Discontinuous Galerkin Methods for Second
Order Elliptic Boundary value Problems
Prof. Amiya K. Pani
Industrial Mathematics Group
Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai-400076.
1
Contents
1 Model Problem and Finite Element Galerkin Method.
2 Discontinuous Galerkin Methods
2.1 Notations and Approximation Properties.
2.2 Weak formulation. . . . . . . . . . . . . .
2.3 Discontinuous Galerkin Methods. . . . . .
2.4 Consistency of the Proposed Schemes . . .
2.5 Local conservation property. . . . . . . . .
3
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12
14
3 Error Estimates for NIPG and SIPG methods
3.1 Boundedness and Coercivity . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A Priori Error Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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DG Methods
4.1 A Priori Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Error Estimates for NCG methods . . . . . . . . . . . . . . . . . . . . . .
5 Numerical Examples
26
6 Nonlinear Elliptic Problems
28
2
In this lecture, we discuss the discontinuous Galerkin methods for approximating the solutions of second order linear elliptic boundary value problems. To start with, we review
quickly the finite element methods and then motivate the use of discontinuous Galerkin
method. Moreover, we plan to derive a priori error estimates.
1
Model Problem and Finite Element Galerkin Method.
Consider the following Neumann problem:
−∆u + u = f
∇u.n = g
in Ω,
on ∂Ω,
(1.1)
(1.2)
where Ω is a bounded domain with Lipschitz continuous boundary ∂Ω, f and g are given
∂u
functions. Here, n is the outward normal vector and ∇u.n = ∂n
.
Since finite element Galerkin method is based on a weak formulation of (1.1) - (1.2), we
derive below a weak form of (1.1) - (1.2).
Weak Formulation. Let L2 (Ω) be a space of square integrable functions define on Ω
with innerproduct
Z
1/2
Z
2
(v, w) =
vw dx and norm kvk =
|v| dx
.
Ω
Ω
Further, let H m (Ω) be a Hilbert Sobolev space of order m, i.e.,
H m (Ω) = {v ∈ L2 (Ω) : D α v ∈ L2 (Ω), |α| ≤ m},
with the innerproduct
X Z
(v, w)H m =
|α|≤m
and norm

D α v D α w dx,
Ω
X Z
kvkH m (Ω) = 
|α|≤m
Ω
1/2
|D α v|2 dx
.
When there is no confusion, we call k · km as norm k · kH m (Ω) The weak form of (1.1)-(1.2)
is to seek u ∈ H 1 (Ω) such that
Z
Z
Z
Z
∇u.∇v dx + u.v dx =
f v dx +
gv ds ∀v ∈ H 1 (Ω).
(1.3)
Ω
Ω
Ω
∂Ω
In abstract form: Find u ∈ H 1 (Ω) such that
∀v ∈ H 1 (Ω),
a(u, v) = L(v)
where the bilinear form
a(v, w) =
Z
(∇v.∇w + vw) dx,
Ω
3
(1.4)
and linear form
L(v) =
Z
f v dx +
Ω
Z
gv ds.
∂Ω
It is easy to check that the bilinear form satisfies
(i) Boundedness. For v and w ∈ H 1 (Ω)
Z
a(v, w) =
(∇v.∇w + vw) dx
Ω
≤
Z
1/2 Z
1/2 Z
1/2 Z
1/2
2
2
2
|∇v| dx
|∇w| dx
+
|v| dx
|w| dx
2
Ω
Ω
Ω
Ω
≤ kvk1 kwk1,
(1.5)
(ii) Coercivity. For v ∈ H 1 (Ω)
a(v, v) =
=
Z
2
|∇v| dx +
Ω
Z
|v|2 dx
Ω
kvk21 .
(1.6)
Further, for g ∈ L2 (∂Ω) and f ∈ L2 (Ω), the linear form satisfies
|L(v)| ≤ kf kkvk + kgkL2 (∂Ω) kvkL2 (∂Ω)
≤ C kf k + kgkL2 (∂Ω) kvkH 1 (Ω) .
(1.7)
Here using trace theorem; we have kvkL2 (∂Ω) ≤ CkvkH 1 (Ω) .
Using Lax-Milgram Theorem, the problem (1.4) has a unique solution u ∈ H 1(Ω) which
satisfies
kukH 1 (Ω) ≤ C kf k + kgkL2 (∂Ω) .
Galerkin Method. Let Th be a finite collection of Nh elements Ki , i = 1 · · · Nh (triangles or rectangles)such that
Ω=
Nh
[
K i and Ki ∩ Kj = φ, i 6= j.
i=1
Further if K i ∩ K j 6= φ, then K i and K j either have a common side or common vertex.
Let hK be the diameter of K and h = maxK∈Th hK .
Define
Sh = {vh ∈ C 0 (Ω) : vh |K ∈ Pr (K), K ∈ Th },
where Pr (K) is a polynomial of degree ≤ r define on K.
It is an easy exercise to see that Sh ⊂ H 1 (Ω). Here, Sh is a set of C 0 − piecewise polynomials
of degree ≤ r on each element K. Note that Sh satisfies the following approximation
property.
inf (kv − χk + hkv − χk1 ) ≤ Chr+1 kvkH r+1(Ω) , v ∈ H r+1 (Ω).
χ∈Sh
Now the finite element Galerkin approximation of u is to seek uh ∈ Sh such that
a(uh , vh ) = L(vh )
∀vh ∈ Sh .
4
(1.8)
Since Sh is finite dimensional, (1.8) leads to a system of linear algebraic equations. For
solvability of (1.8), it is enough to prove uniqueness. If zh and wh are two distinct solutions
of (1.8), then zh − wh satisfies
a(zh − wh , vh ) = 0.
With vh = zh − wh , we use coercivity to obtain
αkzh − wh k21 ≤ a(zh − wh , zh − wh ) = 0,
and hence zh = wh . This leads to a contradiction, and therefore, the solution uh of (1.8)
is unique. Further, with vh = uh in (1.8), the coercivity of the bilinear form a(·, ·) and the
boundedness of the linear form L yield
kuh k1 ≤ C kf kL2 (Ω) + kgkL2(∂Ω) .
This is known as stability result.
Convergence.
Since Sh ⊂ H 1 , we replace v in (1.4) by vh and subtract (1.8) to obtain
a(u − uh , vh ) = 0
∀vh ∈ Sh .
(1.9)
Using Coercivity, Galerkin orthogonality
αku − uh k21 ≤ a(u − uh , u − uh ) = a(u − uh , u − χ) for χ ∈ Sh
≤ ku − uh kH 1 (Ω) ku − χkH 1 (Ω) ,
and hence, as u − uh 6= 0
ku − uh kH 1 (Ω) ≤ inf ku − χkH 1 (Ω) .
χ∈Sh
Theorem ( Cea’s Lemma).
(1.8). Then
(1.10)
Let u and uh , respectively, be the solution of (1.4) and
ku − uh kH 1 (Ω) ≤ Chr kukH r+1(Ω) ,
(1.11)
ku − uh kL2 (Ω) ≤ Chku − uh kH 1 (Ω) ≤ Chr+1 kukH r+1(Ω) .
(1.12)
and
Proof.
In (1.10) use approximation property for the space Sh to complete the proof
of (1.11). For (1.12), we appeal to Aubin-Nitsche duality argument. For ψ ∈ L2 (Ω), let
φ ∈ H 1 (Ω) be the unique solution of
−∆φ + φ = ψ in Ω
∂φ
=0
on ∂Ω
∂n
5
(1.13)
which satisfies the following elliptic regularity result for 0 < δ ≤ 1
kφkH 1+δ (Ω) ≤ CkψkH −1+δ .
