi nj ]]iEho ∪ΓD
pb, [[vj nj ]]iEho ∪ΓD
d(p, ~v ) = −(p, vj,j )Ωh + hb
p
vbj iEho ∪ΓN
= (p,j , vj )Ωh − h[[p nj ]], b
e(p, q) = (ε p, q)Ωh + hκp [[p nj ]], [[q nj ]]iEho ,
where b
rb is obtained from rb by dropping the terms that do not depend on r. The
linear forms are given by
(2.26)
F (S) = huDi , S ij nj iΓD ,
(2.27)
G(~v ) = hκσ uDi , vi iΓD + h
(2.28)
H(q) = huDi ni , qiΓD .
1
g i , v i iΓN ,
2µ
The above identities follow from simple algebraic manipulations. However, the
following result can help to understand why the bilinear forms b(·, ·) and d(·, ·) can
be written in two different ways.
Proposition 2.2. Set
(
{{φ}} + c[[φ nj ]]
φb =
φ
Then
on Eho ,
on ΓN ,
and
b?
ψ =
(
{{ψ}} − c[[ψ nj ]]
ψ
on Eho ,
on ΓD .
b [[ψ nj ]] iE o ∪Γ = −(φ,j , ψ)Ω + h [[φ nj ]], ψb? iE o ∪Γ .
(φ, ψ,j )Ωh − h φ,
N
D
h
h
h
We can immediately see why is it that b(·, ·) can be written in two ways when we
take φ = ui and ψ = S ij . Similarly for the form d(·, ·), it is enough to take φ = p
6
and ψ = vj . To prove this result, we are going to need three trivial but important
identities displayed in the following lemma.
Lemma 2.3. We have
(i)
(φ, ψ,j )Ωh
(ii)
hφ nj , ψi∂Ωh \∂Ω = h1, [[φ nj ψ]]iEho ,
(iii)
= −(φ,j , ψ)Ωh + hφ nj , ψi∂Ωh ,
[[φ nj ψ]]
on Eho .
= {{φ}} [[nj ψ]] + [[φ nj ]] {{ψ}}
Proof of Proposition 2.2. Let Θ be the left-hand side of the identity we want to
prove. Then
b [[nj ψ]]iE o ∪Γ
Θ = − (φ,j , ψ)Ωh + hφ nj , ψi∂Ωh − hφ,
N
h
b nj ψi∂Ω \Γ ,
= − (φ,j , ψ)Ωh + hφ nj , ψi∂Ωh − hφ,
D
h
b nj , ψi∂Ω \∂Ω
= − (φ,j , ψ)Ωh + hφ nj , ψiΓD + h(φ − φ)
h
b nj , ψi∂Ω \∂Ω ,
= − (φ,j , ψ)Ωh + hφ nj , ψb? iΓD + h(φ − φ)
h
by (i),
b Γ = φ,
since φ|
N
since ψb? |ΓD = ψ. It remains to show that
But since
b nj , ψi∂Ω \∂Ω = h[[φ nj ]], ψb? iE o .
Ψ := h(φ − φ)
h
h
b nj ψ]]iE o
Ψ =h1, [[(φ − φ)
h
by (ii),
b [[ψ nj ]] + [[φ nj ]] {{ψ}}iE o
=h1, ({{φ}} − φ)
h
=h[[φ nj ]], {{ψ}} − c[[ψ nj ]]i
o
Eh
by (iii),
by definition of φ|Eho ,
=h[[φ nj ]], ψb? iEho ,
by the definition of ψb? |Eho , the desired identity follows. This completes the proof.
With the above introduced notation, we can rewrite the definition of the approximate solution in a very compact form. Indeed, (h , ~u h , ph ) is the element of the
~ h × Qh such that
space Σh × V
(2.29)
Eh (h , ~u h , ph ; S, ~v , q) = Fh (S, ~v , q)
~
∀ (S, ~v , q) ∈ Σh × V
h
× Qh ,
where
(2.30)
(2.31)
Eh (, ~u, p; S, ~v , q) = a(, S) + b(~u, S)
− b(~v , ) + c(~u, ~v ) + d(p, ~v ) − d(q, ~u) + e(p, q),
Fh (S, ~v , q) = F (S) + G(~v ) + H(q).
We are now ready to prove Proposition 2.1.
Proof of Proposition 2.1. The existence and uniqueness of the approximate solution
follows from the fact that the unique solution of the problem
Eh (h , ~u h , ph ; S, ~v , q) = 0
∀ (S, ~v , q) ∈ Σh × V~
h
× Qh ,
is the trivial solution. Taking (S, ~v , q) := (h , ~u h , ph ) in the above equation, we get
that
a(h , h ) + c(~u h , ~u h ) + e(ph , ph ) = 0.
Discontinuous Galerkin methods for incompressible elastic materials
7
When ε > 0, κσ is a strictly positive function and κp is a non-negative function,
and we immediately obtain that
h = 0,
[[uhi nj ]]|Eh = 0
ph = 0.
and
It remains to show that ~u h = ~0 in Ωh . Since we now have that
~
∀ (S, ~v , q) ∈ Σh × V
Eh (0, ~u h , 0; S, ~v , q) = 0
h
× Qh ,
taking (~v , q) = (~0, 0), we get
b(~u h , S) = 0
∀ S ∈ Σh .
