Delft Aerospace Computational Science ENGINEERING MECHANICS
REPORT
January 2005An H
1(P
h)-Coercive Discontinuous
Galerkin Formulation for the Poisson
Problem: 1-D Analysis
K.G. van der Zee and E.H. van Brummelen
K.G.vanderZee@lr.tudelft.nl
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TUD/LR/EM
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the Poisson Problem: 1-D Analysis
K.G. van der Zee and E.H. van Brummelen
Delft University of Technology, Faculty of Aerospace Engineering P.O. Box 5058, 2600 GB Delft, The Netherlands
ABSTRACT
Discontinuous Galerkin (DG) methods are finite element techniques for the solution of partial differential equations. They allow shape functions which are discontinuous across inter-element edges. In principle, DG methods are ideally suited forhp-adaptivity, as they handle nonconforming meshes and varying-in-space polynomial-degree approximations with ease.
Recently, DG formulations for elliptic problems have been put in a general framework of analysis. Although clarifying basic properties, the analysis does not warrant a clear preference. Specifically, none of the conventional DG formulations possesses a bilinear form that is coercive (and continuous) on an infinite-dimensional broken Sobolev space. Rather, bilinear forms are only weakly coercive or defined on subspaces only and employ stabilization parameters that typically increase unboundedly as the subspace is expanded, e.g., if the polynomial degree is increased. For hp-adaptation, coercivity is a fundamental property: By the classical Lax-Milgram theorem, any conforming discretization of a coercive formulation is stable, i.e., discrete approximations are well-posed and have a unique solution, irrespective of the specifics of the underlying approximation space.
In this work we consider the one-dimensional Poisson problem and present a generic consistent conven-tional DG formulation. We show that convenconven-tional DG formulations are necessarily noncoercive. Moreover, we presents a new symmetric DG formulation which contains nonconventional edge terms based on ele-ment Green’s functions and the data local to the edges. We show that the new DG formulation is coercive onH1(Ph), the space of functions that are piecewise in theH1 Sobolev space. Furthermore, we show
that the new DG formulation and the classical Galerkin formulation are equivalent, that is, in the infinite-dimensional case they yield the same solution.
2000 Mathematics Subject Classification: primary: 65N30. secondary:65N12.
Keywords and Phrases: discontinuous galerkin, finite element method, elliptic problems,hp-adaptivity.
1. Introduction
Discontinuous Galerkin (DG) methods are finite element techniques for the solution of partial differential equations. They allow shape functions which are discontinuous across inter-element edges. The main features of DG methods are the following:
• They are ideally suited for hp-adaptivity as they handle nonconforming meshes and varying-in-space polynomial degree approximations with ease;
• The usual inter-element continuity requirements are enforced weakly; • There is no need for auxiliary variables as used in hybrid or mixed methods.
The development of DG methods for second-order elliptic problems dates back to the early 1970s. After a period of reduced attention, the last decennium has shown renewed interest, partly from the hyperbolic community. Dealing with problems having a dominant convective part, as well as a nonnegligible diffusive part, the need arose to extend the well-acknowledged DG method for hyperbolic problems to elliptic problems. See [7] for a thorough historical overview of DG methods. Recently, DG formulations for elliptic problems have been put in a general framework of analysis in [1]. Although clarifying basic properties, a clear preference is not apparent. The formulations that appear to be most appealing are those having conventional edge terms. However, they have their deficiencies. Particularly, well-posedness of approximate DG problems is not immediate.
For a one-dimensional elliptic model problem, we present a generic conventional DG formu-lation that includes all possible formuformu-lations with conventional edge terms. We show that a conventional DG formulations that avoids the deficiencies is nonexistent. As a remedy, we present a new DG formulation that is nonconventional. This new DG formulation avoids the deficiencies and, in particular, well-posedness of approximate DG problems is immediate.
The contents of this thesis are arranged as follows: Chapter 2 presents the elliptic model problem, viz. the Poisson problem. Furthermore, various mathematical preliminaries for weak formulations of the Poisson problem are given. In Chapter 3, we review elementary existence theorems which form the basis for our analysis in the ensuing chapters. We present the generic conventional DG formulation in Chapter 4 and show its deficiencies. Then, in Chapter 5 we present the new DG formulation and establish important well-posedness results. Finally, Chapter 6 gives conclusions.
2. Problem Statement
In this work, we shall consider the simplest model problem involving an elliptic operator, viz. the linear one-dimensional Poisson problem.
2.1 Poisson Problem
Let Ω ⊂ R be a bounded open interval. Its boundary ∂Ω consists of two disjoint parts, ΓD
(nonempty) and ΓN (possibly empty) on which Dirichlet and Neumann boundary conditions are
imposed, respectively. The unit normal n at the boundary ∂Ω is defined to be outward with respect to the interval Ω.
Within this one-dimensional setting, we formulate the Poisson problem: Find u : Ω→ R such that (2.1a) (2.1b) (2.1c) −d 2u dx2 = f , in Ω , u = gD , on ΓD , un= gN , on ΓN ,
where the subscript (·)n denotes the normal derivative dnd(·) and where the data gD : ΓD → R
and gN : ΓN → R give bounded values in R. Given f ∈ C0(Ω), a classical solution is a function u∈
C2(Ω) which satisfies (2.1) pointwise.
2.2 Weak Formulations of the Poisson Problem
The strong formulation (2.1) stipulates that f ∈ C0(Ω). For more general data f it is necessary
to employ a weak formulation. The generic form of a linear weak formulation is given by the variational problem:
Find u∈ U :
B(u, v) = L(v) , ∀v ∈ V , (2.2)
where the trial space U and test space V are generally infinite dimensional (continuous) Hilbert spaces and B : U × V → R and L : V → R are a bilinear and linear functional, respectively. In the case of U = V , (2.2) is called a Galerkin weak formulation. Examples of weak formulations of (2.1) conforming with (2.2) are the classical weak formulations given in Sec. 2.2.2 as well as the discontinuous Galerkin formulations given in Chapters 4 and 5.
2.2.1 Proper Weak Formulations and Conforming Approximations
A proper weak formulation of the Poisson problem (2.1) has the following properties (cf. [14]): • the weak formulation is well-posed, i.e., there exists a unique solution u ∈ U to the weak
formulation that depends continuously on the data;
Well-posedness of a weak formulation can be established by means of an existence theorem. Several existence theorems are described in Chapter 3. We define a weak formulation to be consistent with another problem (a boundary value problem or another weak formulation) if solutions to the problem satisfy the weak formulation (cf. [1, 8]). Thus, consistency with (2.1) requires that B(uc, v) =L(v), for all v ∈ V , where uc is the classical solution.
A conforming approximation of the continuous weak formulation (2.2) is obtained by replacing U and V by closed, generally finite dimensional, subspaces bU ⊂ U and bV ⊂ V . An approximate solution bu∈ bU is thus obtained by solving the approximate problem:
Find bu∈ bU :
B(bu, v) =L(v) , ∀v ∈ bV . (2.3) The approximate problem is consistent with the continuous problem (2.2). This implies the orthogonality property
B(bu− u, v) = 0 , ∀v ∈ bV , (2.4)
where u∈ U is the continuous solution according to (2.2).
To ensure the existence of a unique approximate solution, well-posedness of (2.3) needs to be established. However, in case of a Galerkin weak formulation, the classical Lax-Milgram existence theorem (see Chapter 3) has the following desirable property:
Well-posedness of the continuous Galerkin problem (2.2) (U = V =: H) implies well-posedness of the approximate Galerkin problem (2.3) ( bU = bV =: bH). (P) We will say that a continuous Galerkin weak formulation is “well-posed with property (P)” if it complies with the conditions for an existence theorem with property (P). That is, if a continuous Galerkin weak formulation is well-posed with property (P) then the approximate problem based on a subspace bH⊂ H is (automatically) well-posed.
In addition, the classical Lax-Milgram theorem yields the following property: Suppose that b
H(2) is a larger approximate space than bH(1)
⊂ bH(2)
⊂ H, then the error (measured in k · kH) of
the corresponding approximate solution bu(2) ∈ bH(2) is at most equal to that of the approximate solution bu(1)
∈ bH(1):
ku − bu(2)
kH≤ ku − bu(1)kH . (2.5)
In particular, this implies that if we consider a sequence of asymptotically dense nested approx-imation spaces bH(1) ⊂ bH(2) ⊂ · · · ⊆ H and bH(m) → H as m → ∞, then the error ku − bu(m)k
H
converges monotonously to 0 as m increases. 2.2.2 Classical Weak Formulations
The classical weak formulations considered below are all proper weak formulations in the afore-mentioned sense. They provide a starting point for DG formulations, since DG formulations are consistent with them. Furthermore, DG formulations constitute nonconforming variants of the classical Galerkin weak formulation in the sense that DG spaces are not generally subspaces of the classical Galerkin space.
Functional Setting A functional setting for classical weak formulations of (2.1) is provided by the Sobolev spaces Hm(Ω) (recall that Ω is a one-dimensional bounded interval). For any positive
integer m, the Sobolev space Hm(Ω) consists of all functions on Ω that reside in L
2(Ω) together
with their generalized derivatives up to and including order m, i.e., Hm(Ω) :={u ∈ L2(Ω) : d
ju
dxj ∈ L2(Ω), 1≤ j ≤ m} .