Taking inner-product between (1.13) and u − uh , we use the integration by parts and (1.9)
to obtain
(u − uh , ψ) = (u − uh , −∆φ + φ) = (∇(u − uh ), ∇φ) + (u − uh , φ)
= a(u − uh , φ − χ),
χ ∈ Sh
≤ Cku − uh k1 kφ − χk1 .
Using approximation property for the space Sh and elliptic regularity result, it follows that
(u − uh , ψ) ≤
≤
ku − uh kH 1 inf kφ − χkH 1
χ∈Sh
Ch ku − uh kH 1 kφkH 1+δ ≤ Chδ ku − uh kH 1 kψkH −1+δ
δ
Thus,
ku − uh kH 1−δ =
sup
ψ∈H −1+δ
(u − uh , ψ)
≤ C hδ ku − uh kH 1 ,
kψkH −1+δ
When δ = 1, we have the desired result as and this complete the rest of the proof.
Remark 1.1 For a convex and bounded polygonal domain Ω, the elliptic regularity result
for the dual problem is valid for δ = 1 and the results are valid for r = 1. When the
regularity of the solution u ∈ H 1+δ only with 0 < δ < 1, we have optimality in H 1−δ -norm,
that is, the error ku − uh kH 1−δ = O(h2δ ). However, using the above duality argument with
0 < δ < 1, we obtain ku − uh k = O(h2δ ), which seems to be sub-optimal.
2
Discontinuous Galerkin Methods
Looking at the construction of the finite element space Sh in section 1, we observe that
the elements in Sh satisfy the inter element continuity requirements that is the element
in Sh are continuous across the inter element boundaries. In 1973, Reed and Hill [11]
introduced the discontinuous Galerkin (DG) methods for hyperbolic problems by relaxing
the continuity requirement. Since that time there has been a great deal of development of
DG methods for hyperbolic and hyperbolic type problems, see for a recent survey Cockburn
et al. [5]. Over the last two decades the DG methods have also been applied to elliptic
and parabolic problems. The motivation for renewed interest in developing these methods
recently may be due to their flexibility in approximating globally rough solutions, their
mass conservation properties, their potential for error control and mesh adaptation etc.
In this section, we describe four discontinuous Galerkin methods for the model problem
(1.1)-(1.2) and discuss a priori error estimates.
6
2.1
Notations and Approximation Properties.
Let Th be a regular partition of Ω. Denote the edges of the Th by {e1 , e2 · · · ePh , ePh+1 , · · · eMh }
where ek ⊂ Ω, 1 ≤ k ≤ Ph and ek ⊂ ∂Ω, Ph+1 ≤ k ≤ Mh . For each edge ek = Ki ∩Kj , i > j
associate a unit normal nk outward from Ki so that n|i = −n|j
For s ≥ 0, let us define broken sobolev spaces H s (Th ) as.
H s (Th ) = {v ∈ L2 (Ω) : v|K ∈ H s (K), K ∈ Th }.
The associated norm on H s (Th ) is given as
X
kφks,h =
kφk2s,K
K∈Th
!1/2
.
For φ ∈ H s (Th ), s > 21 , we now define the average and the jump as follows :
For 1 ≤ k ≤ Ph let ek = ∂Ki ∩ ∂Kj (i > j) with nk as the external normal to Ki . Then the
average value of φ on ek
1
(φ|Ki )|ek + (φ|Kj )|ek ,
{φ} =
2
and the jump of φ on ek
[φ] = (φ|Ki )|ek − (φ|Kj )|ek .
For the integer r, let Dr (Th ) be a discontinuous finite element space given by
Y
Pr (K),
Dr (Th ) =
K∈Th
where Pr (K) is the space of polynomials of degree ≤ r on K.
Approximation Properties. For φ ∈ H s (K), there is a constant CA > 0 independent
of φ, r, h and a sequence zrh ∈ Pr (K), K ∈ Th such that for any 0 ≤ q ≤ s,
kφ − zrh kq,K ≤ CA
and
r−q
hK
s−q kφks,K ,
rK
s ≥ 0,
µ− 21 −l
kφ − zrh kl,ek ≤ CA
hK
s− 21 −l
kφks,K ,
s ≥ 1/2,
rK
where µ = min(r + 1, s) and ek ⊂ ∂K.
Trace Inequalities. Let Th be a family of regular shape triangulation, i.e., there exists
ρ > 0 independent of the triangulation such that hK ≤ ρρK , where ρK is the diameter of
the largest ball inscribed in K. Then for all φ ∈ H 1 (K)
1
2
2
2
kφkL2 (∂K) ≤ CT
kφkL2 (K) + hK k∇φkL2 (K)
hK
and for φ ∈ H 2 (K)
∂φ
k k2L2 (∂K) ≤ CT
∂n
1
2
2
k∇φkL2 (K) + hK |φ|H 2(K) ,
hK
7
where |φ|H m (K) is the seminorm given by

|φ|H m (K) = 
X
|α|=m
1/2
kD α φk2L2 (K) 
.
For a proof, we refer to Prudhomme et al. [10].
Inverse Inequality. For χ ∈ Pr (K), then
kχkL2 (∂K) ≤ CI
rK
1/2
hK
and
k∇χ.nK kL2 (∂K) ≤ CI
Further
k∇χkL2 (K) ≤ CI
kχkL2 (K)
rK
1/2
hK
k∇χkL2 (K) .
2
rK
kχkL2 (K) .
hK
For a proof, we refer to Riviere et al. [12].
2.2
Weak formulation.
For the derivation of weak formulation of (1.1), let us assume for a moment that u is a
sufficiently smooth function. Multiply (1.1) by v ∈ H 2 (Th ) and integrate over the domain
Ω to obtain
Z
Z
{−∇.∇u + u}v dx =
f v dx.
Ω
Ω
Now using Th , decompose the integral as
X Z
X Z
{−∇.∇u + u}v dx =
f v dx.
K∈Th
K
(2.1)
K
K∈Th
Using Gauss divergence theorem for the first term on the right hand side of (2.1), we find
that
X Z
X Z
(∇u.n)v ds
(∇u.∇v + uv)dx −
K∈Th
K
∂K
K∈Th
=
X Z
K∈Th
f v dx.
(2.2)
K
Note that
X Z
K∈Th
(∇u.n)v ds =
∂K
Ph Z
X
k=1
+
Z
Mh
X
k=Ph +1
8
[(∇u.n)v]ds
ek
ek
gv ds,
(2.3)
where [·] is the jump across ek .
If a, b, c and d are real numbers, then write
1
1
ac − bd = (a + b)(c − d) + (a − b)(c + d).
2
2
So in a similar way, we write on each ek
[(∇u.n)v] = {∇u.n}[v] + [∇u.n]{v}.
(2.4)
From (2.3)-(2.4)
X Z
K∈Th
Ph Z
X
(∇u.n)vds =
∂K
k=1
({∇u.n}[v] + [∇u.n]{v}) ds
ek
Z
Mh
X
+
gv ds.
(2.5)
ek
k=Ph +1
Observe that if u ∈ H 2 (Ω), then the fluxes [∇u.n] are continuous almost everywhere in Ω
and hence ,
Ph Z
X
k=1
∀v ∈ H 2 (Th ).