Now, since ~u h is continuous in Ω, the above equation becomes
(uhi,j , S ij )Ωh = 0.
Taking S ij = (uhi,j + uhj,i ), we obtain that
uhi,j + uhj,i = 0 in Ωh ,
and so, ~u h can only be a rigid motion. But since ~u h = ~0 on ΓD and since we
assumed that the area of ΓD is different form zero, we must have that ~u h = ~0 in Ω
as desired. Notice that we have not used any information about the pressure.
When ε = 0, both functions κσ and κp are strictly positive functions, and we
immediately obtain that
h = 0,
[[uhi nj ]]|Eh = 0,
and
[[ph ]]|Eho = 0.
We show that ~u h = ~0 exactly as for the case ε > 0. It remains to show that ph is
equal to zero. Since we now have that
Eh (0, ~0, p; S, ~v , q) = 0
~
∀ (S, ~v , q) ∈ Σh × V
h
× Qh ,
taking (S, q) = (0, 0), we get
d(ph , ~v ) = 0
~ h,
∀ ~v ∈ V
and, since ph is continuous in Ω, the above equation becomes
(ph,i , vi )Ωh − hph , vi ni iΓN = 0
~ h.
∀ ~v ∈ V
We see that we have to take into consideration whether we are in a tetrahedron in
the set
Ωh,N = {K ∈ Ωh : area(∂K ∩ ΓN ) 6= 0)},
or not. Indeed, taking vi = ph,i on Ωh \ Ωh,N , we obtain that p is a constant, say
p, on Ωh \ Ωh,N . Since we are assuming that the area of the Neumann boundary
ΓN is different from zero, the set Ωh,N is not empty and hence there is at least
one tetrahedron K in Ωh,N with a face F` = {~x ∈ K : λ` (~x) = 0} on which the
pressure is constant. Working on this tetrahedron, we are going to show that p = 0
and that ph |K = 0. This immediately implies that the pressure ph is equal to zero
on Ωh . (Notice that if the Neumann boundary is empty, so is the set Ωh,N . As a
consequence, the pressure ph is only defined up to a constant, as expected.)
Thus, on K, we can write
ph = p + λ` p,
8
where p is a polynomial of degree k − 1. Thus, for a test function ~v with support
in K, we have
0 = (ph,i , vi )K − hph , vi ni i∂K∩ΓN
= −(λ` p, vi,i )K + hλ` p, vi ni i∂K − hp + λ` p, vi ni i∂K∩ΓN
= −(λ` p, vi,i )K + hλ` p, vi ni i∂K\(ΓN ∪F` ) − hp, vi ni i∂K∩ΓN .
Since ~v is a generic element of Pk (K)3 , it is possible (by using the BDM projection,
see, for example, [5]) to construct it in such a way that
−vi,i = p, in K,
v i ni = 0
−vi ni = p
As a consequence,
on ∂K \ (ΓN ∪ F` ),
on ∂K ∩ ΓN .
0 = (λ` , p2 )K + h1, p2 i∂K∩ΓN ,
and so, p = 0 and p=0. This implies that ph = 0 on K, as desired. As a consequence,
ph is equal to zero on Ωh,N .
This completes the proof of Proposition 2.1.
2.3. The post-processed displacement ~u h . Next, let us show how to postprocess the approximate solution ( h , ~u h , ph ) to obtain another approximation to
the displacement, ~u h .
The post-processing is a slight modification of the so-called Raviart-Thomas
projection; see [14] and [5]. Indeed, on each tetrahedron K ∈ Th , the displacement
~u h is the only element of the space Vk (K) = Pk (K)3 + ~x Pk (K), such that
(2.32)
(2.33)
(uih − uih , v)K = 0
hu`h −
u
b`h,p , n` viF
v ∈ Pk−1 (K), i = 1, 2, 3,
= 0 v ∈ Pk (F ), for each face F of ∂K.
Proposition 2.4. We have that
(i)
(ii)
~u h ∈ H(div, Ω)
h
εp +
uhi,i
=0
for any Th without hanging nodes,
in Ωh .
Notice that, thanks to this result, the computation of the displacement ~u h is
particularly simple when the material is incompressible and . Indeed, if we have a
hierarchical basis for the spaces P` (K):
P` (K) = span{Bi , i = 0, · · · , `},
we can write
h
and since ~ui,i
Vk (K) = Pk−1 (K)3 + span{Bk }3 + ~x spanBk ,
= 0, we have that ~u h belongs to
Hence, we have that
Vk (K) = Pk−1 (K)3 + span{Bk }3 .
~
~u h = Pk−1 ~u h + U,
~ ∈ span{Bk }3 . Here Pk−1 denotes the componentwise L2 -projection into
for some U
the space of polynomials of degree k−1. Since the equations (2.32) are automatically
~ is determined by solving the equations
satisfied, the vector U
~ ` , n` viF = h−(Pk−1 ~u h )` + u
hU
b h,p , n` viF = 0 v ∈ Pk (F ),
`
Discontinuous Galerkin methods for incompressible elastic materials
9
for any three of the four faces F of ∂K. Thus, we only have to invert a matrix of
order 3 (k + 1) (k + 2)/2.
