Stabilized interior penalty methods for the
time-harmonic Maxwell equations
I. Perugia a,1
D. Schötzau b,2
P. Monk c
a Dipartimento
di Matematica, Università di Pavia,
27100 Pavia, Italy, email: perugia@dimat.unipv.it
b Department
of Mathematics, University of Basel,
4051 Basel, Switzerland, email: schotzau@math.unibas.ch
c Department
of Mathematical Sciences, University of Delaware,
Newark, DE 19716, USA, email: monk@math.udel.edu
Comput. Methods in Appl. Mech. Engrg., Vol. 191,
2002, pp. 4675–4697
Abstract
We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence
of the electric field. We prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality
approach.
Key words: Finite elements, discontinuous Galerkin methods, interior penalty
methods, time-harmonic Maxwell’s equations
1
Supported in part by NSF Grant DMS-9807491 and by the Supercomputing Institute of the University of Minnesota. This work was carried out when the author
was visiting the School of Mathematics, University of Minnesota.
2 Supported in part by NSF Grant DMS-0107609 and by the Supercomputing Institute of the University of Minnesota. This work was carried out while the author
was affiliated with the School of Mathematics, University of Minnesota.
Preprint submitted to Elsevier Science
27 April 2016
1
Introduction
The numerical solution of the time-harmonic Maxwell equations presents a
number of challenges. First, away from boundaries and material interfaces, the
solution is smooth and oscillatory. The need to approximate the oscillations
requires a sufficiently fine grid compared to the wavelength of the solution,
and results in the need for a large number of degrees of freedom at short
wavelengths. This requirement can be loosened (but not entirely avoided)
by the use of high order methods [1,2], so that it is desirable to use high
order methods where the solution is smooth. A second problem is that at
the boundary of the domain the solution can be singular [3]. Indeed, on a
non-convex polyhedral domain the straightforward application of continuous
finite element methods can result in a discrete solution that converges to a
vector function that is not a solution of Maxwell’s equations [4]. While it
is possible to modify the variational form to correct for this failure [5], a
similar problem also occurs at interfaces between different materials. This is
because at discontinuities in the electric properties of the materials, the electric
field is discontinuous. Thus continuous elements need to be modified at such
interfaces.
Considerations of the two problems mentioned above have lead to a widespread adoption of edge finite elements [6,7] for the discretization of the timeharmonic Maxwell equations. For an engineering view of such elements, a good
summary is contained in the books [8,9]. An error analysis of these elements
has been given in [10–12] and the profound connection between these elements
and differential forms has been noted for example in [13–15]. Perhaps the main
problem with such elements is that they become rather complex as the order
of the elements is increased, and like all conforming methods they require a
suitable finite element grid which complicates implementing adaptive solvers.
Nevertheless, adaptive hp-finite element solvers have been implemented and
show considerable promise [16].
In this paper we propose a new way to discretize the indefinite time-harmonic
Maxwell system based on a discontinuous Galerkin method (denoted DG in
the remainder of the paper). In particular, we propose a suitable extension of
the interior penalty methods to the Maxwell system. These methods date back
at least to [17–20] and have been studied for coercive elliptic and convection
diffusion problems more recently in [21–23]. We mention that several other
DG methods for standard coercive elliptic problems can be found in the literature (for instance the LDG method [24,25] or the DG method introduced by
Baumann and Oden [26,27]) and unified analyzes of discontinuous methods in
the context of elliptic problems have been presented in [28,29]. For the timeharmonic Maxwell equations in the low-frequency regime, where the resulting
bilinear forms are coercive, the LDG methods have been recently investigated
2
in [30].
Our goal is to produce a flexible solver in which the order of the scheme
can be changed easily between different regions of the grid. In addition, we
hope to exploit the fact that DG grids do not need to be aligned in order
to improve the efficiency of wave propagation of the method. Thus in regions
with different electromagnetic properties (and hence different wave speeds),
different grid sizes can be used to balance the propagation accuracy of the
scheme in each subdomain. Finally it may be possible to “tune” parameters
in the DG scheme to improve propagation accuracy (this is certainly possible
in one space dimension!).