A characterization of Hm(Ω) is given by Sobolev’s embedding theorem: for every u
∈ Hm(Ω)
there exists an equivalent function in Cm−1(Ω);1 see e.g. [4, 11] for further details. Equipped with
1The function in Cm−1(Ω) is a.e. equivalent to u ∈ Hm
(Ω), i.e., everywhere equal except on a set of measure zero.
the inner product (u, v)Hm(Ω):= m X j=0 (dju dxj, djv dxj) ,
where (·, ·) denotes the standard L2(Ω) inner product, Hm(Ω) is a Hilbert space. The norm
induced by the inner product will be denotedk · kHm(Ω). Moreover, we define H01(Ω) to be the
space of functions in H1(Ω) which vanish on the Dirichlet boundary Γ D, i.e.,
H01(Ω) :={u ∈ H1(Ω) : u = 0 on ΓD} ,
and we define H2
00(Ω) to be the space of functions in H01(Ω)∩ H2(Ω) for which, furthermore, the
derivative vanishes on the Neumann boundary ΓN, i.e.,
H2
00(Ω) :={v ∈ H2(Ω) : v = 0 on ΓD, vn = 0 on ΓN} .
The intersection with H2(Ω) is required to ensure existence of the derivative on the Neumann
edge.
Classical Weak Formulations of the Poisson Problem A classical weak formulation of (2.1) is the following: Given an arbitrary ¯u∈ H2(Ω) satisfying the boundary conditions ¯u = g
D on ΓD and d¯dnu = gN on ΓN, Find u = ¯u + u00∈ ¯u + H2 00(Ω) : B1(u00, v) =L1(v) , ∀v ∈ L2(Ω) , (2.6) where B1(u, v) := Z Ω− d2u dx2v dx , L1(v) := Z Ω f v dx− B1(¯u, v) .
The weak solution of the Poisson problem (2.1) is then given by u = ¯u + u00∈ H2(Ω). Assuming
f ∈ L2(Ω), it can be shown by using the generalized Lax-Milgram theorem (cf. Sec. 3.1) that (2.6)
is well-posed. For f ∈ C0(Ω), we have that (2.6) is consistent with (2.1), i.e.,
B1(uc− ¯u, v) = L1(v) , ∀v ∈ L2(Ω) ,
where ucis the classical solution of (2.1). This can be verified by multiplying (2.1a) with v∈ L2(Ω)
and integrating over Ω. Since (2.6) yields a solution u ∈ H2(Ω)
⊃ C2(Ω) for every f
∈ L2(Ω),
it is a more general formulation of the Poisson problem than the boundary value problem (2.1). Therefore, in the sequel, consistency of a weak formulation is defined with respect to (2.6).
The next weak formulation that we consider, is the classical Galerkin weak formulation: Given an arbitrary ¯u∈ H1(Ω) with ¯u = g D on ΓD, Find u = ¯u + u0∈ ¯u + H01(Ω) : B2(u0, v) =L2(v) , ∀v ∈ H01(Ω) , (2.7) where B2(u, v) := Z Ω du dx dv dx dx , (2.8a) L2(v) := Z Ω f v dx + X e∈ΓN gNve− B2(¯u, v) . (2.8b)
The weak solution of the Poisson problem is then given by u = ¯u + u0∈ H1(Ω). By the classical
Lax-Milgram theorem, it follows [4, 14] that (2.7) is well-posed.2 By property (P), this implies
that for any subspace bU = bV ⊂ H1
0(Ω) the approximate problem is also well-posed.
2This requires that the data f is in the dual of H1
0(Ω), i.e., f ∈ [H01(Ω)] 0
⊃ L2(Ω) and
R
Ωf v dx is understood as
2.2.3 Weak Formulations on Broken Sobolev Spaces
To facilitate the ensuing consideration of DG formulations, we introduce the concept of a finite element partition. Let Ph :=
Ph(Ω) denote such a partition of the interval Ω, i.e.,
Ph is a finite
collection of open nonoverlapping subintervals (elements) K, such that
Ω = int [
K∈Ph
K !
;
see Fig. 1. The mesh parameter h associated withPhwill be defined as:
h:= max
K∈PhhK ,
where hK is the length of element K. The set of all edges, denoted by Γ := Γ(Ph), can be divided
in subsets,
Γ = ΓD∪ ΓN ∪ ΓI,
where ΓI:= ΓI(Ph) consists of all edges in the interior of Ω. We define a unit normal ne at each
edge e∈ Γ. This normal coincides with the unit outward normal of Ω for edges e ∈ ∂Ω = ΓD∪ ΓN
and we define ne:=−1 for edges e ∈ ΓI (see also Fig. 1). For example, if e ∈ ΓI then we have
(un)e= dudn e=− du dx
e. We will further denote byMethe set of elements neighboring edge e∈ Γ,
in other words,
Me:=
K∈ Ph: ∂K∩ e = e .
Note that for e∈ ∂Ω and e ∈ ΓI the setMecontains one element and two elements, respectively.
Functional Setting The functional setting of DG formulations is provided by the so-called (par-titionPhdependent) broken spaces. A broken space is a subspace of L
2(Ω) consisting of functions
of which the restriction to each element K ∈ Phis in an appropriate space. Note that functions
in a broken space are generally discontinuous at the edges e∈ ΓI.
We will work with the broken Sobolev spaces Hm(Ph) [9]. For any positive integer m, the
broken Sobolev space Hm(Ph) is defined as:
Hm(Ph) :={v ∈ L2(Ω) : v K ∈ H m(K), ∀K ∈ Ph} , in other words, Hm(
Ph) consists of functions for which the restriction to each element K
∈ Phis
in Hm(K). Equipped with the inner product
(u, v)Hm(Ph):=
X
K∈Ph
(u, v)Hm(K),
Hm(
Ph) is a Hilbert space. The corresponding norm will be denoted
k · kHm(Ph).
Functions in H1(Ph) have traces on Γ. These are single-valued at edges e∈ ∂Ω and
double-valued at edges e∈ ΓI. To handle the traces, we introduce for each edge e∈ ΓI the trace operator
(·)± e : H1(Ph)→ R, u±e := lim s↓0u(e± s) . PSfrag replacements x ne e
Figure 1: A finite element partition Ph of the interval Ω. We have indicated the direction of the edge
Furthermore, we define the average{·}e: H1(Ph)→ R and the jump [[ · ]]e: H1(Ph)→ R for each
edge e∈ ΓI in the usual manner:
{u}e:= 12(u+e + u−e) ,
[[u]]e:= u+e − u−e .
Finally, for each edge e∈ ∂Ω, we define the trace operator (·)e: H1(Ph)→ R as
ue:= lim
s↓0u(e− nes) .
All the operators just defined are bounded inR for functions in H1(
Ph), that is, trace inequalities
hold.
Discontinuous Galerkin Formulations Let H(Ph) be a broken space and bH(
Ph)
⊂ H(Ph) a
finite-dimensional subspace. A DG formulation is a Galerkin weak formulation posed on H(Ph).
Thus, the generic form of a DG formulation is given by (2.2) with U = V = H(Ph). Clearly,
H(Ph)
⊃ H2(Ω) is a necessary condition for a proper DG formulation of the Poisson problem (2.6).
The generic form of an approximate DG problem is given by (2.3) with bU = bV = bH(Ph). We will
refer to the broken space associated with a particular DG problem as its DG space.
Comparing with the classical Galerkin weak formulation (2.7), (approximate) DG formulations are nonconforming in the sense that a DG space is not a subspace of H1
0(Ω), i.e., H(Ph)* H01(Ω)
( bH(Ph)* H1
0(Ω)). Therefore, the classical Galerkin weak formulation is inappropriate on DG spaces,
and modification of the functionals in (2.7) is necessary to obtain a proper DG formulation. Assuming momentarily that H(Ph)
⊆ H1(
Ph), (2.7) can be heuristically extended to H(
Ph)
by replacing the integralRΩ dudxdvdx dx by a sum of integrals over individual elements: X K∈Ph Z K du dx dv dx dx = Z Ω f v dx + X e∈ΓN gNve, ∀v ∈ H(Ph) . (2.9)
However, this modification alone is not sufficient; Straightforward integration by parts reveals that it is inconsistent. Indeed, for consistency, we must have:
X K∈Ph Z K du dx dv dx dx− X e∈ΓI un[[v]] e− X e∈ΓD unv e= Z Ω f v dx + X e∈ΓN gNve, ∀v ∈ H(Ph) . (2.10) with u∈ H2(Ω) the solution to (2.6). This can be verified by considering (2.6) with v
∈ H(Ph)
⊂ L2(Ω) and integrating by parts, element by element. The analyses in Chapters 4 and 5 convey
that the additional edge terms in (2.10) form an essential complication in constructing proper DG formulations.