Proof of Proposition 2.4. Since the Raviart-Thomas projection is well defined, so
is the displacement ~u h . Now, the property (i) is satisfied if the normal component
of the displacement is continuous across inter-element boundaries. This is ensured
by (2.33) and by the fact that the triangulation Th does not have hanging nodes.
To show that (ii) holds, we only have to insert in the equation (2.6), namely,
(ε ph , q)Ωh − (uhi , q,i )Ωh + hb
uih,p , ni qi∂Ωh = 0
for all q ∈ Qh ,
the definition of ~u h . Indeed, since on each tetrahedron K, q is a polynomial of
degree k, using the definition of the post-processing, (2.32) and (2.33), we can
rewrite the above equation as follows:
(ε ph , q)Ωh − (uhi , q,i )Ωh + huhi , ni qi∂Ωh = 0
for all q ∈ Qh .
Integrating by parts, we get
(ε ph + uhi,i , q)Ωh = 0
for all q ∈ Qh ,
and, since ε ph + uhi,i ∈ Qh , (ii) follows. This completes the proof.
2.4. The a priori error estimates. To state our result, we need to introduce some
notation. First, let us describe the norms and seminorms in which we are going to
measure the approximation errors. The so-called energy seminorm is defined as
|(S, ~v , q)|Eh := (Eh (S, ~v , q; S, ~v , q))
1/2
.
The L2 -norm of a function q in the domain D is denoted by k q k0,D ; when D
coincides with Ω, we simply write k q k0 . For any non-negative integer ` and any
real-valued function q, we define
k q k−` =
where k ϕ k` =
P
`
m=0
| ϕ |2m
1/2
(q, ϕ)Ω
,
ϕ∈C ∞ (Ω) k ϕ k`
sup
and | ϕ |m =
For a vector-valued function ~v , we define
k ~v k−` =
where k φ k` =
S, we define
P
3
i=1
k φi k2`
1/2
sup
~ ∞ (Ω)3
φ∈C
α
∂ 1
|α|=m ∂xα1
P
(vi , φi )Ω
,
k φ k`
∂ α2 ∂ α3
∂y α2 ∂z α3
2 1/2
ϕ
.
0
. Finally, for a symmetric matrix-valued function
k S k−` =
sup
Φ∈C ∞ (Ω)3×3
Φ symmetric
(S ij , Φij )Ω
,
k Φ k`
1/2
P
3
2
.
where k Φ k` =
i,j=1 k Φij k`
We are also going to need information about elliptic regularity results for the
system under consideration as well as about surjectivity properties of the divergence
10
operator. We begin with the elliptic regularity properties. We say that the problem
under consideration is (` + 2)-regular if the solution (R, w,
~ r) of the dual problem
Rij = (wi,j + wj,i )/2
−Rij,j + r,i = φi
in Ω,
in Ω,
ε r + wi,i = 0
in Ω,
with boundary conditions wi = ηi on ΓD and ( Rij − r δ ij ) nj = gi on ΓN , is such
that
~ ~g , ~η ) |||` ,
|(R, w,
~ r)|`+2 ≤ C ||| (φ,
where
1/2
,
|(R, ~v , r)|`+2 = |R|2`+1 + |~v |2`+2 + |r|2`+1
~ ~g , ~η ) |||` = kφk
~ ` + k~gk`+1/2,Γ + k~
||| (φ,
η k`+3/2,ΓD
N
If the boundary of Ω is smooth enough, these results follow from the general elliptic
regularity results in [1]. Moreover, the estimate also holds for exactly incompressible
materials. Indeed, it holds uniformly in the Poisson ratio, as was shown in [16] in
the two-dimensional case; the three dimensional case follows by a similar argument.
Next, we consider the property of surjectivity of the divergence operator. We
say that the domain Ω is `-regular if for all ϕ ∈ L2 (Ω), there is a vector-valued
function w
~ ∈ (H01 (Ω))3 such that
div w
~ =ϕ
and
kwk
~ `+1 ≤ C kϕk` .
The case ` = 0 and Ω Lipschitz is very well known; see [15]. It is not difficult to
show that it also holds for ` > 0 if the boundary of Ω is smooth enough.
Finally, we are going to need the following function defined on the set of faces
of the triangulation Th , Eh :
(
max{hK + , hK − }
on ∂K + ∩ ∂K − ,
h=
hK
on ∂K ∩ ∂Ω.
Here, hK denotes the diameter of the tetrahedron K. As usual, we assume that
the triangulation Th is regular.
We are now ready to state our main result.
Theorem 2.5. Let ` be any non-negative integer. Assume that the domain Ω is
`-regular and that the boundary problem under consideration is (` + 2)-regular. Let
(, ~u, p) be the solution of the problem (1.1), (1.2), (1.3) with boundary conditions
(1.4), (1.5), and let (h , ~u h , ph ) be the solution of the LDG method with
κσ = κp −1 = h.