In this paper we prove basic error estimates for our proposed schemes under
the assumption of smoothly varying material properties. This assumption is
needed for certain a-priori estimates used in the analysis. Ultimately we hope
to extend these results to more general coefficients and boundary conditions.
Perhaps the closest approach in the literature to the DG methods we propose
is the ultra weak variational method of Cessenat [31]. While successful in practice, this method is still incompletely understood on a theoretical level. For
example, convergence is not proved for the standard perfectly electrically conducting boundary condition (or near a singularity), or in general throughout
the domain of computation. However the successful use of this method is one
motivation for proposing the methods in this paper which are convergent globally even in the presence of boundary singularities. Another similar approach
is the mortar finite element method applied to the Maxwell equations [32].
To our knowledge, convergence has not been proved for this method in the
case of wave propagation. However the success of this method applied to low
frequency eddy current problems (in which case the resulting bilinear forms
are coercive) suggests that mortar methods or similar domain decomposition
methods could be useful for scattering problems [33]. Yet another domain decomposition approach is the FETI method applied to the Maxwell equations
[34]. A Lagrange multiplier based version of this method was analyzed in [35]
for the coercive Maxwell problem arising in time stepping. Again to our knowledge, convergence has not been proved for this method in the case of wave
propagation in Maxwell’s equations.
The outline of our paper is as follows. In section 2, we start by detailing the
mixed formulation we shall use as the basis of the DG methods proposed here.
We also summarize some regularity and existence results. Then in section 3 we
propose the DG methods that are the subject of this paper. The main result
of the paper is an optimal a priori error bound that we present in section 4. Its
proof is based on a duality approach and is contained in section 5 and section 6.
We end our presentation with some concluding remarks in section 7.
3
2
A mixed formulation for the time-harmonic Maxwell equations
In this section, we introduce the time-harmonic Maxwell equations and present
a mixed formulation for the continuous problem which will be the basis for
the DG methods introduced here.
2.1 Time-harmonic Maxwell’s equations
Let Ω be a bounded Lipschitz polyhedron in R3 with connected boundary ∂Ω.
The model problem we shall consider is to compute a time-harmonic electric
field E in the cavity Ω with perfectly conducting boundary. Let ω denote the
temporal frequency of the time-harmonic field so that the corresponding time
dependent field E at position x ∈ Ω and time t is given by
E(x, t) = < (E(x) exp(−iωt)) .
Then E : Ω → C3 satisfies the Maxwell system
2
∇ × µ−1
r ∇ × E − k εr E = J
in Ω,
(1)
where µr is the relative magnetic permeability and εr is the relative electric
permittivity of the medium in the cavity Ω. We assume that µr and εr are real,
smooth, and uniformly positive functions of the position in Ω. In addition the
real wave number k is given by
√
k = ω ε 0 µ0 ,
where µ0 is the magnetic permeability and ε0 is the electric permittivity of
free space. The source function J is related to the applied current density
driving the cavity and is assumed to be a given vector function in L2 (Ω)3 .
The assumption that Ω has a perfectly conducting boundary gives the following boundary condition on ∂Ω:
n × E = 0 on ∂Ω.
(2)
Here n denotes the outward normal unit vector to ∂Ω.
Throughout the paper, we will assume that k 2 is not an interior Maxwell
eigenvalue (see also Proposition 1 below), i.e., for any E 6= 0, the pair (λ =
k 2 , E) is not an eigensolution of the problem ∇ × µ−1
r ∇ × E = λ εr E in Ω, n ×
E = 0 on ∂Ω. Note that this assumption would not be necessary if some region
of Ω (containing a ball of non-zero radius) had a non zero conductivity which
would imply that the imaginary part of εr is positive there. Note also that in
4
the special case considered here the real and imaginary parts of the solution
decouple, and hence we can assume that E is real. If εr is complex valued or
if impedance boundary conditions are imposed, the real and imaginary parts
are coupled.