[∇u.n]{v}ds = 0
(2.6)
ek
Therefore, from (2.2), (2.5) and (2.6), we arrive at
XZ
K∈Th
Ph Z
X
(∇u.∇v + uv)dx −
K
{∇u.n}[v]ds
ek
k=1
X Z
=
K∈Th
f v dx +
K
Z
Mh
X
k=Ph +1
gv ds.
ek
We now define a general discontinuous weak formulation of (1.1)- (1.2) as:
B(u, v) − J(u, v) = F (v),
(2.7)
where the bilinear for B(·, ·) on H 2 (Th ) × H 2 (Th ) and the linear form F on H 2 (Th ) are
respectively, defined by
X Z
(∇φ.∇ψ + φψ)dx
(2.8)
B(φ, ψ) =
K∈Th
K
and
F (ψ) =
X Z
K∈Th
f ψ dx +
K
Z
Mh
X
k=Ph +1
9
ek
gψ ds.
(2.9)
Moreover, we set the bilinear form J on H 2 (Th ) × H 2 (Th ) as
J(φ, ψ) =
Ph Z
X
{∇φ.n}[ψ]
(2.10)
ek
k=1
For Various discontinuous Galerkin methods to be defined below, we modify the bilinear
for (2.7) by adding a zero term. However, we observe that in the discrete formulation this
additional term need not be zero.
For u ∈ H 1 (Ω) ∩ H 2 (Th ), the jump [u] = 0 on each interior interfaces ek , i.e,
Z
[u]vds = 0
∀v ∈ L2 (ek ).
ek
Thus, for v ∈ H ( Th )
Z
{∇v.n}[u]ds = 0
k = 1, · · · Ph .
ek
Note that using the definition of J in (2.10)
J(v, u) = 0
2.3
∀v ∈ H 2 (Th ) and u ∈ H 1 (Ω) ∩ H 2(Th ).
Discontinuous Galerkin Methods.
Different DG methods are proposed by adding J(v, u) or −J(v, u) term to the bilinear form
B(u, v) and also by adding penalty terms to impose the weak continuity condition on the
discrete solution across the interelement boundaries.
I. Global Element Methods-(GEM).
Subtract the term J(v, u) to the left hand side of (2.7) to obtain
B− (u, v) = B(u, v) − J(u, v) − J(v, u).
(2.11)
The corresponding Galerkin formulation consists in seeking uGEM
∈ Dr (Th ) such that
h
B− (uGEM
, χ) = F (χ)
h
χ ∈ Dr (Th ).
(2.12)
Note that the bilinear form B− (·, ·) is symmetric and the bilinear form need to be positive
semidefinite. Little can be proved for this method. This method was introduced by Delves
et al.[6, 7].
II. Symmetric Interior Penalty Galerkin Method (SIPG)
Since it is difficult to prove the coercivity of the bilinear form B− (·, ·), the global element
method is of little use. However, the penalty terms can be added to B− (·, ·) in order
to ensure coercivity. Note that the penalty terms are so chosen that the interelement
continuity conditions can be enforced weakly. Now introduce the penalty terms as
σ
J (u, v) =
Ph Z
X
K=1
10
ek
σ[u][v]ds,
where σ denotes the penalty parameter which depends on the length of the edges ek and
the degree of the polynomials used in the elements. Now the symmetric interior penalty
Galerkin (SIPG) approximation uSIPG
is to look for uSIPG
∈ Dr (Th ) satisfying
h
h
σ
B−
(uSIPG
, χ) = F (χ)
h
∀χ ∈ Dr (Th ),
(2.13)
σ
where B−
(φ, ψ) = B− (φ, ψ) + J σ (φ, ψ).
This method is a natural generalization of Nitshe’s method [8] for enforcing non-homogeneous
Dirichlet boundary condition weakly in the variational formulation of the second order elliptic boundary value problems. The method is analyzed by Wheeler [13] and Arnold [1].
Another variant of this method is proposed by Baker- Karakasshian [2, 3]. Based on
ac − bd = a(c − d) + (a − b)d.
The formulation may be modified as
B(u, v) − I(u, v) − I(v, u) + J σ (u, v) = F (v),
where
I(φ, ψ) =
Ph Z
X
k=1
(2.14)
∇φ.n[ψ]ds.
ek
III. Discontinuous hp- Galerkin Method (DGM).
This method differ from GEM by just a sign. The DGM is to find uDGM
∈ Dr (Th ) such
h
that
B+ (uDGM
, χ) = F (χ)
h
∀χ ∈ Dr (Th ),
(2.15)
where B+ (φ, ψ) = B(φ, ψ) − J(φ, ψ) + J(ψ, φ).
Note that B+ (·, ·) is not symmetric. The DG method with this bilinear form was first
introduced by Baumann [4] and Oden et al.[9]. In [4], it was shown that this method is
pointwise conservative and stable in one dimension for polynomial of at least degree three.
However, it is difficult to prove stability for higher dimensions.
A variant of this method as called Non-symmetric Constrained Galerkin (NCG) method
may be defined as: Find the discrete approximation uNCG
∈ Dr⋆ (Th ) such that
h
B+ (uNCG
, χ) = F (χ)
h
∀χ ∈ Dr⋆ (Th ),
(2.16)
where the constrained discrete space
Dr⋆ (Th )
= {v ∈ Dr (Th ) :
Z
[v]ds = 0 ∀k = 1, · · · , Ph }.
ek
This scheme is analyzed by Riviere et al.[8].
IV. Non Symmetric Interior Penalty Galerkin Method (NIPG).
By adding penalty terms in DGM, one obtains the following NIPG method: Find uNIPG
∈
h
Dr (Th ) such that
σ
B+
(uNIPG
, χ) = F (χ)
h
11
∀χ ∈ Dr (Th ),
(2.17)
σ
where B+
(φ, ψ) = B+ (φ, ψ) + J σ (φ, ψ). This method is proposed and analyzed by Rivier
et al. [12].
Remark: All the four schemes discussed above have some similarity except for a plus
and minus sign in front of the term J(v, u) and the addition of a penalty term J σ (u, v) or
not.
2.4
Consistency of the Proposed Schemes
In order to show that the discrete scheme is consistent, we first plug the exact solution
of (1.1)-(1.2)in the discrete formulation and check whether the exact solution satisfies the
discrete formulation at least asymptotically that is as the discretizing parameter tends to
zero. Below, we show this result only for NIPG methods and the remaining three follows
similarly .
Theorem 2.1 Let u ∈ C 2 (Ω) be a solution of (1.1)-(1.2). Then u satisfies the formulation
(2.17). Conversely, if u ∈ H 1 (Ω)∩H 2 (Th ) is a solution of (2.17), then u satisfies (1.1)-(1.2)
in the sense of distribution.
Proof. Let u ∈ C 2 (Ω) and v ∈ Dr (Th ). Multiply (1.1) by v and integrate over K and
then sum over all K ∈ Th . Thus,
X Z
K∈Th
(∇u.∇v + uv)dx −
K
Ph Z
X
k=1
[(∇u.n)vds
ek
Z
Mh
X
−
k=Ph +1
∇u.nv ds =
ek
XZ
K∈Th
f v dx.
K
Since [∇u.n]|ek = 0, k = 1, · · · Ph and ∇u.n = g on ek , k = Ph + 1, · · · Mh , we obtain easily
XZ
K∈Th
(∇u.∇v + uv)dx −
K
Ph Z
X
k=1
=
{∇u.n}[v]ds
ek
X Z
K∈Th
K
f v dx +
Z
Mh
X
k=Ph +1
gv ds.
ek
Since [u] = 0, we add J(v, u) = 0 and J σ (u, v) = 0 on the right hand side to show that u
satisfies
σ
B+
(u, v) = F (v)
∀v ∈ Dr (Th ).