Then, for s ∈ (0, `),
|( − h , ~u − ~u h , p − ph )|Eh ≤ C hmin{k,s+1} ||| (f~/2µ, ~g/2µ, ~uD ) |||s ,
k~u − ~u h k−` ≤ C hmin{k,s+1}+min{k,`+1} ||| (f~/2µ, ~g/2µ, ~uD ) |||s ,
k − h k−` + kp − ph k−` ≤ C hmin{k,s+1}+min{k,`} ||| (f~/2µ, ~g/2µ, ~uD ) |||s ,
Moreover, if ~u h is the post-processed displacement, we have that
k~u − ~u h k0 ≤ C hmin{k,s+1}+1 ||| (f~/2µ, ~g/2µ, ~uD ) |||s ,
kui,i − ui,i k−` ≤ ε C hmin{k,s+1}+min{k,`+1} ||| (f~/2µ, ~g/2µ, ~uD ) |||s .
Discontinuous Galerkin methods for incompressible elastic materials
11
The constant C is independent of the Poisson ratio ν.
Notice that when the exact solution is smooth, the above result states that the
displacement converges with the optimal order convergence of k +1 in the L2 -norm;
the strain and pressure converge with order k in the L2 -norm which is suboptimal
by one. Notice also that the displacement converges with order 2 k in the H −k+1 norm. This implies that, if the triangulation is locally translation invariant and
the exact solution locally very smooth, a local post-processing of the displacement
results in a new approximation that converges with order 2 k; see [4]. A similar
remark can be made for the strain and the pressure as both converge with order
2 k in the H −k -norm.
Concerning the post-processed displacement ~u h , the above result states that it
has the same convergence rate in L2 as the displacement ~u h . Let us recall that, by
Proposition 2.4, it satisfies
ε ph + uhi,i = 0 in Ωh .
and belongs to the space H(div, Ω) whenever Th does not have hanging nodes.
We thus see that in the exactly incompressible case, this approximation of the
displacement is exactly incompressible.
3. Proof of Theorem 2.5
To prove this result, we use the technique adopted to analyze a wide class of DG
methods for the Laplacian in [6]. In this paper, we extend the approach to a much
more complex elliptic problem and we further refine it. This approach allows us to
obtain error estimates not only in the energy norm and in the L2 -norms but also
in negative-order norms without any extra effort. Since it uses a duality technique,
it also allows us to bypass having to prove an inf − sup condition for the discrete
problem; instead, it completely relies on the corresponding inf − sup condition for
the original continuous problem; see also [10]. In contrast, the approaches used, for
example, in [2] and [13] (for second order equations), in [9] (for the Stokes system),
and more recently in [8] (for the Navier-Stokes equations) all rely on the elimination
of one of the variables and cannot easily obtain error estimates in negative-order
norms for all the variables.
To prove our result, we proceed as follows. First, we describe the main steps
of the error analysis and show that all the error estimates for the approximation
(h , ~u h , ph ) follow from three simple properties related to a single functional denoted by K(·, ·, ·). This functional captures the continuity properties of the bilinear
form Eh as well as the approximation properties of our finite element spaces. Once
this has been established, we actually find such an functional and conclude. Finally,
we prove the estimates for the post-processed displacement ~u h .
3.1. The main steps of the error analysis. In this section we sketch the main
steps of our error analysis. The approach we take here is closely related to the one
developed in [6] where the analysis of the LDG method for the Laplacian was carried
out. The main reason of using such an approach is that it allows us to obtain error
estimates in all the variables not only in the energy norm but in negative-order
norms as well.
As it is customary, we split the error (e , ~eu , ep ) := ( − h , ~u − ~u h , p − ph ) as
the sum
(e , ~eu , ep ) = (ξ , ξ~u , ξp ) + (Pe , P~eu , Pep ),
12
where P is the componentwise L2 -projection into Qh . Notice that the projection
error (ξ , ξ~u , ξp ) = ( − P, ~u − P~u, p − Pp), can be easily estimated. To estimate
the other part of the error, we first use an energy argument and then a duality
argument.
To do so, we use five essential properties. The first is the classical Galerkin
orthogonality property which in this case reads
(3.1)
Eh (e , ~eu , ep ; S, ~v , q) = 0
∀(S, ~v , q) ∈ Σh × V~ h × Qh .
The second is a reflection of the definition of the bilinear form Eh (·, ·) and tells us
how to switch around its arguments:
(3.2)
Eh (, ~u, p; S, ~v , q) = Eh (S, −~v , q; , −~u, p),
for all (, ~u, p), (R, w,
~ r) ∈ H 1 (Ωh )3×3 × H 1 (Ωh )3 × H 1 (Ωh ).
The next two properties are continuity properties of the bilinear form Eh (·; ·).
We assume that there is a functional K such that
(3.3)
| Eh (ξ , ξ~u , ξp ; ξ , ξ~w , ξr ) | ≤ K(ξ , ξ~u , ξp ) K(ξ , ξ~w , ξr ),
R
R
for all (, ~u, p), (R, w,
~ r) ∈ H 1 (Ωh )3×3 × H 1 (Ωh )3 × H 1 (Ωh ), and such that
(3.4)
| Eh (S, ~v , q; ξ R , ξ~w , ξr ) | ≤ |(S, ~v , q)|Eh K(ξ R , ξ~w , ξr ),
for all (S, ~v , q) ∈ Σh × V~ h × Qh and (R, w,
~ r) ∈ H 1 (Ωh )3×3 × H 1 (Ωh )3 × H 1 (Ωh ).