Conventional DG Formulations A cogent approach to construct DG formulations consists in adding conventional edge terms to (2.9). Conventional edge terms are loosely defined here as products of edge values and/or edge derivatives, for instance,{un}e[[v]]e for e∈ ΓI or (unv)e for
e∈ ΓD. A precise definition will be given in Chapter 4.
Most DG formulations in literature are the result of such an augmentation. We will refer to such formulations as conventional DG formulations. They differ from one another only by the number and the precise form of the added conventional edge terms. An example of a conven-tional DG formulation with a symmetric bilinear funcconven-tional is the interior penalty DG formulation (IPDG) [1]. Nonsymmetric conventional DG formulations are the Baumann and Oden DG for-mulation (BODG) [2, 9], the stabilized DG forfor-mulation (SDG) [13] and the nonsymmetric IP DG formulation (NIPDG) [12].
Conventional DG formulations exhibit several deficiencies. Firstly, for symmetric formulations, approximate problems are well-posed under approximation-space-dependent stability parameters.
These have to be established a-priori. Moreover, these parameters increase unboundedly if the approximation space is refined, e.g., for broken polynomial spaces the stability parameters are typically proportional with a positive power of the polynomial degree; see [1]. Secondly, for nonsymmetric formulations, the continuous problem, although well-posed, is not well-posed with property (P). Therefore, in principle, for each approximate problem, well-posedness has to be established separately. Furthermore, the error in L2(Ω) converges suboptimally. This has been
conjectured to emanate from the nonsymmetry of the formulation; see [1].
In chapter 4, we show that all aforementioned formulations form instances of a generic con-ventional DG formulation. Based on this generic concon-ventional formulation, we will establish that a conventional DG formulation which avoids the aforementioned problems is nonexistent. Specifi-cally, we show that although the continuous problem of symmetric and antisymmetric formulations is well-posed, conventional DG formulations cannot be well-posed with property (P).
Other DG Formulations Alternatively, consistency of DG formulations can be enforced by in-troducing nonconventional edge terms into (2.9). We remark that the support of such terms is not necessarily restricted to the edges. Examples of such DG formulations are the lift-operator-based schemes in [1]. These include, among others, the Bassi and Rebay DG formulation (BRDG) [3] and the local DG formulation (LDG) [6]. Their lift-operators originate from the flux formulation and they are based on the finite dimensional approximate DG space. Furthermore, these operators are undefined for the continuous problem. Therefore, trivially, these DG formulations cannot be well-posed with property (P).
In Chapter 5, we present a new symmetric DG formulation with nonconventional edge terms that is well-posed with property (P), i.e., our new DG formulation does not incur the afore-mentioned difficulties inherent to conventional DG formulations and previous nonconventional DG formulations.
3. Existence of Unique Weak Solutions
In this chapter, we review elementary existence theorems pertaining to the well-posedness of linear variational problems. These theorems form the basis for our analysis in Chapters 4 and 5. Furthermore, a-priori error estimates are given for Galerkin approximations.
Sec. 3.1 is concerned with the generalized Lax-Milgram theorem. This theorem provides the fundament for the Galerkin existence theorems which are discussed in Sec. 3.2.
3.1 The Generalized Lax-Milgram Theorem
The generalized Lax-Milgram theorem gives necessary and sufficient conditions for the well-posedness of a generic linear variational problem. Unlike the theorems discussed in Sec. 3.2, this theorem holds in the general case in which the test and trial space are distinct. Its proof can be found in [11, 14].3
Let U and V be Hilbert spaces4 with corresponding norms
k · kU andk · kV, respectively.
Theorem 3.1 (Generalized Lax-Milgram). Consider a continuous bilinear functional A : U× V → R and continuous linear functional F : V → R, i.e., there exist constants ca, cf > 0 such
that
|A(u, v)| ≤ ca kukU kvkV , ∀u ∈ U , v ∈ V ,
|F(v)| ≤ cf kvkV , ∀v ∈ V .
If and only if there exists a γ > 0 such that inf u∈U \{0} v∈V \{0}sup A(u, v) kukUkvkV ≥ γ , (3.1a) sup u∈UA(u, v) > 0 , ∀v ∈ V \ {0} , (3.1b)
3The necessity of (3.1) is shown in [14].
4The generalized Lax-Milgram theorem actually requires U and V to be reflexive Banach spaces only (note that
then a unique u∗∈ U exists such that
A(u∗, v) =F(v) , ∀v ∈ V .
Eq. (3.1a) is known as the inf-sup condition. A bilinear form satisfying (3.1) is said to be weakly coercive on U× V .
3.2 Galerkin Weak Formulations: Continuous and Approximate Problem
In this section, we consider existence theorems for Galerkin problems. These theorems can be derived as special cases of the generalized Lax-Milgram theorem discussed previously.
Let H be a (possibly infinite dimensional, i.e., continuous) Hilbert space with normk · kH. We consider the generic (linear) Galerkin weak formulation:
Find u∗∈ H :
B(u∗, v) =L(v) , ∀v ∈ H . (3.2)
Since it is required by all the existence theorems that follow, we assume throughout this chapter that the bilinear and linear functionalB : H × H → R and L : H → R are continuous, i.e., there exist constants cb, c`> 0 such that
|B(u, v)| ≤ cbkukH kvkH , ∀u, v ∈ H , (3.3a)
|L(v)| ≤ c`kvkH , ∀v ∈ H . (3.3b)
Next to the continuous problem, we also consider the approximate problem. Let bH ⊂ H be a closed subspace, which we call the approximation space. The approximate Galerkin problem is given by:
Find bu∈ bH :
B(bu, v) =L(v) , ∀v ∈ bH . (3.4) Below, we consider three different existence theorems concerning the continuous and approx-imate Galerkin problem. Each poses different conditions on B(·, ·). Furthermore, a-priori error estimates are given for the approximations. Finally, we will see that although well-posed problems can be established by the theorems, only the theorems discussed in Sec. 3.2.2 have the property (P) presented in Sec. 2.2.1.
3.2.1 Weak Coercivity on Continuous and Approximation Space
By applying the generalized Lax-Milgram theorem to the continuous and approximate Galerkin problem, we obtain the following theorem:
Theorem 3.2. If and only if B(·, ·) is weakly coercive on H × H, i.e., there exists a γ > 0 such that inf u∈H\{0} v∈H\{0}sup B(u, v) kukHkvkH ≥ γ , (3.5a) sup u∈HB(u, v) > 0 , ∀v ∈ H \ {0} , (3.5b)
then, the continuous Galerkin problem (3.2) is well-posed. Additionally, if and only if B(·, ·) is weakly coercive on bH× bH, i.e., there exists a bγ := bγ( bH) > 0 such that
inf u∈ bH\{0} sup v∈ bH\{0} B(u, v) kukHkvkH ≥ bγ , (3.6a) sup u∈ bH B(u, v) > 0 , ∀v ∈ bH\ {0} , (3.6b)
then, the approximate Galerkin problem (3.4) is well-posed. Moreover, the following a-priori esti-mate holds: ku∗− bukH ≤ (1+cb/bγ) inf u∈ bHku ∗ − ukH .
For a proof of the a-priori estimate see e.g. [11, 14]. It is to be noted that conditions (3.6) are not implied by (3.5). Therefore, Theorem 3.2 does not have property (P). Also, since the constant bγ generally depends on the approximation space bH, choosing a larger approximation space does not necessarily decrease the error in the approximate solution.
3.2.2 Coercivity on Continuous and Approximation Space
A bilinear functional B : H × H → R is said to be (strongly) coercive on H, if there exists a constant κ > 0 such that
|B(u, u)| ≥ κkuk2H, ∀u ∈ H . (3.7)
Note that coercivity on H is a sufficient condition for weak coercivity on H× H as well as weak coercivity on bH × bH. The notion of coercivity underlies the classical Lax-Milgram theorem on well-posedness of Galerkin weak formulations (see e.g. [11, 14]):
Theorem 3.3 (Classical Lax-Milgram). If B(·, ·) is coercive on H according to (3.7), then the continuous Galerkin problem (3.2) and the approximate Galerkin problem (3.4) are well-posed. Moreover, the following a-priori estimate holds:
ku∗− bukH ≤ cb/κ inf u∈ bHku
∗
− ukH. (3.8)
Note that the coercivity condition (3.7) is sufficient but not necessary for well-posedness. As a sidenote, we mention that if B(·, ·) is symmetric and |B(u, u)| = B(u, u) ≥ κkuk2H for all u ∈ H, thenB(·, ·) is an innerproduct on H and the solution u∗ of (3.2) is the unique minimizer over H
of the quadratic functional
J (u) :=1
2B(u, u) − L(u) ,
see, e.g., Ref. [4].
Theorem 3.3 has property (P). This is because coercivity on the continuous space carries over to the approximation space. Also, since the constants in the a-priori estimate (3.8) are independent of the approximation space, the aforementioned error inequality (2.5) holds for all conforming approximations. Not only is coercivity sufficient for property (P) to hold, it is also necessary. This is summarized in the following:
Proposition 3.4. If and only ifB(·, ·) is coercive on H, then well-posedness of the continuous Galerkin problem (3.2) implies well-posedness of the approximate Galerkin problem (3.4), i.e., property (P) holds.