Hence, we obtain first part of this proof.
To prove the converse, let D(K) ⊂ H 2 (K) be the infinitely differentiable functions with
compact support on K and let v ∈ D(K). Then (2.17) implies
Z
Z
(∇u.∇v + uv)dx =
f v dx
∀v ∈ D(K).
K
K
12
Using integration by parts, we obtain as u ∈ H 2(K)
Z
Z
(−∆u + u)vdx =
f v dx
K
∀v ∈ D(K).
K
Since v is an arbitrary element in D(K), we find that
−∆u + u = f
a.e inK.
(2.18)
Next we consider an interior edge ek where ek = K i ∩ K j , i > j. Set v ∈ H02 (Ki ∪ Kj ) ⊂
H 2 (Ki ) × H 2 (Kj ) extended by zero outside. Note that [u] = [v] = 0 on ek and hence
J(u, v), J(v, u) and J σ (u, v) vanish. Thus, the weak formulation (2.17) reduces to
Z
Z
(∇u.∇v + uv)dx =
f v dx.
(2.19)
Ki ∪Kj
Ki ∪Kj
Integrate the first term on the left hand side of (2.19) and use v ∈ H02 (Ki ∪ Kj ) to find that
Z
(−∆u + u)v dx −
Ki ∪Kj
Z
[∇u.n]v ds =
ek
Using (2.18) on Ki and Kj , we obtain
Z
[∇u.n]v ds = 0
Z
f v dx.
Ki ∪Kj
∀v ∈ H02 (Ki ∪ Kj ).
ek
Since v is arbitrary, [∇u.n] = 0 on each interior edges ek and ∇u ∈ H(div, Ω)⋆ .
1
Hence
−∆u + u = f.
a. e. in Ω
(2.20)
To recover the boundary condition, multiply (2.20) by v ∈ H 2 (Ω) ⊂ H 2 (Th ) and integrate
by parts to obtain
Z
Z
Z
(∇u∇v + uv)dx −
gv ds =
f v dx ∀v ∈ H 2 (Ω).
Ω
∂Ω
Ω
From the weak formulation (2.16) as the terms containing the internal boundaries vanish,
we arrive at
Z
Z
X Z
∂u
(∇u∇v + uv)dx −
v ds =
f v dx ∀v ∈ H 2 (Ω).
∂n
K
∂Ω
Ω
K∈T
h
Subtracting both the equations, we find that
Z
(∇u.n − g)vds = 0
∂Ω
and hence ∇u.n = g on ∂Ω. This completes the proof.
1
d
⋆ H(div, Ω) = {φ ∈ L2 (Ω) : ∇.φ ∈ L2 (Ω)}
13
∀v ∈ H 2 (Ω),
Theorem 2.2 If r ≥ 1, the discrete solutions of the last two methods (NCG, NIPG) exists
and is unique.
Proof. Since Dr (Th ) is finite dimensional, the corresponding discrete method leads to a
system of linear algebraic equations. So it is enough to prove uniqueness ( as uniqueness
in this case implies existence ).
For NCG method, choose f = 0, g = 0 and χ = uNCG
in (2.16), then
h
X Z 2
NCG |2 dx = 0,
|∇uNCG
|
+
|u
h
h
K∈Th
K
and hence uNCG
= 0 a.e on each K. The constraint on the discrete space implies that
h
NCG
uh
= 0 a.e in Ω and this completes the proof for NCG method.
in (2.17) to obtain
For NIPG method, choose f = 0, g = 0. Set χ = uNIPG
h
X Z 2
NIPG |2 dx + J σ uNIPG , uNIPG = 0.
|∇uNIPG
|
+
|u
h
h
h
h
K
K∈Th
The first term implies uNCG
= 0 a.e in K ∈ Th and from the second term
h
Z h
i2
NCG
uh
ds = 0,
ek
h
i
that is uNIPG
= 0. Therefore, uNIPG
= 0 a.e in Ω and this completes the proof.
h
h
2.5
Local conservation property.
One of the beauty of the DG methods described earlier is that on each element K, the mass
conservation property holds. For NIPG method, the mass conservation property can be
written as
Z
K
uNIPG
h
−
Z
∂K\∂Ω
{∇uNIPG
h
· n∂K } +
Ph Z
X
k=1
ek
σ[uNIPG
][1]ds
h
=
Z
f+
K
Z
g.
(2.21)
∂K∩∂Ω
Similarly, one can discuss conservation property for other DG methods.
3
Error Estimates for NIPG and SIPG methods
Let us recall that the SIPG (uSIPG
) and NIPG (uNIPG
) approximations are described, reh
h
SIPG
spectively, as the discrete solution uh
∈ Dr (Th ) satisfying
σ
B−
(uSIPG
, χ) = F (χ),
h
χ ∈ Dr (Th ),
14
(3.1)
and the discrete solution uNIPG
∈ Dr (Th ) satisfying
h
σ
B+
(uNIPG
, χ) = F (χ),
h
χ ∈ Dr (Th ).
(3.2)
For our error analysis, we need the following mesh dependent norms: For v ∈ H 1 (Th ),
define the energy norm
X
||v||21,K )1/2
||v||1,Th = (
k∈Th
and norm proposed by Baumann et al. [4, 9] and by Backer and Karakashian [3]:
|||v|||21,Th
=
||v||21,Th
+
Mh Z
X
k=1
3.1
2
σ[v] ds +
ek
Mh Z
X
k=1
ek
1
{∇v · n}2 ds.
σ
Boundedness and Coercivity
In order to apply a variant of Cea’s lemma, we need the boundedness and coercivity propσ
erties of the bilinear forms B±
(·, ·).
σ
Lemma 3.1 Let B±
(·, ·) be the bilinear forms defined either in (3.1) or in (3.2). Then for
2
all V and W ∈ H (Th )
σ
|B±
(v, w)| ≤ |||v|||1,Th |||w|||1,Th .
(3.3)
σ
Proof. From the definition of B±
(., .), we obtain
σ
B±
(v, w) = |B(v, w) − J(v, w)±J(w, v) + J σ (v, w)|
≤ |B(v, w)| + |J(v, w)| + |J(w, v)| + |J σ (v, w)|
Note that using Cauchy-Schwarz inequality
X Z
(|∇v||∇w|) + |v||w|)dx
|B(v, w)| ≤
K∈Th
K
= ||v||1,Th ||w||1,Th .
For the term J(v, w), we observe that
|J(v, w)| ≤
Ph Z
X
|{∇v · n}[w]|ds
k=1 ek
Ph Z
X
≤ (
k=1
ek
P
h
X
1
{∇v · n}2 ds)1/2 (
σ
k=1
Z
σ[w]2 ds)1/2 .
ek
Similarly,
|J(w, v)| ≤ (
Ph Z
X
k=1
ek
Ph Z
X
1
2
1/2
σ[v]2 ds)1/2 .
{∇w · n} ds) (
σ
k=1 ek
15
Moreover,
σ
|J (v, w)| =
Ph Z
X
|σ[v][w]|ds
ek
k=1
Ph Z
X
2
σ[v] ds)
≤ (
k=1
ek
1/2
(
Ph Z
X
k=1
σ[w]2 ds)1/2 .
ek
Altogether, we obtain the required bound for B± (·, ·) and this complete the proof.