2.2 Mixed formulation
Our DG method is based on a mixed formulation of the Maxwell boundary
value problem (1)–(2). Such formulations have been used previously for edge
element discretizations of Maxwell’s equations to improve stability [16], and
to handle coercive problems in which meshes are not aligned at a material
boundary [35]. We can derive this formulation by using a Helmholtz decomposition.
Given a domain D in R2 or R3 , we denote by H s (D)d , d = 1, 2, 3, the Sobolev
space of real or complex scalar- or vector-valued functions with regularity
exponent s ≥ 0, endowed with the usual norm k · ks,D and seminorm | · |s,D .
We write H01 (D) for the subspace of H 1 (D) of functions with zero trace and
set L2 (D)d = H 0 (D)d . For the computational domain Ω ⊂ R3 , we let
n
H(div0εr ; Ω) = u ∈ L2 (Ω)3 | ∇ · (εr u) = 0 in Ω
n
o
H(div0 ; Ω) = u ∈ L2 (Ω)3 | ∇ · u = 0 in Ω ,
o
equipped with the L2 (Ω)3 -norm. We also use the standard spaces
n
H(curl; Ω) = u ∈ L2 (Ω)3 | ∇ × u ∈ L2 (Ω)3
o
H0 (curl; Ω) = {u ∈ H(curl; Ω) | n × u = 0 on ∂Ω} ,
endowed with the norm
kuk2curl,Ω = kuk20,Ω + k∇ × uk20,Ω .
Furthermore, let L2εr (Ω)3 denote the space of square integrable functions on Ω
equipped with the inner product
(u, v)εr =
Z
Ω
εr u · v dx.
We consider the L2εr (Ω)3 -orthogonal Helmholtz decomposition of the function
E ∈ H0 (curl; Ω) given by E = u + ∇p, where u ∈ H0 (curl; Ω) ∩ H(div0εr ; Ω)
and p ∈ H01 (Ω), see [36].
Using this decomposition, problem (1)–(2) can be reformulated as follows: find
5
u ∈ H(curl; Ω) and p ∈ H 1 (Ω) such that
2
2
∇ × µ−1
r ∇ × u − k εr u − k εr ∇p = J
∇ · (εr u) = 0
n×u=0
p=0
in Ω
in Ω
on ∂Ω
on ∂Ω.
(3)
(4)
(5)
(6)
We start by showing well-posedness of problem (3)–(6).
Proposition 1 Assume that k 2 is not a Maxwell eigenvalue. Then problem
(3)–(6) has a unique solution (u, p) ∈ H0 (curl; Ω) ∩ H(div0εr ; Ω) × H01 (Ω), with
µ−1
r ∇ × u ∈ H(curl; Ω), and we have the stability estimates
kpk1,Ω ≤ k −2 Cell kJ k0,Ω ,
kukcurl,Ω ≤ Cstab kJ k0,Ω ,
with positive constants Cstab and Cell , Cell independent of k.
Proof Consider the Helmholtz decomposition of J as a function in L2 (Ω)3 ,
J = J0 + ∇j with J0 ∈ H(div0 ; Ω) and j ∈ H01 (Ω). Owing to the orthogonality of this decomposition, problem (3)–(6) decouples into two independent
subproblems, namely into the Maxwell problem with divergence free data
2
∇ × µ−1
r ∇ × u − k εr u = J 0
∇ · (εr u) = 0
n×u=0
in Ω
in Ω
on ∂Ω,
(7)
p = 0 on ∂Ω,
(8)
and the elliptic problem
−k 2 ∇ · (εr ∇p) = F
in Ω,
with right hand side F ∈ H −1 (Ω) defined by F (q) = − Ω ∇j ·∇q dx, for all q ∈
H01 (Ω). Existence and uniqueness of solutions to (7) follow
now in a standard
R
way from Fredholm theory and the coercivity of the form Ω µ−1
r ∇×u·∇×v dx
on the space H0 (curl; Ω)∩H(div0εr ; Ω) (see [36, Proposition 2.7]). Furthermore,
µ−1
r ∇ × u ∈ H(curl; Ω) and
R
kukcurl,Ω ≤ Cstab kJ0 k0,Ω ≤ Cstab kJ k0,Ω ,
with a stability constant Cstab > 0 depending on Ω, µr , εr and on the wave
number k 2 . For problem (8), existence and uniqueness follows from standard
elliptic theory, and we have
kpk1,Ω ≤ k −2 Cell kF k−1,Ω ≤ k −2 Cell k∇jk0,Ω ≤ k −2 Cell kJ k0,Ω ,
with Cell > 0 only depending on Ω and εr .