Proof. i) Forward implication: By theorem 3.3, coercivity on H ensures that the continuous Galerkin problem (3.2) is well-posed as well as the approximate Galerkin problem (3.4).
ii) Reverse implication: We show the proof by contradiction. Assume thatB(·, ·) is not coercive and simultaneously property (P) holds. As H is a closed space, noncoercivity implies the existence of a ¯
u∈ H such that B(¯u, ¯u) = 0 and k¯ukH= 1. Taking the approximate space as the one-dimensional space bH = span{¯u}, we have
inf u∈ bH\{0} sup v∈ bH\{0} B(u, v) kukHkvk =B(¯u, ¯u) k¯uk2H = 0 ,
in other words, weak coercivity does not hold on bH × bH. By theorem 3.2 weak coercivity on b
H× bH is necessary for well-posedness of the approximate Galerkin problem, thus the approximate Galerkin problem is ill-posed, contradicting property (P).
3.2.3 Weak Coercivity on Approximation Space Only
There exist weak formulations that are well-posed for the approximate problem (3.4) but might not be well-posed for the continuous problem (3.2). We assume that for such formulations, the continuous problem (3.2), although ill-posed, is nevertheless consistent with a different continuous problem which is well-posed. Denoting by u∗∗ ∈ H the solution of the well-posed continuous
problem, consistency implies that
B(u∗∗, v) =L(v) , ∀v ∈ H . From this we obtain the orthogonality relation
B(bu− u∗∗, v) = 0 , ∀v ∈ bH , (3.9)
which allows an a-priori error estimate with respect to u∗∗. We summarize this in the following theorem:
Theorem 3.5. Let (3.2) be consistent with a well-posed problem having solution u∗∗∈ H, i.e.,
B(u∗∗, v) =L(v) , ∀v ∈ H .
If and only if B(·, ·) is weakly coercive on bH× bH, i.e., there exists a bγ := bγ( bH) > 0 such that inf u∈ bH\{0} sup v∈ bH\{0} B(u, v) kukHkvkH ≥ bγ , (3.10a) sup u∈ bH B(u, v) > 0 , ∀v ∈ bH\ {0} , (3.10b)
then the approximate Galerkin problem (3.4) is well-posed. Moreover, the following a-priori esti-mate holds:
ku∗∗− bukH ≤ (1+cb/bγ) inf u∈ bHku
∗∗
− ukH .
Proof. The proof consists of showing the a-priori estimate. It can be found in, for instance, Ref. [1], but will be repeated here. First note that by the triangle inequality we have
ku∗∗− bukH≤ inf u∈ bH ku∗∗− ukH+kbu− ukH .
Using the inf-sup condition, (3.10a), gives
ku∗∗− bukH≤ inf u∈ bH ku∗∗− ukH+ 1/bγ sup v∈ bH\{0} B(bu− u, v) kvkH .
On account of consistency, the orthogonality relation (3.9) holds and, hence,
ku∗∗− bukH≤ inf u∈ bH ku∗∗− ukH+ 1/bγ sup v∈ bH\{0} B(u∗∗− u, v) kvkH .
Finally, applying continuity ofB(·, ·), (3.3a), we obtain the a-priori estimate ku∗∗− bukH≤ (1+cb/bγ) inf
u∈ bHku ∗∗
− ukH.
Trivially, Theorem 3.5 does not have property (P). Also, since the constant bγ generally depends on the approximation space bH, if we consider a sequence of asymptotically-dense nested subspaces H(1)⊂ H(2) ⊂ · · · ⊆ H, H(m)→ H as m → ∞, then the corresponding approximations u(m)need
4. Conventional Discontinuous Galerkin Formulations
This chapter is concerned with an analysis of the properties of conventional DG formulations on the basis of a generic consistent conventional DG formulation. The generic formulation is presented in Sec. 4.1. Important well-posedness results based on theorems of Chapter 3 are presented in Sec. 4.2.
4.1 Weak Formulation
Conventional DG formulations emanate from a cogent selection of the terms required for consis-tency; See Sec. 2.2.3. In this section we first phrase a generic conventional DG formulation and, subsequently, derive the consistent generic conventional formulation. Based on this generic form, we then consider symmetric and antisymmetric formulations, and establish well-posedness results in Sec. 4.2.
4.1.1 A Generic DG Formulation with Conventional Edge Terms
Consider the following generic conventional DG formulation, set on the DG space H(Ph):
Find u∈ H(Ph) : BΛ(u, v) =LΛ(v) , ∀v ∈ H(Ph) , (4.1) where5 BΛ(u, v) := X K∈Ph Z K du dx dv dxdx + X e∈Γ uTΛve, (4.2a) LΛ(v) := Z Ω f v dx + X e∈ΓD gDΛ¯¯ve+ X e∈ΓN gNΛ¯¯ve, (4.2b)
and where ueand ve, ¯veare (column-) vectors containing values of u and v, respectively, at edge e
in the following way:
ve:= h−12{v}, h−12[[v]], h12{vn}, h12[[vn]]T e , e∈ ΓI , h−1 2v, h 1 2 vnT e , e∈ ∂Ω , (4.3a) ¯ ve:= h−1v, v n T e , e∈ ΓD, v, h vn T e , e∈ ΓN . (4.3b)
The definition of ue is similar to ve. The matrices Λe ∈ R4×4, for e ∈ ΓI, and Λe ∈ R2×2,
¯
Λe ∈ R1×2, for e ∈ ∂Ω specify bilinear relations between edge values and edge derivatives of u
and v. Conventional edge terms can now be precisely defined as any term in the bilinear functional conforming to (uTΛv)
efor all e∈ Γ, and any term in the linear functional conforming to (gDΛ¯¯v)e
or (gNΛ¯¯v)efor e∈ ΓD or e∈ ΓN, respectively.
The constants h : Γ→ R in (4.3) are local mesh parameters introduced to minimize the mesh dependence of the matrices. Typically, for e∈ ΓI, he is set to the average of the lengths of the
elements neighboring edge e and for e ∈ ∂Ω it is set to half the length of the element next to edge e, i.e.,
he= 12
X
K∈Me
hK , ∀e ∈ Γ . (4.4)
To ensure continuity of the bilinear functionalBΛ(·, ·) on H(Ph)× H(Ph) and the linear
func-tionalLΛ(·) on H(Ph), we define H(Ph) to be the broken Hilbert space of functions v∈ H1(Ph) for
which the vector vecontains finite values. Furthermore, it holds that H2(Ph)⊂ H(Ph)⊂ H1(Ph).
A precise definition of the DG space H(Ph) for specific Λ pertaining to symmetric and
antisym-metric formulations will be given in Sec. 4.1.3.
5Here and in the equations that follow, for the sake of clarity, we often suppress the subscripts (·)e at the
relevant variables and put them outside the largest enclosing brackets. For example, (gDΛ¯¯v)eshould be interpreted
4.1.2 Consistency with Poisson Problem
A particular conventional DG formulation is obtained by specifying the matrices Λe and ¯Λe.
The proposition below specifies conditions on the matrices Λe and ¯Λe which ensure consistency
with (2.6).
Proposition 4.1. If and only if the matrices Λe and ¯Λe are of the following form:
Λe= 0 0 0 0 α1 α2 α3 α4 0 −1 0 0 β1 β2 β3 β4 e , ∀e ∈ ΓI, (4.5a) Λe= α1 α2 −1 0 e , Λ¯e= α1 α2e , ∀e ∈ ΓD , (4.5b) Λe= 0 0 β1 β2 e , Λ¯e= β1+1 β2e , ∀e ∈ ΓN , (4.5c)
for certain fixed parameters αi
e, βei ∈ R (i = 1, 2, 3, 4 for e ∈ ΓI and i = 1, 2 for e ∈ ∂Ω), then
the conventional DG formulation (4.1) is consistent with (2.6), i.e., if u∈ H2(Ω)
⊂ H(Ph) is the
solution of (2.6), then
BΛ(u, v) =LΛ(v) , ∀v ∈ H(Ph) . (4.6)
Proof. Let u ∈ H2(Ω) be the solution of (2.6). Taking a v ∈ H(Ph) ⊂ L
2(Ω) in (2.6) and
integrating by parts, element by element, gives (see also (2.10)): X K∈Ph Z K du dx dv dx dx = Z Ω f v dx + X e∈ΓI un[[v]] e+ X e∈∂Ω unv e. (4.7)
i) Forward implication: Using (4.7) inBΛ(u, v) results in
BΛ(u, v) = Z Ω f v dx + X e∈ΓI un[[v]] e+ X e∈ΓD∪ΓN unv e+ X e∈Γ uTΛve.