σ
Although boundedness of the bilinear forms B±
(·, ·) is proved for elements in H 2 (Th ), but
σ
it seems difficult to prove coercivity of B±
(·, ·) for elements in H 2 (Th ). Below, we prove
coercivity only in the discrete spaces Dr (Th ).
κr 2
Lemma 3.2 (NIPG method). Let σ =
, where κ is a +ve constant. Then for all κ > 0,
h
there is a positive constant α > 0 independent of h, r such that
σ
B+
(vh , vh ) ≥ α|||vh |||21,Th
∀vh ∈ Dr (Th ).
(3.4)
Proof. For vh ∈ Dr (Th ) and α arbitrary real number
σ
B+
(vh , vh )
= B(vh , vh ) +
Ph Z
X
k=1
= (1 − α)
X
σ[vh ]2 ds
ek
||∇vh ||20,K + ||vh ||2 +
K∈Th
+
α|||vh|||21,Th
−α
Ph Z
X
k=1
Ph Z
X
k=1
ek
σ[vh ]2 ds
ek
!
1
{∇vh · n}2 ds.
σ
Let ek = K̄i ∩ K̄j , (i > j), Ki , Kj ∈ Th . Since {∇vh · n} be the average value of the flux
at ek , so we can split the integral
Z
Z
1
1
1
2
{∇vh · n} ds =
(∇vh · n|Ki + ∇vh · n|Kj )2 ds
4 ek σ
ek σ
Z
Z
1
1
1
1
2
≤
(∇vh · n|Ki ) ds +
(∇vh · n|Kj )2 ds.
2 ek σ
2 ek σ
With K = Ki or Kj , we obtain using trace inequality and inverse property
Z
Z
∂vh 2
1
1
2
|
(∇vh · n) ds =
| ds
σ ek ∂n
ek σ
Cr −1
(hK |vh |21,K + hK |vh |22,K )
≤
σ 2
1
rK
Cr
+ CI
||∇vh ||21,K .
≤
σ hK
hK
16
With σ =
2
κrK
, we obtain
hK
−
Z
ek
C
1
(∇vh · n)2 ds ≥ − ||∇vh ||21,K .
σ
κ
Since mesh size hKi and hKj and the polynomial degrees rKi and rKj are different in the
two elements Ki and Kj sharing the edge ek . We can choose σ as
σ=
2
2
max(rK
, rK
)
i
j
min(hKi , hKj )
to obtain
2
1
C rK
i
||∇vh ||20,Ki
(∇vh · n|Ki )2 ds ≤
σ
σ hKi
2
C min(hKi , hKj ) rK
i
||∇vh ||20,Ki
≤
2
2
κ max(rKi , rKj ) hKi
C
||∇vh ||20,Ki .
≤
κ
Z
ek
Similar estimate can be derived for Kj .
On summing up over ek
−α
Ph Z
X
k=1
ek
1
C
C
{∇vh · n}2 ds ≥ −α ||∇vh ||20,Th ≥ −α ||∇vh ||21,Th .
σ
κ
κ
Therefore,
C
σ
B+
(vh , vh ) ≥ α|||vh |||21,Th + (1 − α − α )|||vh |||21,Th .
κ
Choose α such that (1 − α − α Cκ ) > 0 that is
0<α<
1
C
1+
κ
.
σ
Then B+
(vh , vh ) ≥ α|||vh |||21,Th
∀vh ∈ Dr (Th ) and this completes the proof of coercivity
of the NIPG method.
Remark. It is straight forward to see that the bilinear form B+ (·, ·) in NIPG method
satisfies
B+ (v, v) = ||vh ||21,Th + J σ (v, v)
∀v ∈ H 2 (Th ).
κr 2
, where κ is a +ve constant. Then for all
Lemma 3.3 (SIPG method). Let σ =
h
κ > κ0 , there is a positive constant α > 0 independent of h, r such that
σ
B−
(vh , vh ) ≥ α|||vh |||21,Th
∀vh ∈ Dr (Th ).
17
(3.5)
σ
Proof. From the definition of B−
(·, ·), we note that for vh ∈ Dr (Th ) and for arbitrary real
α
σ
B−
(vh , vh ) = B(vh , vh ) − 2J(vh , vh ) + J σ (vh , vh )
= (1 − α)(||vh ||21,Th + J σ (vh , vh )) + α|||vh |||21,Th
Ph Z
Ph Z
X
X
1
{∇vh · n}[vh ]ds − α
− 2
{∇vh · n}2 ds.
σ
k=1 ek
k=1 ek
1
Using Cauchy-Schwarz inequality along with 2ab ≤ ǫa2 + b2 , ǫ > 0 we find that
ǫ
Ph Z
Ph Z
Ph Z
X
X
X
1
2
1/2
{∇vh · n}[vh ]ds ≤ 2(
σ[vh ]2 ds)1/2
2
{∇vh · n} ds) (
σ
k=1 ek
k=1 ek
k=1 ek
Z
P
h
X
1
1
≤ ǫ
{∇vh · n}2 ds + J σ (vh , vh ).
σ
ǫ
k=1 ek
(3.6)
(3.7)
As in the proof of Lemma (3.2), we now obtain
ǫ
Ph Z
X
k=1
ek
C
C
1
{∇vh · n}2 ds ≤ ǫ ||∇vh ||20,Th ≤ ǫ ||vh ||21,Th .
σ
κ
κ
(3.8)
From (3.6)-(3.8), we arrive at
C
1
σ
B−
(vh , vh ) ≥ (1 − α − (α + ǫ) )||vh ||21,Th + (1 − α − )J σ (vh , vh ) + α|||vh|||21,Th .
κ
ǫ
(3.9)
1
C
Now we need to find α > 0 such that (1 − α − (α + ǫ) ) > 0 and (1 − α − ) > 0.
κ
ǫ
1
Note that from the second inequality 0 < α ≤ 1− , i.e., ǫ must be greater than 1. Moreover
ǫ
from the first inequality, we arrive at
C
1−
κ ≤
0<α≤
C
1+
1+
κ
1−ǫ
C
κ = κ− C.
C
κ+C
κ
Choose κ sufficiently large, so that for κ > κ0 and κ > C. Then we obtain from (3.9) the
required estimate and this complete the rest of the proof.
3.2
A Priori Error Estimates.
In this subsection, we discuss a priori error estimates for both SIPG and NIPG methods.
Theorem 3.1 Let u ∈ H 1 (Ω) ∩ H s (Th ), s ≥ 2 be a solution of
σ
B±
(u, v) = F (v)
∀v ∈ H 2(Th ),
18
(3.10)
and let uh ∈ Dr (Th ) be the discrete solution of
σ
B±
(uh , χ) = F (χ)
∀χ ∈ Dr (Th ).
(3.11)
κr 2
, where κ > 0 for NIPG and κ ≥ κ0 > 0 for SIPG methods. Then the error
Let σ =
h
e = u − uh satisfies
hµ−1
≤ C s−3/2 ||u||H s(Th ) ,
r
||e||1,Th
(3.12)
where µ =min(r + 1, s) and r ≥ 1.
Proof. Using an interpolant Zrh of u in Dr (Th ), we now split e as e := u − uh = (u −
zrh ) − (uh − zrh ) = η − ξ. Since η estimate can be found out easily (see section 2), it
is sufficient to derive an estimate for ξ. Since Dr (Th ) ⊂ H 2 (Th ), we have the following
Galerkin orthogonality,
σ
B±
(u − uh , χ) = 0
χ ∈ Dr (Th )
(3.13)
Using coercivity (Lemmas 3.2-3.3) and Galerkin orthogonality (3.13) we now arrive at
|||ξ|||21,Th ≤
1 σ
1 σ
B± (ξ, ξ) = B±
(η, ξ)
α
α
σ
Using boundedness for the bilinear form B±
(·, ·) (see Lemma 3.1), we obtain
|||ξ|||21,Th ≤
1
|||η|||1,Th |||ξ|||1,Th .