The last property is a reflection of the approximation properties of the finite
element spaces:
(3.5)
K(ξ , ξ~u , ξp ) ≤ C hmin{k,s+1} |(, ~u, p)|s+2 ,
for s ≥ 0. We are now ready to display the main steps of the error analysis.
Step 1: Estimate of the energy seminorm of the error. We have the
following result.
Lemma 3.1. We have
| (Pe , P~eu , Pep ) |Eh ≤ K(ξ , ξ~u , ξp ),
| (e , ~eu , ep ) |Eh ≤ 2 K(ξ , ξ~u , ξp ).
Proof. Since | · |Eh is a seminorm,
| (e , ~eu , eq ) |Eh ≤| (ξ , ξ~u , ξq ) |Eh + | (Pe , P~eu , Peq ) |Eh
≤K(ξ , ξ~u , ξp ) + | (Pe , P~eu , Peq ) |Eh ,
by assumption (3.3), the second estimate follows from the first. Let us prove the
first estimate. Set Θ = | (Pe , P~eu , Peq ) |2Eh , then
Θ =Eh (Pe , P~eu , Peq ; Pe , P~eu , Peq )
=Eh (−ξ , −ξ~u , −ξq ; Pe , P~eu , Peq )
=Eh (Pe , −P~eu , Peq ; −ξ , ξ~u , −ξq )
by Galerkin orthogonality (3.1),
by the property (3.2),
≤| (Pe , P~eu , Pep ) |Eh K(ξ , ξ~u , ξp ),
by assumption (3.4), and so
| (Pe , P~eu , Peq ) |Eh ≤ K(ξ , ξ~u , ξp ).
This completes the proof.
Discontinuous Galerkin methods for incompressible elastic materials
13
Step 2: Estimate of the error in the velocity. We have the following result.
Lemma 3.2. Let ` be a non-negative integer. Then we have
K(ξ R , ξ~w , ξr )
k ~eu k−` ≤ 2 K(ξ , ξ~u , ξp )
sup
,
~ k`
~ ∞ (Ω)3
kφ
φ∈C
where (R, w,
~ r) is the solution of the dual problem
Rij = (wi,j + wj,i )/2
−Rij,j + r,i = φi
in Ω,
in Ω,
ε r + wi,i = 0
in Ω,
with boundary conditions wi = 0 on ΓD and ( Rij − r δ ij ) nj = 0 on ΓN .
Proof of Lemma 3.2. By definition of the solution of the dual problem and definition of the form Eh (·, ·), we have
1
for all (S, ~v , q) ∈ H (Ωh )
we get
Eh (R, w,
~ r; S, ~v , q) = (φi , vi )Ω ,
3×3
× H 1 (Ωh )3 × H 1 (Ωh ). Taking (S, ~v , q) = (e , ~eu , ep ),
(φi , vi )Ω =Eh (R, w,
~ r; e , ~eu , ep )
=Eh (e , −~eu , ep ; R, −w,
~ r)
by the property (3.2),
=Eh (e , −~eu , ep ; ξ R , −ξ~w , ξr ),
by Galerkin orthogonality (3.1). Then
(φi , vi )Ω =Eh (Pe , −P~eu , Pep ; ξ R , −ξ~w , ξr ) + Eh (ξ , −ξ~u , ξp ; ξR , −ξ~w , ξr )
≤|(Pe , −P~eu , Pep )|Eh K(ξ R , −ξ~w , ξr ) + K(ξ , −ξ~u , ξp ) K(ξ R , −ξ~w , ξr ),
by assumptions (3.3) and (3.4). The result follows from the first inequality of
Lemma 3.1 and the definition of the norm under consideration. This completes the
proof.
Step 4: Estimate of the error in the strain. We have the following result.
Its proof is analogous of that of the previous lemma.
Lemma 3.3. Let ` be a non-negative integer. Then we have
K(ξ R , ξ~w , ξr )
,
k e k−` ≤ 2 K(ξ , ξ~u , ξp )
sup
k Φ k`
Φ∈C ∞ (Ω)3×3
Φsymmetric
where (R, w,
~ r) is the solution of the dual problem
Rij = (wi,j + wj,i )/2 + Φij
−Rij,j + r,i = 0
ε r + wi,i = 0
in Ω,
in Ω,
in Ω,
with boundary conditions wi = 0 on ΓD and ( Rij − r δ ij ) nj = 0 on ΓN .
14
Step 5: Estimate of the error in the pressure.