For the solution u ∈ H2(Ω)
⊂ C1(Ω), we have u
e = (gD)e for e ∈ ΓD, (un)e = (gN)e for
e∈ ΓN and [[u]]e= [[un]]e= 0,{un}e= (un)e for e∈ ΓI. Invoking this and the definitions of the
matrices Λe gives BΛ(u, v) = Z Ω f v dx + X e∈ΓD α1gDv + α2gDvne+ X e∈ΓN gNv + β1gNv + β2gNvne =LΛ(v) .
ii) Reverse implication: Using (4.7) in (4.6) gives X e∈ΓI un[[v]] e+ X e∈∂Ω unv e+ X e∈Γ uTΛve= X e∈ΓD gDΛ¯¯v e+ X e∈ΓN gNΛ¯¯v e, ∀v ∈ H(P h) .
Invoking the same relations for the solution u∈ H2(Ω) as before, results in
X e∈ΓI un[[v]] + (h− 1 2u, 0, h12u n, 0)Λv e + X e∈ΓD unv + (h− 1 2gD, h12u n)Λv e+ X e∈ΓN gNv + (h− 1 2u, h12 gN)Λv e = X e∈ΓD gDΛ¯¯ve+ X e∈ΓN gNΛ¯¯ve, ∀v ∈ H(Ph) . (4.8)
Taking, for example, a v∈ H(Ph) such that [[v]]
e= 1 for an edge e∈ ΓI and such that all other
edge terms vanish, gives the equation
(un)e+ (u)eΛ(1,2)e /he+ (un)eΛ(3,2)e = 0 .
Therefore, Λ(1,2)e = 0 and Λ(3,2)e = −1. Similarly, by making appropriate choices for the test
function v ∈ H(Ph) in (4.8), the precise form (4.5) of the matrices Λ
e and ¯Λe for consistent
formulations can be established.
4.1.3 Symmetric and Antisymmetric Formulations
In the analysis of the well-posedness of conventional DG formulations, we will restrict ourselves to symmetric and antisymmetric formulations. In particular, in Sec. 4.2.2 we will show that coercive consistent conventional DG formulations are nonexistent. We will prove this for the symmetric and antisymmetric formulations, but the proof extends mutatis mutandis to any consistent conventional DG formulation.
Definition of Λ± Matrices The symmetric and antisymmetric formulations are represented by matrices Λ− and Λ+, respectively. The symmetric formulations yield a symmetric bilinear
func-tional. An example in the literature is provided by, e.g., the IPDG formulation. The antisymmetric formulations correspond to matrices with an antisymmetric off-diagonal part. Antisymmetric for-mulations include BODG, NIPDG and SDG.
The matrices Λ± are defined as follows:
Λ±e = 0 0 0 0 0 α ±1 ±γ 0 −1 0 0 0 −γ 0 β e , ∀e ∈ ΓI, Λ±e = α ±1 −1 0 e , Λ¯±e = α ±1 e , ∀e ∈ ΓD, Λ±e = 0 0 0 β e , Λ¯±e = 1 β e , ∀e ∈ ΓN ,
where αe ≥ 0 (for e ∈ ΓI ∪ ΓD), βe ≥ 0 (for e ∈ ΓI∪ ΓN) and γe ∈ R (for e ∈ ΓI). The
parameters αe and βe correspond with jumps of u, v and jumps of un, vn at edge e, respectively.
Similar terms appear in classical penalty formulations which help stabilize the formulation (such that the resulting formulation is well-posed). This explains the names of several DG formulations. If present, the parameters αe and βealso stabilize the DG formulations, thus negative values of
these parameters are not considered. DG Space H(Ph) With the matrices Λ±
e and ¯Λ±e defined as above, we can give a precise definition
of the DG space H(Ph). We introduce the normk · k
H(Ph)defined by: kuk2H(Ph):= X K∈Ph |u|21,K+ X e∈ΓI (1+α+|γ|)[[u]] 2 h + h{un} 2+ (β + |γ|) h [[un]]2 e + X e∈ΓD (1+α)u 2 h + h u 2 n e+ X e∈ΓN β h u2n e,
where the seminorm| · |1,K is defined by: |u|21,K:= Z K du dx 2 dx .
We now define the space H(Ph) as the completion of H2(Ph) under normk · k
H(Ph), i.e.,
H(Ph) := H2(Ph)k·kH(Ph)
A complete characterization of the space obtained in this way is given in Sec. B.1. A similar characterization of the DG space H(Ph) with α
e, βe, γe = 0 can be found in [2]. Note that
functions in H(Ph) belong to H1(
Ph). Thus, there exists a constant C > 0, dependent on the
parameters and the partitionPh, such that6
kukH1(Ph)≤ CkukH(Ph), ∀u ∈ H(Ph) . (4.10)
Equipped with the innerproduct (·, ·)H(Ph) (associated withk · kH(Ph)),
(u, v)H(Ph):= X K∈Ph Z K du dx dv dx dx + X e∈ΓI (1+α+|γ|)[[u]][[v]] h + h{un}{vn} + (β+|γ|) h [[un]][[vn]] e + X e∈ΓD (1+α)u v h + h unvn e+ X e∈ΓN β h unvn e,
the space H(Ph) is a Hilbert space.
Symmetric and Antisymmetric Formulations The DG formulation associated with Λ± is given
by:
Find u∈ H(Ph) :
BΛ±(u, v) =L
Λ±(v) , ∀v ∈ H(Ph) , (4.11)
where, by evaluating the matrix products, the functionals can be expanded as: BΛ±(u, v) = X K∈Ph Z K du dx dv dx dx + X e∈ΓI α[[u]][[v]]/h± [[u]]{vn}+γ[[vn]] −{un}+γ[[un]] [[v]] + βh[[un]][[vn]] e + X e∈ΓD α u v/h± u vn− unv e+ X e∈ΓN β h unvn e, (4.12a) LΛ±(v) = Z Ω f v dx + X e∈ΓD α gDv/h± gDvn e+ X e∈ΓN gNv + β h gNvn e. (4.12b)
In Table 1, we have summarized the± sign and parameter choices for several conventional DG for-mulations that have appeared in the literature.
4.2 Well-Posedness Results
This section addresses the well-posedness of (4.11). For this purpose, we employ the theorems from Chapter 3. First, continuity of the functionals BΛ±(·, ·) and LΛ±(·) is established. Next,
we show that BΛ±(·, ·) is not coercive on H(Ph). Rather, it is shown that weak coercivity on
H(Ph)
× H(Ph) holds. We can then establish well-posedness on account of Theorem 3.2.
4.2.1 Continuity of Bilinear and Linear Functional
The theorems of Chapter 3 require continuity of the bilinear and linear functional with respect to the DG space H(Ph).7 We summarize the results concerning continuity in the following
proposi-tions.
Proposition 4.2 (Continuity ofBΛ±). The bilinear functional BΛ±(·, ·) given in (4.12a) is
continuous on H(Ph)× H(Ph), i.e.,
|BΛ±(u, v)| ≤ cbkuk
H(Ph)kvkH(Ph), ∀u, v ∈ H(Ph) , 6In the following, we denote with C and C
i, i = 1, 2, . . ., positive constants, which, in general, take on different
values for each usage.
7Essentially, since the space H(Ph) and its norm k · k
Table 1: The ± sign and parameter choices in Λ±for several conventional DG formulations (note that + corresponds with a antisymmetric formulation and − with a symmetric formulation).
DG formulation ± α β γ
IPDG [1] − α 0 0
BODG [2] + 0 0 0
NIPDG [12] + α 0 0
SDG [13] + 0 β 0
with continuity constant cb= 1.
Proof. First note that |BΛ±(u, v)| ≤ X K∈Ph Z K dudxdvdx dx + X e∈Γ uTΛ±ve .
Applying the Schwarz inequality (see (A.1)) on the first term, and expanding the matrix terms, we obtain |BΛ±(u, v)| ≤ X K∈Ph |u|1,K|v|1,K+ X e∈ΓI α [[u]][[v]] /h + [[u]]{vn} + |γ| [[u]][[vn]] + {un}[[v]] + |γ| [[un]][[v]] + βh [[un]][[vn]] e + X e∈ΓD α u v /h + u vn + unv e+ X e∈ΓN βh unvn e.
Then, an application of the discrete Schwarz inequality (see (A.2)) yields
|BΛ±(u, v)| ≤ X K∈Ph |u|21,K+ X e∈ΓI (1+α+|γ|)[[u]] 2 h + h{un} 2+ (β + |γ|)h[[un]]2 e + X e∈ΓD (1+α)u 2 h + h u 2 n e+ X e∈ΓN β h u2n e !1/2 · · · !1/2 ,
where the dots (· · · ) represent an identical term with u replaced by v. The proof then follows immediately from the definition of the normk · kH(Ph).
Proposition 4.3 (Continuity of LΛ±). For f ∈ [H(Ph)]0 ⊃ L2(Ω), the linear functional
LΛ±(·) given in (4.12b) is continuous on H(Ph), i.e.,
|LΛ±(v)| ≤ c`kvk
H(Ph), ∀v ∈ H(Ph) ,
with continuity constant c`> 0, dependent on the data f, gD and gN, partitionPhand parameters
αefor e∈ ΓD and βe for e∈ ΓN.
Proof. First note that |LΛ±(v)| ≤ Z Ω f v dx + X e∈ΓD gD (α|v|/h + |vn|) e+ X e∈ΓN gN (β h|vn| + |v|) e.