6
2
In our duality approach in section 6 we will also make use of the following
regularity result.
Proposition 2 For smooth coefficients µr and εr , there exists a regularity
exponent σ = σ(Ω) > 21 such that the solution u in (3)–(6) satisfies u ∈
H σ (Ω)3 and ∇ × u ∈ H σ (Ω)3 . Furthermore,
kukσ,Ω + k∇ × ukσ,Ω ≤ Creg kJ k0,Ω ,
with a positive constant Creg depending on Ω, k 2 , µr and εr .
Proof By decoupling problem (3)–(6) into (7) and (8), we see that the solution
u of (7) satisfies ∇ × u ∈ L2 (Ω)3 , ∇ · u ∈ L2 (Ω)3 (here we use the assumption
that εr is smooth, so that εr ∇ · u = ∇ · (εr u) − ∇εr · u holds true) and
n × u = 0 on ∂Ω . From [37, Proposition 3.7], it follows that u ∈ H σ1 (Ω)3 for
σ1 > 12 and kukσ1 ,Ω ≤ C(Ω, εr )kukcurl,Ω with an embedding constant C(Ω, εr )
just depending on Ω and εr . Thus, from Proposition 1, we have kukσ1 ,Ω ≤
C1 kJ k0,Ω .
Now set w = µ−1
r ∇ × u. From the first equation in (7), we have ∇ × w =
J0 +k 2 εr u ∈ L2 (Ω)3 . Furthermore, ∇·(µr w) = 0 and µr w·n = ∇×u·n = 0 on
∂Ω. Since µr is smooth, using again [37, Proposition 3.7], it follows that w ∈
H σ2 (Ω)3 for σ2 > 12 and kwkσ2 ,Ω ≤ C(Ω, µr )kwkcurl,Ω ≤ C(Ω, µr )(k∇ × uk20,Ω +
1
kJ0 + k 2 εr uk20,Ω ) 2 . Hence, from the triangle inequality and Proposition 1, we
conclude that kwkσ2 ,Ω ≤ C2 kJ k0,Ω . Choosing σ := min{σ1 , σ2 } and Creg =
max{C1 , C2 } completes the proof.
2
Remark 3 If the polyhedron Ω is convex and µr = εr = 1, the parameter σ
in Proposition 2 can be chosen as σ = 1, see [37].
3
Discontinuous Galerkin discretization
In this section, we introduce stabilized interior penalty discontinuous Galerkin
discretizations for the Maxwell system (3)–(6).
3.1 Triangulations
Let Th be a regular triangulation of the domain Ω into tetrahedra. We denote
by hK the diameter of the element K and set h = maxK∈Th hK . The diameter
of the face f is denoted by hf . We also assume the triangulation to be shape
7
regular, that is, there is a positive constant κ such that, for any K ∈ Th ,
hK
≤ κ,
ρK
where ρK is the diameter of the biggest ball contained in K (see [38, p. 124]).