α
Since ξ 6= 0 (if ξ = 0, the result holds trivially), we arrive at
|||ξ|||1,Th ≤ C|||η|||1,Th .
Using triangle inequality, we find that
|||η|||1,Th = |||u − uh |||1,Th ≤ C|||η|||1,Th .
(3.14)
To estimate |||η|||1,Th , we note that
|||η|||21,Th
=
X Z
K∈Th
+
Ph Z
X
k=1
2
2
(|∇η| + |η| )dx +
K
Ph Z
X
k=1
ek
1
{∇η · n}2 ds
σ
σ[η]2 ds.
(3.15)
ek
From the approximation properties stated in section 2, we obtain
Z
h2µ−2
2
(|∇η|2 + |η|2 )ds ≤ CA K
2s−2 ||u||s,K , s ≥ 1.
r
K
K
19
(3.16)
For the second term on the right hand side of (3.15), if ek = K̄i ∩ K̄j (i > j), then
Z
Z
1
1
1
2
{∇η · n} ds ≤
((∇η · n|Ki )2 + (∇η · n|Kj )2 )ds.
σ
2
σ
ek
ek
Let K is either Ki or Kj . So using trace inequality
Z
Z
1
1
C
∂η
2
2
(∇η · n) ds ≤
| |2 ds ≤ (h−1
K ||∇η||0,K + ||∇η||0,K |η|2,K ).
σ
σ
∂n
σ
ek
ek
Using approximation properties
Z
µ−2 1
hµ−1
C h2µ−3
2
K hK
K
||u||2s,K
(∇η · n) ds ≤
2s−2 + s−1 s−2
σ
σ
r
r
r
ek
K
K
K
C h2µ−3
2
K
≤
2s−3 ||u||s,K .
σ rK
Using the form of σ, we obtain
Z
1
C h2µ−2
2
K
(∇η · n)2 ds ≤
2s−1 ||u||s,K ,
σ
κ
r
ek
K
s ≥ 2.
(3.17)
For the last term on the right hand side of (3.15), we again rewrite it as
Z
Z
Z
Z
2
2
2
σ[η] ds =
σ(η|Ki − η|Kj ) ds ≤ 2
σ(η|Ki ) ds + 2
σ(η|Kj )2 ds.
ek
ek
ek
ek
With K as Ki or Kj , we obtain using approximation properties
Z
ση 2 ds ≤ Cσ
ek
h2µ−1
h2µ−2
2
2
K
K
||u||
≤
Cκ
s,K
2s−1
2s−3 ||u||s,K
rK
rK
(3.18)
On substitution of (3.16)-(3.18) in (3.15) and (3.14), we arrive at
|||e|||1,Th ≤ C|||η|||1,Th
X h2µ−2 h2µ−2 h2µ−2 1/2
K
K
K
||u||s,K
≤C
2s−2 + 2s−1 2s−3
r
r
r
K
K
K
K∈T
h
≤ C
X hµ−1
K
||u||s,K ≤ C
s−3/2
K∈Th
rK
hµ−1
||u||s,Th .
r s−3/2
As ||e||1,Th ≤ |||e|||1,Th , we now complete the rest of the proof.
r
Remarks. (i) For NIPG method using penalty parameter σ|k = σk
β on each ek ,
|ek |
k = 1, · · · , Ph ,, β ≥ (n − 1)−1 , where Ω ⊂ Rn . Riviere et al.[12] have derived optimal
estimates in both h and r with respect to norm || · ||1,Th that is
||u − uh ||1,Th ≤ C
20
hµ−1
||u||s,Th ,
r s−1
for pure Neumann problem (1.1)-(1.2).
(ii) For SIPG method, the optimal rate of convergence with respect to H 1 and L2 -norm in
h only is derived by Backer [2] and Arnold [1].
(iii) Using Aubin-Nitsche duality arguments and strengthening the penalty, i.e., using
β ≥ 3 for n = 2. Riviere et al. [12] have derived L2 -estimates which are optimal in h and
suboptimal in r. However, it has been observed experimentally that the L2 -estimates are
sub optimal and it is due to inconsistency of the adjoint problem which is require for the
Aubin-Nitche duality argument.
4
DG Methods
For the global element(GE) method, the bilinear form is not guaranteed be positive semidefinite and hence, it is difficult to prove any result. So in the rest of this section, we
concentrate on discontinuous hp-Galerkin methods(DGM). Compared to NIPG methods,
the penalty parameter σ is equated to be zero in these DG methods. Therefore, it is difficult
to prove coercivity and boundedness of the bilinear form B+ (·, ·) simultaneously using the
same norm. Note that for v ∈ H 2 (Th )
B+ (v, v) = kvk1,Th ,
and for v and w ∈ H 2 (Th )
|B+ (v, w)| ≤ |||v|||1,Th |||w|||1,Th .
4.1
A Priori Error Estimates
The main result of this section is stated in the following theorem.
Theorem 4.1 Let u ∈ H 1 (Ω) ∩ H s (Th ), s ≥ 2 be a solution of
∀v ∈ H 2 (Th ),
B+ (u, v) = F (v)
(4.1)
and let uh ∈ Dr (Th ) be the discrete DG solution of
B+ (uDG
, χ) = F (χ)
h
∀χ ∈, Dr (Th ).
(4.2)
Then the error e := u − uh satisfies
kek1,Th ≤ C
hµ−2
kvkH s (Th ) ,
r s−3/2
where µ = min(r + 1, s) and r ≥ 1.
Proof. Using an interpolation zrh of u in Dr (Th ), we split the error e as
e := u − uDG
= (u − zrh ) − (uDG
− zrh ) = η − ξ.
h
h
21
(4.3)
To obtain and estimate for ξ, we use Galerkin orthogonality and boundedness to obtain
kξk21,Th = B+ (ξ, ξ) = B+ (η, ξ)
= B(η, ξ) − J(η, ξ) + J(ξ, η)
≤ |B(η, ξ)| + |J(η, ξ)| + |J(ξ, η)|.
(4.4)
Note that
|B(η, ξ)| ≤ ||η||1,Th ||ξ||1,Th ,
(4.5)
and for J(ξ, η), we apply Cauchy-Schwartz inequality to find that
Z
Ph Z
X
2
1/2
( {∇ξ · n} ds) ( σ[η]2 ds)1/2 .
|J(ξ, η)| ≤
k=1
ek
(4.6)
ek
For ξ ∈ Dr (Th ) and the common edge ek = K̄i ∩ K̄j , we use inverse inequality with K = Ki
or Kj to obtain
Z
r2
{∇ξ · n}2 ds ≤ C K ||∇ξ||20,K ,
(4.7)
hK
ek
and using approximation property
Z
h2µ−1
2
2
2
[η] ds ≤ ||η||0,ek ≤ C K
2s−1 ||u||s,K
r
ek
K
(4.8)
Substituting (4.7)-(4.8) in (4.6), we arrive at
|J(ξ, η)| ≤ C
hµ−1
||u||s,Th ||ξ||1,Th .