Assuming that f is in [H(Ph)]0, the first term is bounded by C
kvkH(Ph) with C > 0 dependent
PSfrag replacements u1 u2 u3 ΓD x1 x2 x3 e
Figure 2: Sample functions ui
of the Cauchy sequence {ui
} used to show noncoercivity on H(Ph).
respectively, dependent on gD, gN, h, αe(for e∈ ΓD) and β (for e∈ ΓN). Hence
|LΛ±(v)| ≤ Ckvk H(Ph)+ C0 X e∈ΓD kvkH(Ph)+ C1 X e∈ΓN kvkH(Ph)+|v| e.
The trace |v|e, for e ∈ ΓN, can be bounded by a trace inequality since v ∈ H(Ph) ⊂ H1(Ph).
Using (4.10), we have
|v|e≤ C2kvkH1(Ph)≤ CkvkH(Ph).
The proof then follows by substitution.
4.2.2 Noncoercivity on the Continuous SpaceH(Ph)
At this point, it would be desired to show that the bilinear functionalBΛ±(·, ·) given in (4.12a) is
coercive on H(Ph). Then, by the classical Lax-Milgram, Theorem 3.3, the DG formulation would
be well-posed with property (P). However, the next proposition shows that BΛ±(·, ·) cannot be
coercive on H(Ph).
Proposition 4.4 (Noncoercivity). The bilinear functional BΛ±(·, ·) given in (4.12a) is not
coercive on H(Ph), i.e., a positive constant κ such that
|BΛ±(u, u)| ≥ κkuk2
H(Ph), ∀u ∈ H(Ph) , (4.13)
is nonexistent.
Proof. There exists sequences of functions from which we can show thatBΛ±(·, ·) is not coercive
on H(Ph). Such a sequence is the following. Consider an edge e
∈ ΓD and the element K ∈ Me
at this edge. Let xi
∈ K, i = 1, 2, . . . denote successive points which converge to e in the following way:
∀ε > 0 ∃Nε<∞ : hi:=|xi− e| < ε , ∀i ≥ Nε. (4.14)
The open bounded interval defined by xi and e will be denoted Ki and its length has been
denoted hi. The function ui
∈ H2(Ω)
⊂ H(Ph) is the unique C1(Ω) function defined by:
• supp ui= Ki, • ui Ki∈ P 2(Ki) , • (ui n)e= h−1/2e ,
wherePm(K) is the space of polynomial functions of degree at most m on K. Sample functions ui
are presented in Fig. 2. For these functions it holds that |ui|21,K= hi 3he and (ui)2e= (hi)2 4he . (4.15)
kui k2H(Ph)=|ui| 2 1,K+ (1+αe)(ui)2e/he+ he(uin)2e > C hi+ (hi)2+ 1 > 1 , (4.16)
where C is independent of hi. Furthermore, the sequence {ui} is a Cauchy sequence in H(Ph)
since for all δ > 0 we can choose ε = ε(δ) such that∀i, j > Nε(δ)
kui − uj kH(Ph)=|ui− uj| 2 1,K+ (1+αe)(uie− uje)2/he ≤ C |ui |21,K+|u j |21,K+ (u i)2 e+ (uj)2e
by the triangle ineq. ≤ C hi+ hj+ (hi)2+ (hj)2 by (4.15)
< C(ε + ε2) by (4.14)
< δ .
Noting that the Cauchy sequence {ui} has the property that for all δ∗ > 0, we can choose an
ε = ε(δ∗) such that∀i > Nε(δ∗)
|BΛ±(ui, ui)| = |ui|21,K+ αe(u i)2 e/he+ (±1−1)uie(uin)e ≤ C(hi+ (hi)2) by (4.15) < C(ε + ε2) by (4.14) < δ∗. (4.17)
We show the proof by contradiction. Suppose that (4.13) holds for some κ > 0. The coercivity inequality must thus also hold for all functions ui
∈ H(Ph) of the Cauchy sequence
{ui }. By (4.16), (4.17) and (4.13), we have δ∗>|BΛ±(ui, ui)| ≥ κkuik 2 H(Ph)> κ > 0 , ∀i > Nε(δ∗).
This inequality cannot hold for all δ∗> 0. Hence, we have a contradiction.
Further to the sequence{ui
} used in the proof above, there are other sequences that can be used to show noncoerciveness. Any sequence{vi
} with vi
∈ H(Ph) such that
kvikH(Ph)→ C > 0 and |BΛ±(vi, vi)| ↓ 0
for i→ ∞ is suitable. By comparing kvk2H(Ph) withBΛ±(v, v), it can be seen that kvk2
H(Ph) has
the quadratic terms {vn}2e (e∈ ΓI) and (vn)2e (e∈ ΓD) whereasBΛ±(v, v) has these terms only
in case of a symmetric formulation and then merely in a linear combination with [[v]]e (e∈ ΓI)
and ve (e ∈ ΓD). It is this difference that makes the sequences possible. Thus, the
nonposi-tive terms [[u]]e{vn}e, {un}e[[v]]e for e ∈ ΓI and ue(vn)e, (un)eve for e∈ ΓD are responsible for
the noncoercivity. Note that these terms are inherently present in any consistent conventional DG formulation, see Proposition 4.1. Although formally the norm is slightly different, the nonco-erciveness result generalizes to any consistent conventional DG formulation. Hence, a consistent conventional DG formulation that is coercive on the continuous DG space H(Ph) is nonexistent.
In Chapter 3 it was shown that coercivity on the continuous space is necessary for a Galerkin weak formulation to be well-posed with property (P). In conclusion, there are no conventional DG formulations that are well-posed with property (P).
4.2.3 Well-Posedness of Continuous and Approximate Problems
To elaborate further on well-posedness, we first consider well-posedness of the continuous problem and, subsequently, the approximate problems for the symmetric and antisymmetric formulations seperately.
Well-Posedness of Continuous Problem To study well-posedness without coercivity, we have to show weak coercivity and revert to Theorem 3.2. Indeed, it can be shown that if |γe| ≤ Cβe
for all e ∈ ΓI and the mesh parameter h : Γ → R is given by (4.4), then the symmetric and
antisymmetric formulations are well-posed on the continuous DG space H(Ph). The proof of this
is rather involved and is therefore transferred to Appendix B.
Well-Posedness of Approximate Problems, Symmetric Formulations For the IPDG formulation (βe, γe = 0), well-posedness has been established in, for instance, [1, 8] for broken polynomial
spaces. Here, a disadvantage is that the parameter αe depends on the approximation space and
must be chosen large enough. In obtaining coercivity on the approximation space, the penalty-like parameter αe compensates the nonpositive terms discussed at the end of Sec. 4.2.2. This is
possible for finite dimensional subspaces, as use can be made of inverse inequalities. We note that for broken polynomial spaces, the parameter must be chosen proportional with the square of the polynomial degree.
Well-posedness of approximate problems based on any symmetric formulation can be estab-lished similarly. The additional parameters βeand γedo not change significantly the requirements
on αe. Since the continuous Λ− DG formulation is consistent with (2.6), we could revert to the
existence theory of Sec. 3.2.3, which doesn’t require the well-posed of the continuous Λ− DG for-mulation. Indeed, this has been the usual approach in the literature. However, well-posedness of the continuous Λ− DG formulation is proven in Appendix B, thus we can revert to the existence theory of Sec. 3.2.1 and to Theorem 3.2 for a-priori error estimates.
Well-Posedness of Approximate Problems, Antisymmetric Formulations In [2, 12, 13] well-posedness of approximate problems based on the antisymmetric formulations of Table 1 is established. We refer to these references for more details. Unlike the symmetric formulations, approximate prob-lems based on antisymmetric formulations can be well-posed without any requirements on its parameters. However, a disadvantage is that for broken polynomial spaces, the polynomial degree must be at least equal to two. Another disadvantage is that the error measured in the L2(Ω) norm
is suboptimal for even polynomial orders (unless αeis chosen mesh-dependent, proportional with
a negative power of he, thereby negatively affecting the conditioning of the discrete problem,
see [5]). The L2(Ω) suboptimality has been thought to originate from the nonsymmetry of the
formulation [1].
Well-posedness of approximate problems of any antisymmetic formulation can be established similarly. The L2(Ω) suboptimality will not change by additional parameters. Since the
anti-symmetric formulations fall within the existence theory of Sec. 3.2.1, we refer to Theorem 3.2 for a-priori error estimates.
5. A New Symmetric Discontinuous Galerkin Formulation: H
1(
P
h)-Coercivity
As demonstrated in the previous chapter, none of the consistent conventional DG formulations is well-posed with property (P). Consequently, a DG formulation that is well-posed with prop-erty (P), if existent, must be nonconventional. In this chapter we present a new nonconventional symmetric coercive DG formulation which is endowed with property (P) on account of coercivity on H1(Ph).
First, in Sec. 5.1, its weak formulation is given and we demonstrate consistency. Then in Sec. 5.2, its well-posedness is addressed. We show that the corresponding bilinear functional is coercive on the continuous space H1(
Ph). This important result allows us to conclude that the
new DG formulation is well-posed with property (P). 5.1 Weak Formulation
In conventional DG formulations, noncoercivity is caused by the nonpositive terms involving the product of edge values with edge derivative values (see the discussion at the end of Sec. 4.2.2). However, these terms are essential for consistency; see (2.10).