Let E be the union of all the faces of Th , and EI the union of the internal
faces. We define the function h in L∞ (E) by
if x ∈ f.
h = h(x) = hf
3.2 Trace operators
First, we need to define some notation concerning functions in H s (Th ) := {v :
v|K ∈ H s (K), K ∈ Th }, for s > 21 . The elementwise traces of such functions
belong to TR(E) := ΠK∈Th L2 (∂K); they are double-valued on EI and singlevalued on E \ EI . The space L2 (E) can be identified with the functions in
TR(E) for which the two trace values coincide.
Next, we introduce certain trace operators. To this end, fix w ∈ TR(E)3 and
ϕ ∈ TR(E), and let e ⊂ EI be an interior face shared by the elements K1 and
K2 . Let ni be the normal unit vector pointing exterior to Ki and wi = w|∂Ki ,
ϕi = ϕ|∂Ki (i = 1, 2). Then we define for x ∈ e the average, the tangential
jump and the normal jump of w as follows:
1
{{w}} = (w1 +w2 )
2
[[w]]T = n1 ×w1 +n2 ×w2
[[w]]N = w1 ·n1 +w2 ·n2 .
Similarly, we define for x ∈ e the average and the normal jump of ϕ by
1
{{ϕ}} = (ϕ1 + ϕ2 )
2
[[ϕ]]N = ϕ1 n1 + ϕ2 n2 .
Then, on any boundary face e ⊂ E \ EI , we set for x ∈ e
{{w}} = w
[[w]]T = n × w
[[ϕ]]N = ϕn.
Since we will not require either of the quantities {{ϕ}} and [[w]]N on the boundary E \ EI , we leave them undefined.
If w ∈ H(curl; Ω), then, for all e ⊂ EI , the jump condition n1 ×w1 +n2 ×w2 = 0
−1
−1
holds true in H00 2 (e)3 , and thus also in L2 (e)3 (for the definition of H00 2 (e), see,
e.g., [39]). Therefore, [[w]]T is equal to zero on EI . Similarly, for w ∈ H(div; Ω),
we have that [[w]]N is well-defined and equal to zero on EI . Furthermore, for
the exact solution u ∈ H0 (curl; Ω) ∩ H(div0εr ; Ω), we have [[u]]T = 0 in L2 (e)3
for any boundary face e, in addition to [[u]]T = 0 and [[εr u]]N = 0 on EI .
8
3.3 Stabilized interior penalty discontinuous Galerkin methods
We approximate u and p in the discontinuous finite element space Vh × Qh
where
Vh = {v ∈ L2 (Ω)3 : v|K ∈ P ` (K)3 , ∀K ∈ Th }
Qh = {q ∈ L2 (Ω) : q|K ∈ P ` (K), ∀K ∈ Th },
for an approximation order ` ≥ 1, with P ` (K) denoting the space of polynomials of degree at most ` on K.
We consider the following discontinuous Galerkin method: find (uh , ph ) ∈ Vh ×
Qh such that, for any (v, q) ∈ Vh × Qh ,
a(uh , v)+c(uh , v)+d(uh , v)−k 2 (uh , v)εr +b(v, ph ) = F (v)
b(uh , q) −e(ph , q)
= 0.
(9)
(10)
Here,
a(u, v) =
Z
µ−1
r ∇h
ΩZ
−
b(v, p) =k 2
ZE
c(u, v) =α
Z
× u · ∇h × v dx −
[[v]]T ·
E
[[u]]T · {{µ−1
r ∇h × v}} ds
× u}} ds
p ∇h · (εr v) dx − k 2
Ω
Z
EI
{{p}} [[εr v]]N ds
h−1 m−1 [[u]]T · [[v]]T ds
EZ
d(u, v) =k 2 β
EI
+ k2β
h [[εr u]]N [[εr v]]N ds
X
K∈Th
e(p, q) =k 2 γ
{{µ−1
r ∇h
Z
Z
E
h2K
Z
K
∇h · (εr u) ∇h · (εr v) dx
h−1 e[[p]]N · [[q]]N ds + α
Z
E
h−1 m−1 [[∇h p]]T · [[∇h q]]T ds,
with ∇h ×, ∇h · and ∇h denoting the elementwise curl, divergence and gradient,
respectively, and the functions m and e are defined on E as the restriction to
E of µr and εr , respectively. The parameters α, β and γ in the forms c, d,
and e are positive. The purpose of these forms is to stabilize the method. The
functional F at right-hand side of (9) is
F (v) =
Z
Ω
J · v dx.