r s−3/2
(4.9)
For J(η, ξ), we again use Cauchy-Schwartz inequality to obtain
Z
Ph Z
X
2
1/2
( {∇η · n} ds) ( σ[ξ]2 ds)1/2 ,
|J(η, ξ)| ≤
k=1
ek
(4.10)
ek
and for ek = K̄i ∩ K̄j , we find using approximation property
Z
{∇η · n}2 ds ≤ C
ek
h2µ−3
K
2s−3 ||u||s,K .
rK
Moreover from trace inequality, it follows that
Z
1
2
2
2
2
[ξ] ds = 2||ξ||0,ek ≤ C
||ξ||0,K + hK ||∇ξ||0,K
hK
ek
C
≤
||ξ||21,K .
hK
22
(4.11)
(4.12)
Substitute (4.11)-(4.12) in (4.10) to obtain
|J(η, ξ)| ≤ C
hµ−2
||u||s,Th ||ξ||1,Th .
r s−3/2
(4.13)
From (4.4), (4.5), (4.9) and (4.13), we arrive at
µ−1
h
hµ−1
hµ−2
||ξ||1,Th ≤ C
+ s−3/2 + s−3/2 ||u||s,Th
r s−1
r
r
µ−2
h
≤ C s−3/2 ||u||s,Th .
r
An appeal to triangle inequality with an estimate for ||η||1,Th , we complete the rest of the
proof.
Remark. The estimate of e in H 1 (Th )-norm is suboptimal in both h and r. If we trace
back, we note that the estimate in (4.13) is responsible in decreasing the overall order
of convergence. A new interpolant Πh of u with Πh u ∈ PrK (K), rK ≥ 2 is proposed in
Prudhomme et al. [10] which is defined as
Z
∇(u − Πh u) · nds = 0
∀ek ⊂ ∂K.
ek
With η = u − Πh u, we note that
||η||0,K + hK ||∇η||0,K ≤ C
hµK
s−3/2
rK
||u||s,K ,
s≥2
and
|η|2,K ≤ C
hµ−2
K
s−2 ||u||s,K ,
rK
where µ = min(rK + 1, s). For a proof, we again refer to Prudhomme et al. [10, pp. 26-30].
Thus using this new interpolant, and the estimates
|J(ξ, η)| ≤ C
hµ−1
||u||s,Th ||ξ||1,Th
r s−5/2
|J(η, ξ)| ≤ C
hµ−1
||u||s,Th ||ξ||1,Th ,
r s−7/2
and
we can easily prove the following result.
Theorem 4.2 Let u ∈ H 1 (Ω) ∩ H s (Th ), s ≥ 2 be a solution of (4.1) and uh be the discrete
solution of (4.2). Then there is a positive constant C independent of h and r such that
||e||1,Th ≤ C
hµ−1
||u||s,Th ,
r s−5/2
23
(4.14)
where µ =min(r + 1, s), r ≥ 2.
Remark. While the rate of convergence in (4.14) is optimal in h, but the rate of convergence in r is worse than the estimate in (4.3). Therefore, there is a possibility improving
the order of convergence in r, perhaps using a better interpolant. In Riviere et al. [12,
pp. 917-927], a special interpolant which is local to each element and whose average flux
is orthogonal to constants on each edge is introduced and the following estimate is proved:
||e||1,Th ≤ C
hµ−1
||u||s,Th ,
r s−5/2
where µ =min(r + 1, s). Now, we move to indicate the error analysis of Non symmetric
Constrained Galerkin (NCG) method.
4.2
Error Estimates for NCG methods
Let us recall that the non symmetric constrained Galerkin approximation uNCG ∈ Dr⋆ (Th )
is a solution of
B+ (uNCG
, χ) = F (χ) ∀χ ∈ Dr⋆ (Th ),
h
where
Dr⋆ (Th )
= {v ∈
Dr⋆ (Th )
:
Z
(4.15)
[v]ds = 0 ∀k = 1, 2, . . . , Ph }
ek
The consistency and local conservation properties are the consequences of the results
described in section 2. Following the analysis of Riviere et al.[12, pp. 913-915], we prove
the following theorem,
Theorem 4.3 Let u ∈ H s (Th ) and let uNCG
be a solution of (4.15). Then the error
h
e : u −→ uh satisfies
hµ−1
kek1,Th ≤ C s−5/2 kukH s(Th ) ,
(4.16)
r
where µ = min(r + 1, s) and r ≥ 1, s ≥ 2.
Proof: Not that the following Galerkin orthogonality relation holds:
∀χ ∈ Dr⋆ (Th ).
B+ (e, χ) = 0
(4.17)
Using an interpolant zrh of u in Dr⋆ (Th ), (see, section 2 for approximation properties), we
now split e as e : u − uNCG
= (u − zrh ) − (uNCG
− zrh ) = η − ξ.
h
h
A use of (4.17) with definition of B+ (·, ·) yields
kξk21,Th = B+ (ξ, ξ) = B+ (η, ξ)
≤ |B(η, ξ)| + |J(η, ξ)| + |J(ξ, η)|.
24
(4.18)
Since the estimate of |B(η, ξ)| and |J(ξ, η)| follows from (4.5) and (4.9), respectively, it
remains to estimate |J(η, ξ)|.
Note that
Ph
Ph Z
X
X
[ξ − ξ k ]ds,
{∇.n}[ξ]ds =
J(η, ξ) =
k=1
ek
k=1
where ξ k is the average value of ξ on ek = K i ∩ K j defined by
Z
1
ξk =
ξ|Ki ds.
|ek | ek
Observe that as ξ ∈ Dr⋆ (Th ), then
Z
ξ|Ki ds =
ek
Z
ξ|Kj ds.
ek
Using Cauchy-Schwarz inequality and approximation property we obtain
!1/2
Ph
X
hµ−3/2
,
|J(η, ξ)| ≤ C s−3/2 kukH s (Th )
k[ξ − ξ k ]k20,ek
r
k=1
with ek = K i ∩ K j ,
k[ξ]k0,ek = k[ξ − ξ k ]k0,ek ≤ k(ξ − ξ k )|Ki k0,ek + k(ξ − ξ k )|Kj k0,ek .
Thus it is sufficient to estimate k(ξ − ξ k )|Ki k0,ek . Passing to the reference element K̂, ek is
R
R
transformed to ê. Hence ξ k = 1ê ê ξˆ dŝ. Since the mapping fˆ → fˆ − 1ê ê fˆ dŝ is continuous
on H 1 (ê) and vanishes on constant functions, using Bramble- Hilbert Lemma, we obtain
ˆ 0,ê ,
ˆ ê ξk
kξˆ − ξ k0,ê ≤ k∇
k
ˆ ê is the tangential gradient on ê. Using inverse hypothesis (as h = 1) on the
where ∇
K̂
reference element
ˆ
ˆ ê ξk
kξˆ − ξ k k0,ê ≤ ĈK k∇
0,K̂ .
Thus,
k(ξ − ξ k )|Ki k0,ek ≤ Ĉ|ek |1/2 hKi |Ki |−1/2 rKi k∇ξk0,Ki
1/2
≤ ĈhKi rKi k∇ξk0,Ki .
On summing on k, we find that
Ph
X
k=1
On combining
k[ξ]k20,ek
=
Ph
X
k[ξ − ξ k ]k20,ek ≤ C
2
hK rK
k∇ξk0,K .
K∈Th
k=1
|J(η, ξ)| ≤ C
X
hν−1
kukH s (Th ) kξk1,Th ,
r s−5/2
and this completes the rest of the proof.
Remark. The result proposed in Theorem 4.2 is optimal in h, but suboptimal in r with
respect to broken H 1 - norm. Using Aubin-Nitsche duality argument, it is possible to discuss
L2 - error estimates which is optimal in h, but suboptimal in r.