Lemma 3.4. Let ` be a non-negative integer. Then we have
k ep k−` ≤ 2 K(ξ , ξ~u , ξp )
+ k eh k−`
inf
1
sup
~
0 (Ω)
ϕ∈C ∞ (Ω) w∈H
div w=ϕ
~
sup
inf
1
~
0 (Ω)
ϕ∈C ∞ (Ω) w∈H
div w=ϕ
~
3
3
K(0, ξ~w , 0)
k ϕ k`
kwk
~ `+1
.
k ϕ k`
Proof. Since
(ep , φ)Ω = (ep , div w)
~ Ω = −d(ep , w),
~
by definition of d(·, ·) (2.23), we easily obtain, by definition of Eh , that
(ep , φ)Ω = − Eh (e , ~eu , ep ; 0, w,
~ 0) + c(~eu , w)
~ − b(w,
~ e )
= − Eh (e , ~eu , ep ; 0, ξ~w , 0) + c(~eu , w)
~ − b(w,
~ e ),
by Galerkin orthogonality (3.1). Hence
(ep , φ)Ω = −Eh (Pe , P~eu , Pep ; 0, ξw , 0) − Eh (ξ , ξ~u , ξp ; 0, ξ~w , 0) + c(~eu , w)
~ − b(w,
~ e ).
But, since w
~ ∈ (H01 (Ω))3 ,
c(~eu , w)
~ = 0,
and, by definition of the form b(·, ·), (2.20),
~ `+1 k eh k−` .
−b(w,
~ e ) = (wi,j , (eh )ij )Ωh ≤ kwk
Moreover,
−Eh (ξ , ξ~u , ξp ; 0, ξ~w , 0) ≤ K(ξ , ξ~u , ξp ) K(0, ξ~w , 0),
by assumption (3.4). Finally,
−Eh (Pe , P~eu , Pep ; 0, ξw , 0) ≤ K(ξ , ξ~u , ξp ) K(0, ξ~w , 0),
by assumption (3.3) and by the first inequality of Lemma (3.1). This completes the
proof.
Step 6: Conclusion. We are now ready to prove the error estimates. Let
us being with the estimate in the energy seminorm. By Lemma 3.1 and by the
estimate (3.5), we get
| (e , ~eu , ep ) |Eh ≤ C hmin{k,s+1} |(, ~u, p)|s+2
≤ C hmin{k,s+1} ||| (f~/2µ, ~g/2µ, ~uD ) |||s
since the problem is s-regular for s ≤ `.
The remaining estimates follow in a similar way. For the error in the velocity
and in the strain, we have to use that
|(R, w,
~ r)|`+2 ≤ C kφk` ,
where ` ≥ 0, for the dual problem of Lemma 3.2, and
|(R, w,
~ r)|`+1 ≤ C kΦk` ,
where ` ≥ 0, for the dual problem of Lemma 3.3. The estimate for the pressure,
follows from Lemma 3.4, the estimate of the strain and the fact that Ω is `-regular.
Discontinuous Galerkin methods for incompressible elastic materials
15
In this way, to prove Theorem 2.5, we only have to prove the properties involving
the functional K, that is, the inequalities (3.3), (3.4), and (3.5). This is what we
are going to do in the next section.
3.2. The functional K.
a. Continuity properties. The equalities and inequalities that follow can be
easily obtained from the definitions of the bilinear forms under consideration; see
(2.19), (2.20), (2.21), (2.22), (2.23), (2.24), and (2.25).
For all (, ~u, p), (S, ~v , q) ∈ H 1 (Ωh )3×3 × H 1 (Ωh )3 × H 1 (Ωh ), we have
a(, S) ≤ {a(, ) a(S, S)}
c(~u, ~v ) ≤ {c(~u, ~u) c(~v , ~v )}
1/2
1/2
e(p, q) ≤ {e(p, p)) e(q, q)}
,
,
1/2
.
Moreover,
b(~u, S) ≤ k u k1 k S k0 + { c(~u, ~u) }
√ b
b k0,E \Γ ,
k hS
N
h
√
k hb
pb k0,Eh \ΓN .
1/2
d(p, ~v ) ≤ k p k0 k ~v k1 + { c(~v , ~v ) }1/2
Moreover, for (R, w,
~ r) ∈ H 1 (Ωh )3×3 × H 1 (Ωh )3 × H 1 (Ωh ) and (S, ~v , q) ∈ Σh ×
~ h × Qh , we have
V
b
ξ R )ij iEh \ΓN ,
b(~v , ξ R ) = h[[vi ni ]], (b
b
b(ξ~w , S) = −h(ξbw )i , [[S ij nj ]]iEh \ΓD .
As a consequence, we have
√ b
hb
ξR k0,Eh \ΓN ,
1 b
1/2
b(ξ~w , S) ≤ Cinv { a(S, S) }
k √ ξbw k0,Eh \ΓD ,
h
where we have used the following result.
1/2
b(~v , ξ R ) ≤ {c(~v , ~v )}
k
Lemma 3.5 (cf. [7]). There exists a positive constant Cinv that solely depends on
the shape regularity of the tetrahedra and the polynomial degree k such that for all
q ∈ Pk (K) we have
−1
kqk0,∂K ≤ Cinv hK 2 kqk0,K ,
for all K ∈ Th .
Similarly, we have
b
b
d(q, ξ~w ) = h[[q ni ]], (ξ~w )i iEh \∂Ω ,
and so,
b
d(ξr , ~v ) = −hξbr , [[vi ni ]]iEh \ΓN ,
d(q, ξ~w ) ≤ { e(q, q) }
d(ξr , ~v ) ≤ {c(~v , ~v )}
1/2
1/2
1 b
k √ ξbw k0,Eh \ΓD ,
h
√ b
k h ξbr k0,Eh \ΓN .