Let us discuss the following points about this method:
• The form a(·, ·) + c(·, ·) corresponds to the interior penalty discretization
of the curl-curl operator, cf. [28]; it is symmetric and stable provided that
9
the parameter α is large enough (see Lemma 14 below). The nonsymmetric
variant of the interior penalty discretization is obtained by changing the sign
in front of the second term in the definition of the form a, that is, by choosing
a(u, v) =
Z
µ−1
r ∇h × u · ∇h × v dx +
ΩZ
−
E
[[v]]T ·
{{µ−1
r ∇h
Z
E
[[u]]T · {{µ−1
r ∇h × v}} ds
(11)
× u}} ds.
Then the form a(·, ·) + c(·, ·) is nonsymmetric, but stable for any α > 0 (see
Remark 15 below). In the following we will only present the analysis for the
symmetric method in (9)-(10), but emphasize that the error estimates so obtained hold true verbatim for its nonsymmetric variant.
• The form b(·, ·) discretizes the divergence constraint in the mixed formulation
(3)–(6) by means of DG techniques, similar to the forms used in [40] for the
Stokes system. Notice that, after integration by parts, the form b(·, ·) can also
be expressed by
b(v, p) = −k
2
Z
Ω
εr v · ∇h p dx + k
2
Z
E
[[p]]N · {{εr v}} ds,
(v, p) ∈ Vh × Qh .
• The forms d(·, ·) and e(·, ·) provide stabilization. While d(·, ·) is related
to the divergence constraint, the form e(·, ·) provides stability via control of
jumps of the scalar potential p. We found it necessary to include these forms
in order to be able to prove optimal error estimates with our techniques of
analysis. Whether or not similar results can actually be obtained without
these stabilization forms remains an open question and will be investigated
numerically in a forthcoming work.
• By elementary manipulations the third term in the form a(·, ·) can be expressed by
−
Z
E
[[v]]T ·
{{µ−1
r ∇h
× u}} ds =
X Z
K∈K ∂K
−
Z
EI
v · nK × [µ−1
r ∇h × u] ds
{{v}} ·
[[µ−1
r ∇h
(12)
× u]]T ds,
for all u, v ∈ Vh , where nK is the outward normal unit vector to ∂K. For the
exact solution u we have u ∈ H0 (curl; Ω) and µ−1
r ∇ × u ∈ H(curl; Ω). Thus,
[[u]]T = 0 on E and [[µ−1
∇
×
u]]
=
0
on
E
,
and,
with (12), a(u, v) has to be
h
T
I
r
understood as
a(u, v) =
Z
Ω
µ−1
r ∇h
× u · ∇h × v dx +
X Z
K∈K ∂K
v · nK × [µ−1
r ∇h × u] ds,
for v ∈ Vh , where the boundary integrals are in fact duality pairings.
Let us now address the consistency of the method.
10
Proposition 4 The discontinuous Galerkin method in (9)–(10) is consistent,
i.e., the exact solution (u, p) of problem (3)–(6) satisfies (9)–(10), for all test
functions (v, q) ∈ Vh × Qh .
0
Proof We have u ∈ H0 (curl; Ω), µ−1
r ∇ × u ∈ H(curl; Ω) and u ∈ H(divεr ; Ω).
Thus, [[u]]T = 0 on E, as well as [[εr u]]N = 0 and [[µ−1
r ∇h × u]]T = 0 on EI .