25
5
Numerical Examples
In this section, we discuss the computations of DG approximate solution to the following
elliptic PDE.
−△u + u = f in Ω,
u = = 0 on ∂Ω,
where Ω = (0, 1) × (0, 1). we chose f in such a way that the analytical solution takes the
form u(x, y) = 10x(1 − x)y(1 − y). The domain Ω is devided into open rectangles K. On
each K, the local basis functions are taken to be 1, x, y, xy, ...., xr y r (0 ≤ r ≤ 3). Then it
is natural to define the global basis functions as
i j
φl = φij
on K, (0 ≤ i, j ≤ r)
K = x y
= 0 elsewhere.
h
Since {φl }N
j=1 forms a basis for Dr (Th ), the approximate solution takes the form
Nh
X
uh =
αj φ j .
j=1
Then, the bilinear form for NIPG and SIPG method takes the form
Nh
X
αj
j=1
XZ
K
∇φj · ∇φl + φj φl −
K
Mh Z
X
!
Mh Z
Mh Z
X
X
∂φj
∂φl
{
{
σ[φj ][φl ]
}[φl ] + θ
}[φj ] +
∂ν
ek ∂ν
e
e
k
k
k=1
k=1
k=1
and the linear form takes the form
Z
f φl ,
Ω
where φl is an arbitrary basis function of Dr (Th ). This leads to the following linear system
of equations
Aα = b
where A = (aij ), α = (αj ), b = (bj ) and
aij =
XZ
K
∇φj · ∇φl + φj φl −
K
Mh Z
X
k=1
!
Mh Z
Mh Z
X
X
∂φj
∂φl
{
{
σ[φj ][φl ]
}[φl ] + θ
}[φj ] +
∂ν
ek ∂ν
e
e
k
k
k=1
k=1
bj =
Z
f φj
Ω
We investigate the order of convergence in energy norm for NIPG(θ = 1) and SIPG(θ = −1)
methods. For fixed r, we compute |||u − uhi |||, hi = 1/4, 1/6, 1/8, 1/10 and the slope
26
2
NIPG
SIPG
1.8
1.6
1.4
1.2
0.4
1
0.4
0.8
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Figure 1: Order of convergence in ’h’ for NIPG and SIPG methods for r=1.
8
NIPG
SIPG
7.5
7
6.5
6
0.75
5.5
0.35
5
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Figure 2: Order of convergence in ’h’ for NIPG and SIPG methods for r=2.
27
7.4
7.2
NIPG
SIPG
7
6.8
6.6
6.4
0.5
6.2
6
0.2
5.8
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Figure 3: Order of convergence in ’h’ for NIPG and SIPG methods for r=3.
of the line log(hi ) → log(|||u − uhi |||). The slope is the order of convergence in h. The
computed order of convergence in h matches with the order of convergence obtained from
the theoretical results, see Fig(1), Fig(2), Fig(3). In a similar way, for a fixed h = 1/3, we
compute |||u−uri |||, ri = 1, 2, 3 and the slope of the line log(1/ri ) → log(|||u−uri |||). The
slope is the order of convergence in r. The computed order of convergence in r matches
with the order of convergence obtained in theoretical results, see Fig(4).
6
Nonlinear Elliptic Problems
In this section, we present the results on Symmetric Interior Penalty Galerkin (SIPG) and
Non-symmetric Penalty Galerkin (NIPG) methods for the following quasi-linear elliptic
problems on non-monotone type :
−∇ · (a(x, u)∇u) = f (x)
u(x) = g(x)
in Ω,
on ∂Ω,
(6.19)
(6.20)
where Ω is a bounded domain in IR2 with smooth boundary ∂Ω. There exists positive
constants α, M such that 0 < α ≤ a(x, v) ≤ M, x ∈ Ω̄, v ∈ IR. a(x, v) ∈ Cb2 (Ω̄ × IR),
where Cb2 (Ω̄ × IR) is the class of twice continuously differentiable functions on Ω̄ × IR with
all the derivatives of a(x, v) through second order are bounded in Ω̄ × IR. Further for some
δ ∈ (0, 1), f ∈ C δ (Ω) and g can be extended to Ω to be in C 2+δ (Ω),
28
4.6
NIPG
SIPG
4.4
4.2
4
0.2
0.2
3.8
3.6
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Figure 4: Order of convergence in ’r’ for NIPG and SIPG methods (h=1/3).
Acknowledgment. I would like to thank my Ph. D. students Mr. Thirupathi Gudi,
Mr. Sarvesh Kumar and Mr. Debasish Pradhan for providing their help during the preparation of this manuscript.
References
[1] D.N.ARNOLD, An interior penalty finite element method with discontinuous elements,
SIAM J. Numer. Anal., 19(1982), pp. 742-760.
[2] A. BAKER, Finite element methods for elliptic equations using non-confirming elements
, Math. Comp., 31(1977), pp. 45-59.
[3] A. BAKER, W.N. JUREIDINI AND O.A. KARAKASHIAN, Piecewise solenoidal vector fields
and the stokes problems , SIAM J. Numer. Anal., 26(1990), pp. 1466-1475.
[4] C.E. BAUMANN, An hp- adaptive discontinuous finite element method for computational
fluid dynamics, Ph.D Thesis, Univ. Texas Austin (1997).
[5] B. COCKBURN AND C. W. SHU, Runge-Kutta discontinuous Galerkin methods for convection dominated problems, J. Sci. Comput., 16(2001), pp. 173-261.
[6] L. M. DELVES AND C. A. HALL, An implicit matching principle for global element calculations, J. Inst. Math. Appl., 23(1979).
29
u
uh
1
1
0.5
0.5
0
1
0
1
1
0.5
0 0
1
0.5
0.5
0 0
uh
x
0.5
u
5
5
0
0
−5
1
−5
1
1
0.5
0 0
x
1
0.5
0.5
0 0
uhy
0.5
u
y
5
5
0
0
−5
1
−5
1
1
0.5
0 0
0.5
1
0.5
0 0
0.5
Figure 5: Approximate solution (first column) and exact solution (second column) along
with their partial derivatives with respect to x (second row) and with respect to y (third
row). (r=2, h=1/10)
30
[7] J. A. HENDRY AND L. M. DELVES , The Global element method applied to a harmonic
mixed boundary value problem, J. Comput. Phys, 33(1979).
[8] J.A. NITSCHE, ÜBER EIN VARIATION SPRINZIP ZUR LÖSUNG , Dirichlet- Problem bee
Verwendung von Teilräuman, die keinnen randbedingungen unteworfen sind, Abh.
Math. Sem. Univ. Hamburg., 36(1971), pp. 9-15.
[9] J.T.ODEN, I.BABUSKA AND C.E.BAUMANN, A discontinuous hp finite element method
for diffusion problems, J. Comput. Phys, 146(1998), pp. 491-519.
[10] S.PRODHOMME, F.PASCAL, AND J.T.ODEN, Review of error estimation for discontinuous galerkin methods, TICAM REPORT 00-27,October 17, 2000.
[11] 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, Los Alamos, NM, 1973.
[12] B.RIVIERE, M.F.WHEELER, AND V.GIRAULT, A priori error estimates for finite element
methods based on discontinuous approximation spaces for elliptic problems, SIAM J.
Numer. Anal., 39(2001), pp. 902-931.
[13] M.F.WHEELER, An elliptic collocation-finite element method with interior penalties,
SIAM J. Numer. Anal., 15(1978), pp. 152-161.
31