16
Hence, it is a simple matter to verify that a functional that satisfies the inequalities (3.3) and (3.4) is
n
√
K(ξ T , ξ~u , ξp ) =C k ξT k20,Ωh + k h ξ T k20,∂Ωh
1
+ k ξ~u k21,Ωh + k √ ξ~u k20,∂Ωh
h
o1/2
√
2
,
+(1 + ) k ξp k0,Ωh + k h ξp k20,∂Ωh
where C is a constant depending on Cinv only.
b. Approximation properties.
Now, a direct application of the lemma below gives us the the inequality (3.5).
Lemma 3.6 (cf. [7]). Let q ∈ H r+1 (K), r ≥ 0. Let P be the L2 -projection into
Pk (K). Then for m = 0, 1 we have
min{r,k}+1−m
|q − Pq|m,K ≤ ChK
kq − Pqk0,∂K ≤
kwkr+1,K ,
min {r,k}+ 12
kwkr+1,K ,
ChK
for some constant C that solely depends on the shape regularity of the tetrahedra
and the polynomial degree k.
To complete the proof of Theorem 2.5, it only remains to obtain the estimates
for the post-processed displacement.
3.3. Estimate of the error for the post-processed displacement. From Proposition 2.4, we have that
ph + ui,i = 0
in Ωh .
Hence, the estimate of the error in the divergence follows from that of the pressure.
To obtain the estimate of the error in the L2 -norm, we proceed as follows. On
the tetrahedron K, we have that
(uih − uih , v)K = 0
hu`h
−
This implies that
u
b`h,p , n` viF
v ∈ Pk−1 (K), i = 1, 2, 3,
= 0 v ∈ Pk (F ), for each face F of ∂K.
1/2
b
k~u h − ~u h k0,K ≤ C hK k~u
h,p
− ~u h k0,∂K ,
where C depends only on the shape regularity of the tetrahedron K. Hence, by the
b h,p ,
definition of the numerical trace ~u
k~u h − ~u h k0 ≤ C |(0, ~eu , ep )|Eh h ≤ C |(eT , ~eu , ep )|Eh h.
Hence, by the triangle inequality
k~u − ~u h k0 ≤ k~eu k0 + C |(eT , ~eu , ep )|Eh h,
and the result follows.
This completes the proof of Theorem 2.5.
Discontinuous Galerkin methods for incompressible elastic materials
17
4. Numerical experiments
In this section, we display several numerical experiments devised to test the
sharpness of the orders of convergence predicted by the theory, to verify that the
DG method performs as well for incompressible materials as it does for compressible
ones, and to compare the quality of the approximation of the displacement ~u h to
that of its post-processing ~u h .
To do this, we consider two test problems. One is associated with a compressible
material for which the Poisson ratio ν is taken to be 2/5. The other is associated
with an incompressible material. For both materials, we take the Young modulus E
equal to 8. For the compressible material, the displacement field and the pressure
are given by
1
sin2 (x),
2
u2 (x, y, z) = sin(x) sin(y),
u1 (x, y, z) =
u3 (x, y, z) = sin(x) sin(z),
1
p(x, y, z) = sin(x) (cos(x) + cos(y) + cos(z)) ,
ε
whereas for the compressible material, by
2
u1 (x, y, z) = − sin3 (x),
3
u2 (x, y, z) = sin2 (x) (y cos(x) − z sin(x)) ,
u3 (x, y, z) = sin2 (x) (z cos(x) + y sin(x)) ,
p(x, y, z) = sin(x).
The meshes we use are obtained by successive refinement; a mesh obtained by
refining ` times the coarsest mesh is called a mesh a level `. In Fig. 1, we display
the coarsest mesh (level 0) and its first refinement (level 1). In all our experiments,
we took
ci = di = 0.
We have experimented with other choices of these parameters and of the parameters
κσ and κp , but since we did not find any significant difference, we do not display
the corresponding results.
In Table 4, we display the history of convergence of the DG method for the
compressible material. The ratio
R :=
k~u − ~u h k0
,
k~u − ~u h k0
gives a measure of the relative quality of the two approximations we have for the
displacement. The quantity σ is the stress and is defined as
σ ij = 2 µ (ij − p δ ij ) .
We see that the method convergences with the orders predicted by the theory. We
also see that the approximation given by the post-processing of the displacement
field is not worse than that without it. In fact, it is interesting to see that the postprocessed displacement is better by a factor of around 1.7 for P1 and of around 2.0
for P3 ; no improvement is noticed for P2 .
18
Z
Z
X
X
Y
Y
Figure 1. The triangulations Th : The level 0 mesh (of only 6elements) (left) and the level 1 mesh (of 48-elements) (right). The
elements are tetrahedra.
In Table 4, we display the history of convergence of the DG method for the
incompressible material. We see results similar to those for the compressible material. The theoretical orders of convergence seem to be achieved. The post-processed
displacement, which in this case is exactly incompressible, seems to be better by a
factor 1.66 for P1 and by a factor 1.5 for P3 . For P2 the factor is bigger than one,
but seems to be going down to one as the mesh is refined.