1
Moreover, p ∈ H0 (Ω) and ∇p ∈ H0 (curl; Ω) and thus [[p]]N = 0 and [[∇p]]T = 0
on E. The second equation (10) is then trivially satisfied for all q ∈ Qh . From
(12), the first equation (9) reduces to
Z
Ω
µ−1
r ∇ × u · ∇h × v dx +
− k2
Z
Ω
εr u · v dx + k 2
Z
Ω
X Z
K∈K ∂K
v · nK × [µ−1
r ∇h × u]ds
p ∇h · (εr v) dx − k 2
Z
EI
{{p}}[[εr v]]N ds =
Z
Ω
J · v dx,
for v ∈ Vh . Integration by parts over each element K, taking into account the
boundary conditions for u and p, yields
Z
Ω
2
2
∇ × µ−1
r ∇ × u − k εr u − k εr ∇p · v dx =
Z
Ω
J · v dx,
which is satisfied for all v ∈ Vh .
2
Remark 5 Note that the regularity of the solution stated in Proposition 2 as
consequence of the smoothness assumption on the coefficients is needed neither
in the definition of the method, nor in the proof of Proposition 4. As a matter
of fact, the method is defined and consistent for piecewise smooth coefficients
µr and εr . In this case, the functions m and e have to be adjusted by taking the
corresponding averages, i.e., by taking m = {{µr }} and e = {{εr }} on E.
Remark 6 The analysis developed in the following sections makes use of conforming projection operators, and therefore only covers the case of meshes
that do not contain hanging nodes. On the other hand, the DG method is
well-defined for general non-matching grids. In this case, the interior faces
are understood as the (non-empty) interiors of the intersections between two
adjacent elements and the function h has to be redefined on E as
h = h(x) =





4
1
(hK
2
+ hK 0 )
if x ∈ e = ∂K ∩ ∂K 0
hK
if x ∈ e ⊂ ∂K ∩ ∂Ω.
The main result
In this section, we present and discuss our main result – an optimal a priori
error estimate for the DG method in (9)–(10). The proof of this bound is
11
developed in section 5 and section 6; it is based on a suitable duality argument
that heavily relies on the regularity result of Proposition 2, and therefore the
assumption of smooth coefficients µr and εr is essential. Moreover, in order to
simplify the presentation, we restrict ourselves to the case µr = εr = 1. The
extension to general smooth coefficients is straightforward.
Define the spaces
V(h) := Vh + H0 (curl; Ω) ∩ H(div0 ; Ω))
Q(h) := Qh + H01 (Ω),
and the broken norm ||| (u, p) |||h given by
||| (u, p) |||2h = k 2 kuk20,Ω + k 2 k∇h pk20,Ω + | (u, p) |2h,
where the seminorm | (u, p) |h is given by
1
1
| (u, p) |2h = k∇h × uk20,Ω + α kh− 2 [[u]]T k20,E + k 2 β kh 2 [[u]]N k20,EI
+ k2β
X
K∈Th
1
1
h2K k∇ · uk20,K + k 2 γ kh− 2 [[p]]N k20,E + α kh− 2 [[∇h p]]T k20,E .
It is easy to see that ||| (·, ·) |||h is actually a norm in V(h) × Q(h).
Our main result establishes error estimates in the norm ||| (u, p) |||h .
Theorem 7 Assume that the exact solution (u, p) of the continuous problem
(3)–(6) satisfies
u ∈ H s (Ω)3
∇ × u ∈ H s (Ω)3
p ∈ H s+1 (Ω)
1
s> ,
2
(13)
and let (uh , ph ) be the discrete solution obtained by the DG method (9)–(10).
Then, there exist positive constants α0 and β0 , with α0 = α0 (κ, `) and β0 =
β0 (κ, `, Cell), such that for stabilization parameters α > α0 and β > β0 we
have the error bound
||| (u − uh , p − ph ) |||h ≤ C hmin{`,s} kuks,Ω + k∇ × uks,Ω + kpks+1,Ω ,
provided that h ≤ h0 for some h0 = h0 (κ, `, k, Creg , α, β, γ, Ω, s), with a constant C > 0 independent of the meshsize.