As a remedy, we construct our new symmetric DG formulation by augmenting (2.9) with nonconventional terms, making use of element Green’s functions and, additionally, of the data f
x
J K
e f
φJ,e
φK,e φK,f
Figure 3: Several solutions of auxiliary problem (5.1).
local to the edge under consideration. These terms enable coercivity of the bilinear functional, and simultaneously, when considering consistency, they reduce to the essential nonpositive terms. 5.1.1 Edgewise Auxiliary Problem for Element Green’s Functions
For each edge e∈ ΓI∪ΓD, we obtain (un)eby considering auxiliary problems. The auxiliary
prob-lems are local probprob-lems posed on elements K∈ Me neighboring the edge e under consideration.
Thus, fixing e, we select an element K ∈ Meand denote its two-point boundary by ∂K ={e, f}.
Now, let the function φK,e: K→ R solve the following Laplace problem on element K:
(5.1a) −d 2φ K,e dx2 = φK,e = 0 , −nenK, 0 , in K , on e , on f . (5.1b)
The solution φK,e ∈ C2(K) of the auxiliary problem (5.1) is a linear function on K. See Fig. 3
for several solutions of (5.1). We will need the| · |1,K norm of φK,e:
|φK,e|21,K= 1/hK . (5.2)
Specifically, φK,ecorresponds with the element Dirichlet-to-Neumann Green’s function for the 1-D
Laplacian. To see this, multiply (5.1a) with u∈ H2(K) and integrate over K. Performing twice
an integration by parts and invoking the boundary values of φK,e, we obtain
(un)e= Z K− d2u dx2φK,e dx− X {e,f } udφK dx nK , (5.3)
which shows that the ‘Neumann’ value of u at e is readily expressed in terms of the Laplacian and ‘Dirichlet’ values of u at e and f .
Due to the number of elements inMe, for each Dirichlet edge e∈ ΓDand interior edge e∈ ΓI,
we have one and two auxiliary problems, respectively. Thus, for an interior edge, (un)e can be
obtained from both elements by an arbitrary partition of unity. Let’s define for convenience the operators Φe: H1(Ph)→ R and ¯Φe: [H01(Ω)]0→ R for edges e ∈ ΓI∪ ΓD by:8
Φe(u) := X K∈Me θK,e Z K du dx dφK,e dx dx , (5.4a) ¯ Φe(f ) := X K∈Me θK,e Z K f φK,edx , (5.4b)
where the φK,e are the Green’s functions defined by auxiliary problem (5.1) and θK,e ∈ R (K ∈
Me) form a partition of unity, i.e., X
K∈Me
θK,e = 1 .
8The reason for taking [H1
Note that for e∈ ΓD we have θK,e= 1 (K∈ Me).
For the solution u of the Poisson problem (2.6), we can now obtain (un)e using the Green’s
functions φK,e and data f local to the edge e. This is stated in the following lemma:
Lemma 5.1. Consider an edge e∈ ΓI∪ ΓD. Let u∈ H2(Ω) be the solution of (2.6), then,
(un)e=−Φe(u) + ¯Φe(f) .
Proof. Multiply (5.1a) with u and integrate over K. An integration by parts gives 0 = Z K du dx dφK,e dx dx− X {e,f } udφK,e dx nK .
Combining this with (5.3) gives (un)e=− Z K du dx dφK,e dx dx + Z K f φK,edx ,
where (2.6) was used with a v∈ L2(Ω) such that v|K = φK,e and v is zero on Ω\ K. Multiplying
with θK,e, summing over all K∈ Meand employing the assumed partition of unity of θK,e gives
the lemma.
5.1.2 Consistent Weak Formulation Employing Element Green’s Functions
The operators Φe(·) and ¯Φe(·) associated with the element Green’s functions introduced above,
can be used to construct nonconventional terms. Eq. (2.9) can then be augmented with these nonconventional terms. It is the property stated in Lemma 5.1, that insures consistency.
Specifically, a consistent symmetric DG formulation posed on H1(Ph) can be constructed. The
weak formulation of this new DG formulation is given by: Find u∈ H1( Ph) : BΦ(u, v) =LΦ(v) , ∀v ∈ H1(Ph) , (5.5) where BΦ(u, v) := X K∈Ph Z K du dx dv dx dx + X e∈ΓI
α[[u]][[v]]/h + [[u]]Φ(v) + Φ(u)[[v]]
e
+ X
e∈ΓD
αuv/h + uΦ(v) + Φ(u)v
e, (5.6a) LΦ(v) := Z Ω f v dx + X e∈ΓI ¯ Φ(f )[[v]] e + X e∈ΓD αgDv/h + gDΦ(v) + ¯Φ(f )v e+ X e∈ΓN gNv e. (5.6b)
The parameters αe∈ R (e ∈ ΓI∪ΓD) are associated with jumps of u and v at edge e. The rationale
for adding these terms will be clarified by the coercivity analysis in Sec. 5.2.2. Unlike the αe in
the symmetric conventional DG formulation, in the new symmetric DG formulation (5.5), they need not be chosen approximation-space-dependent.
The following proposition establishes consistency of (5.5).
Proposition 5.2. The DG formulation (5.5) is consistent with (2.6), i.e., if u ∈ H2(Ω) is
the solution of (2.6), then
BΦ(u, v) =LΦ(v) , ∀v ∈ H1(Ph) .
Proof. Let u∈ H2(Ω) be the solution of (2.6). Recalling (2.10),
X K∈Ph Z K du dx dv dx dx = Z Ω f v dx + X e∈ΓI un[[v]] e+ X e∈ΓD unv e+ X e∈ΓN gNve,
for all v∈ H1(
Ph), and using this in
BΦ(u, v) results in BΦ(u, v) = Z Ω f v dx + X e∈ΓI un[[v]] e+ X e∈ΓD unv e+ X e∈ΓN gNve + X e∈ΓI
α[[u]][[v]]/h + [[u]]Φ(v) + Φ(u)[[v]]
e+
X
e∈ΓD
αuv/h + uΦ(v) + Φ(u)v
e.
For the solution u∈ H2(Ω)
⊂ C1(Ω), we have u
e = (gD)e for e∈ ΓD and [[u]]e= 0 for e∈ ΓI,
thus, BΦ(u, v) = Z Ω f v dx + X e∈ΓI un+ Φ(u) [[v]] e + X e∈ΓD αgDv/h + gDΦ(v) + un+ Φ(u) v e+ X e∈ΓN gNv e.
Finally, invoking Lemma 5.1, we obtain BΦ(u, v) = X K∈Ph Z K f v dx + X e∈ΓI ¯ Φ(f )[[v]] e + X e∈ΓD αgDv/h + gDΦ(v) + ¯Φ(f )v e+ X e∈ΓN gNv e =LΦ(v) .
To facilitate the well-posedness analysis in the next section, we equip H1(Ph) with the energy
norm||| · ||| (equivalent with k · kH1(Ph)), defined as
|||u|||2:= X K∈Ph |u|21,K+ X e∈ΓI [[u]]2/he+ X e∈ΓD u2/he. (5.7) 5.2 Well-Posedness Results
This section addresses well-posedness of (5.5). After establishing continuity of the function-als BΦ(·, ·) and LΦ(·), we show that BΦ(·, ·) is coercive on H1(Ph). Conclusions with respect
to well-posedness of the continuous and approximate problems are discussed last.
To assist the analysis, we first obtain a bound for he(Φe(u))2. For this, we consider a local
mesh parameter h : Γ→ R satisfying he≤12
X
K∈Me
hK , ∀e ∈ ΓI∪ ΓD. (5.8a)
Furthermore, for each e ∈ ΓI ∪ ΓD, we consider the following partition of unity for the θK,e
appearing in (5.4):
θK,e = hK/
X
J∈Me
hJ, ∀K ∈ Me. (5.8b)
Lemma 5.3. Consider an edge e∈ ΓI∪ ΓD. Under the conditions in (5.8), it holds that
he Φe(u) 2 ≤12 X K∈Me |u|21,K .
Proof. Recalling the definition of Φe(·), we first note that
he Φe(u) 2 = he X K∈Me θK,e Z K du dx dφK,e dx dx 2 .
Invoking the θ inequality, see (A.3), we obtain he Φe(u) 2 ≤ he X K∈Me θK,e Z K du dx dφK,e dx dx 2 .
Subsequently using the Schwarz inequality and (5.2), we obtain
he Φe(u) 2
≤ he
X
K∈Me
θK,e|u|21,K|φK,e|21,K ≤ he
X K∈Me θK,e hK |u| 2 1,K.
Finally, using conditions (5.8b) and (5.8a), gives the lemma:
he Φe(u) 2 ≤P he J∈MehJ X K∈Me |u|21,K≤ 12 X K∈Me |u|21,K.