It is interesting to actually see how the quality of the approximate displacement
improves when the mesh is refined and when the polynomial degree is increased. In
Fig. 1, we display the configuration of our incompressible material before deformation and the tetrahedral meshes. In Fig. 2, we show the material after deformation.
Notice that in that figure the meshes only have a graphical purpose; the meshes
used for the actual computations are the ones displayed in Fig. 1. We can easily
see that it is better to raise the polynomial degree than to refine the mesh.
5. Concluding remarks
Let us point out that although we have worked here with the LDG method, we
could have worked as well with any other DG method whose numerical traces are
consistent and conservative, as defined in [2]. We have to point out, however, that
for incompressible materials, exactly incompressible approximations can only be
obtained if the DG method has a structure similar to the LDG method considered
here. This cannot be achieved, for example, for DG methods that lack suitable
integrals in the border of the elements.
References
[1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic
partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl.
Math. 17 (1964), 35–92.
[2] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous
Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), 1749–1779.
[3] P. Bastian and B. Rivière, Superconvergence and H(div) projection for discontinuous
Galerkin methods, Internat. J. Numer. Methods Fluids 42 (2003), 1043–1057.
Discontinuous Galerkin methods for incompressible elastic materials
19
Table 1. History of convergence for the compressible material.
mesh
k~u − ~u h k0
order
ratio R
kp − ph k0
order
kσ − σ h k0
order
1.39
1.67
1.73
1.67
6.26e-01
2.91e-01
1.37e-01
6.76e-02
3.38e-02
1.11
1.08
1.02
1.00
1.79
2.45
2.53
3.32
8.08e-02
2.35e-02
5.30e-03
1.16e-03
2.42e-04
1.78
2.15
2.20
2.26
2.43
3.26
3.28
6.24
1.40e-02
2.91e-03
3.60e-04
4.38e-05
4.86e-06
2.27
3.01
3.04
3.17
P1
0
1
2
3
4
1.43e-01
4.78e-02
1.53e-02
4.52e-03
1.21e-03
1.58
1.64
1.76
1.90
1.24
1.73
1.65
1.66
1.65
1.15e-01
4.40e-02
1.38e-02
4.18e-03
1.31e-03
P2
0
1
2
3
4
1.13e-02
2.04e-03
2.12e-04
2.26e-05
2.69e-06
2.46
3.27
3.23
3.07
1.67
1.41
1.23
1.08
1.04
1.54e-02
4.47e-03
8.18e-04
1.42e-04
1.42e-05
P3
0
1
2
3
4
2.01e-03
2.80e-04
1.90e-05
1.23e-06
7.63e-08
2.84
3.88
3.95
4.01
2.29
2.04
1.99
1.99
1.99
2.51e-03
4.65e-04
4.85e-05
4.99e-06
6.58e-08
Table 2. History of convergence for the incompressible material.
mesh
k~u − ~u h k0
order
ratio R
kp − ph k0
order
kσ − σ h k0
order
0.88
1.19
1.31
1.35
3.69e-01
2.16e-01
1.24e-01
6.80e-02
3.58e-02
0.77
0.81
0.86
0.93
1.75
2.07
2.29
2.36
1.48e-01
5.77e-02
1.53e-02
3.45e-03
7.63e-04
1.36
1.91
2.15
2.18
2.07
2.63
2.83
13.36
3.87e-02
1.10e-02
1.75e-03
2.38e-04
2.46e-05
1.82
2.65
2.88
3.27
P1
0
1
2
3
4
1.27e-01
5.52e-02
2.12e-02
7.21e-03
2.15e-03
1.20
1.38
1.55
1.75
1.71
1.60
1.64
1.67
1.68
3.74e-02
2.03e-02
8.86e-03
3.59e-03
1.41e-03
P2
0
1
2
3
4
3.15e-02
8.04e-03
9.81e-04
9.91e-05
1.16e-05
1.97
3.04
3.31
3.09
1.70
1.53
1.31
1.07
0.97
2.22e-02
6.60e-03
1.57e-03
3.21e-04
6.23e-05
P3
0
1
2
3
4
7.71e-03
1.67e-03
1.17e-04
7.33e-06
4.55e-07
–
2.21
3.83
3.99
4.01
2.75
1.55
1.54
1.52
1.53
6.15e-03
1.46e-03
2.37e-04
3.33e-05
3.17e-09
20
Z
Z
X
Y
X
Y
Z
Z
X
Y
X
Y
Z
Z
X
Y
X
Y
Figure 2. Approximation of the domain after deformation: Results with the 6-element mesh (left column) and with the 48element mesh (right column). Polynomial degree of the displacement: k = 1 (top), k = 2 (middle) and k = 3 (bottom). (The
meshes we see here are refinements of the 6-element and the 48element meshes; they have been introduced for graphical purposes
only.)
Discontinuous Galerkin methods for incompressible elastic materials
21
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Bernardo Cockburn, School of Mathematics, University of Minnesota, Minneapolis,
MN 55455, USA
E-mail address: cockburn@math.umn.edu
Dominik Schötzau, Department of Mathematics, University of British Columbia, Vancouver, Canada
E-mail address: schoetzau@math.ubc.ca
Jing Wang, I.M.A., University of Minnesota, Minneapolis, MN 55455, USA
E-mail address: jwang@ima.umn.edu