5.2.1 Continuity of Bilinear and Linear Functional
We summarize the results concerning continuity in the following two propositions:
Proposition 5.4 (Continuity of BΦ). Let the conditions in (5.8) hold. The bilinear
func-tionalBΦ(·, ·) given in (5.6a) is continuous on H1(Ph)× H1(Ph), i.e.,
|BΦ(u, v)| ≤ cb|||u||||||v||| , ∀u, v ∈ H1(Ph) ,
with continuity constant cb= max2, 1+ max e∈ΓI∪ΓD
αe .
Proof. First note that |BΦ(u, v)| ≤ X K∈Ph Z K dudxdvdx dx + X e∈ΓI
α [[u]][[v]] /h + [[u]]Φ(v) + Φ(u)[[v]]
e
+ X
e∈ΓD
α uv /h + uΦ(v) + Φ(u)v
e.
Applying the Schwarz inequality on the first term then gives |BΦ(u, v)| ≤ X K∈Ph |u|1,K|v|1,K+ X e∈ΓI
α [[u]][[v]] /h + [[u]]Φ(v) + Φ(u)[[v]]
e
+ X
e∈ΓD
α uv /h + uΦ(v) + Φ(u)v
e.
Then, an application of the discrete Schwarz inequality yields
|BΦ(u, v)| ≤ X K∈Ph |u|21,K+ X e∈ΓI (1+α)[[u]]2/h + h Φ(u)2 e + X e∈ΓD (1+α)u2/h + h Φ(u)2 e !1/2 · · · !1/2 ,
where the dots (· · · ) represent an identical term with u replaced by v. Finally, invoking Lemma 5.3 gives the proposition:
|BΦ(u, v)| ≤ X K∈Ph |u|21,K+ X e∈ΓI (1+α)[[u]]2/h +12PK∈Me|u| 2 1,K e + X e∈ΓD (1+α)u2/h +12PK∈M e|u| 2 1,K e !1/2 · · · !1/2 ≤ 2 X K∈Ph |u|21,K+ X e∈ΓI (1+α)[[u]]2/h e+ X e∈ΓD (1+α)u2/h e !1/2 · · · !1/2 ≤ cb|||u||| |||v||| .
Before addressing the continuity of LΦ(·), we mention a useful lemma involving the
oper-ator ¯Φe(·). For this, we introduce the following definitions. Consider a v ∈ H1(Ph). Using
the Green’s functions φK,e defined by auxiliary problem (5.1), we can subtract a ‘discontinuous
part’ vd := vd(v) ∈ H1(Ph) of v and retrieve a continuous function vc ∈ H1
0(Ω). We define the ‘discontinuous part’ vdas vd:= X e∈ΓI [[v]]ePK∈M eθK,eEK(−φK,e) + X e∈ΓD vePK∈M eθK,eEK(−φK,e) , (5.9)
where we have introduced the extension operators EK : H1(K)→ H1(Ph),
EK(φ) = φ , in K , 0 , in Ω\ K .
Note that vd actually contains the ‘discontinuous part’ of v since we have
[[vd]]e= [[v]]e, ∀e ∈ ΓI,
ved= ve, ∀e ∈ ΓD ,
ved= 0 , ∀e ∈ ΓN .
It thus holds that
H01(Ω)3 vc:= vc(v) = v− vd, ∀v ∈ H1(Ph) , (5.10)
and, with C dependent onPh, we have the following inequality:
kvc
kH1(Ω)≤ C|||v||| , ∀v ∈ H1(Ph) . (5.11)
Note also that, in particular,
(vc)e= (v)e, ∀e ∈ ΓN . (5.12)
In Fig. 4, we illustrate for an example function v, the corresponding vd and vc. We now state the
lemma involving ¯Φe(·). Lemma 5.5. Consider an f∈ [H1 0(Ω)]0 ⊃ [H1(Ph)]0. It holds that Z Ω fv dx + X e∈ΓI ¯ Φ(f)[[v]] e+ X e∈ΓD ¯ Φ(f)v e= Z Ω fvcdx , ∀v ∈ H1(Ph) , with vc∈ H1
PSfrag replacements x ΓN ΓD v vd vc
Figure 4: Illustration of vdand vcfor an example function v ∈ H1(Ph) on a domain for which the left
boundary is ΓDand the right boundary is ΓN.
Proof. Consider a v ∈ H1(Ph). First note that
X e∈ΓI ¯ Φ(f )[[v]] e+ X e∈ΓD ¯ Φ(f )v e= X e∈ΓI [[v]]e P K∈MeθK,e R Kf φK,e dx + X e∈ΓD vePK∈MeθK,e R Kf φK,edx = Z Ω f X e∈ΓI [[v]]ePK∈M eθK,eEK(φK,e) + X e∈ΓD vePK∈MeθK,eEK(φK,e) ! dx .
Thus, using the definition of vd, we have
X e∈ΓI ¯ Φ(f )[[v]] e+ X e∈ΓD ¯ Φ(f )v e= Z Ω f (−vd) dx . As vc= v
− vd, the lemma follows immediately.
Next, we establish continuity ofLΦ(·).
Proposition 5.6 (Continuity of LΦ). Let the conditions in (5.8) hold. If and only if f ∈
[H1
0(Ω)]0, the linear functionalLΦ(·) given in (5.6b) is continuous on H1(Ph), i.e.,
|LΦ(v)| ≤ c`|||v||| , ∀v ∈ H1(Ph) ,
with continuity constant c`> 0, dependent on the data f, gDand gN, partitionPhand parameter αe
for e∈ ΓD.
Proof. Using Lemma 5.5, we have |LΦ(v)| = Z Ω f vcdx + X e∈ΓD αgDv/h + gDΦ(v) e+ X e∈ΓN gNv e ≤ Z Ω f vcdx + X e∈ΓD gD (α|v|/h + |Φ(v)|) e+ X e∈ΓN |gN||v| e (5.13)
with vc = vc(v)∈ H1
0(Ω) defined in (5.10). Thus, if f ∈ [H01(Ω)]0, then the first term is bounded
by Ckvc
kH1(Ω)with C dependent on f . The reverse implication can be shown by considering the
Poisson problem with homogeneous boundary conditions, i.e., gD = 0 on ΓD and gN = 0 on ΓN.
This yields
LΦ(v) =
Z
Ω
f vc dx , and f must necessarily be in [H1
0(Ω)]0 to obtain a continuous LΦ(·). We proceed with
bound-ing terms in (5.13). Using Lemma 5.3, the term |Φe(v)|, for e ∈ ΓD, can be bounded by
C1PK∈Me|v|1,K, with C1 dependent on h. The ΓN term can be bounded by C2kvkH1(Ph)using
a trace inequality, with C2 dependent on gN andPh. Substituting the bounds, we have
|LΦ(v)| ≤ CkvckH1(Ω)+ X e∈ΓD gD α|v|/h + C1PK∈Me|v|1,K e+ X e∈ΓN C2kvkH1(Ph).
Finally, invoking (5.11) and equivalence of||| · ||| with k · kH1(Ph), all terms can be bounded by c`|||v|||
and the lemma follows.
5.2.2 Coercivity on the Continuous SpaceH1(
Ph)
Next, we establish thatBΦ(·, ·) is coercive on H1(Ph).
Proposition 5.7 (Coercivity). Let the conditions in (5.8) hold. If the parameter αe> 1 for
all e∈ ΓI∪ ΓD, then the bilinear functional BΦ(·, ·) given in (5.6a) is coercive on H1(Ph), i.e.,
|BΦ(u, u)| ≥ κ|||u|||2, ∀u ∈ H1(Ph) , (5.14)
with, in particular, coercivity constant κ = min e∈ΓI∪ΓD 1 2 (αe−1) + 2 − p (αe−1)2+ 4 ∈ (0, 1). (5.15)
Note that αe can be chosen such that κ in (5.15) is bounded away from 0.
Proof. Let u∈ H1(Ph) be given. We consider 0 < κ < 1 and show that there exists a κ in this
interval such thatBΦ(u, u)− κ|||u|||2≥ 0. First note that
BΦ(u, u)− κ|||u|||2= (1−κ) X K∈Ph |u|21,K+ X e∈ΓI (α−κ)[[u]]2/h + 2[[u]]Φ(u) e + X e∈ΓD (α−κ)u2/h + 2uΦ(u) e.
Applying the elementary inequality, Eq. (A.4), gives
BΦ(u, u)− κ|||u|||2≥ (1−κ) X K∈Ph |u|21,K+ X e∈ΓI (α−κ)[[u]]2/h− [[u]] 2 (1−κ)h− (1−κ)h Φ(u) 2 e + X e∈ΓD (α−κ)u2/h − u 2 (1−κ)h− (1−κ)h Φ(u) 2 e .
We now invoke Lemma 5.3 to obtain BΦ(u, u)− κ|||u|||2≥ (1−κ) X K∈Ph |u|21,K+ X e∈ΓI α−κ−1−κ1 [[u]]2/h−12(1−κ)PK∈M e|u| 2 1,K e + X e∈ΓD α−κ− 1 1−κ u2/h−1 2(1−κ) P K∈Me|u| 2 1,K e.