Remark 8 Theorem 7 guarantees optimal a priori error bounds provided that
the stabilization parameters α and β are large enough. Restrictions of this type
are typically encountered in interior penalty methods (see, e.g., [28,41]). It is
worth noting that α0 and β0 are independent of the wave number k 2 .
Remark 9 Note that the smoothness assumptions u ∈ H s (Ω)3 and ∇ × u ∈
H s (Ω)3 follow from Proposition 2, whereas the assumption p ∈ H s+1 (Ω) does
not seem to hold true for general source terms J in L2 (Ω)3 . This lack of
12
smoothness in the potential p for general right hand sides will be the major
difficulty in our duality argument in section 6 below and is the reason why we
introduced the stabilization forms d(·, ·) and e(·, ·). We also point out that for
divergence free source terms J (often encountered in practice) we have p = 0
and the assumption p ∈ H s+1 (Ω) is trivially satisfied.
Proceeding along the lines of [42], we can conclude from the a priori error
estimates in Theorem 7 existence and uniqueness of discrete solutions.
Corollary 10 For stabilization parameters α > α0 and β > β0 , the DG
method (3)–(4) admits a unique solution, provided that h ≤ h0 .
Proof We only need to establish that if J = 0, then the only solution is
uh = 0 and ph = 0. But if J = 0, then u = 0, p = 0 and the estimate of
Theorem 7 implies ||| (uh , ph ) |||h ≤ 0 for h ≤ h0 . Since ||| (·, ·) |||h is a norm on
Vh × Qh , we conclude that uh = 0 and ph = 0.
2
The proof of Theorem 7 is developed in section 5 and section 6. First, we
rewrite the DG method in a non-conforming fashion, using lifting operators
similar to the ones introduced in [28], and prove an inf-sup condition. Then, we
derive an abstract error estimate which can be viewed as a variant of Strang’s
lemma. This is done in section 5. Finally, in section 6, we make explicit the
error estimates for the principal part of the problem and then for the L2 -norm
of the error by a duality approach where we need β to be large enough.
5
Abstract error estimates
In this section, we prove an abstract error estimate for our DG method. The
key ingredient to obtain this estimate is an inf-sup condition which we prove in
Proposition 17, following an argument often used in the analysis of stabilized
finite elements in Stokes flow (see, e.g., the survey article [43] and the references
therein).
5.1 A global bilinear form
For the purpose of our analysis, we replace the integrals over interelement
boundaries by volume integrals given in terms of the lifting operators L :
13
L2 (E)3 → Vh and M : L2 (EI ) → Qh defined by
Z
L(v) · w dx =
Ω
Z
M(v) q dx =
Ω
Z
ZE
v · {{w}} ds
∀w ∈ Vh
v {{q}} ds
∀q ∈ Qh .
EI
We also need the lifting operator N : L2 (EI ) → Vh defined by
Z
Ω
N (v) · w dx =
Z
EI
v [[w]]N ds
∀w ∈ Vh .
Consider the forms alift (·, ·) and blift (·, ·) given by
alift (u, v) :=
Z
Ω
−
blift (v, p) :=k
2
∇h × u · ∇h × v dx
Z h
Z
Ω
Ω
i
L([[u]]T ) · (∇h × v) + L([[v]]T ) · (∇h × u) dx
p ∇h · v dx − k 2
Z
Ω
M([[v]]N ) p dx.
Again, by integration by parts, we have
blift (v, p) = −k 2
Z
Ω
v · ∇h p dx + k 2
Z
Ω
L([[p]]N ) · v dx,
∀(v, p) ∈ Vh × Qh .
For discrete test and trial functions, the forms alift (·, ·) and blift (·, ·) coincide
with a(·, ·) and b(·, ·). However, this is no longer true for continuous functions,
due to the discrete nature of the lifting operators. Nevertheless, we carry out
our analysis in terms of the forms alift (·, ·) and blift (·, ·) since they have favorable
continuity and coercivity properties and take into account the inconsistency
of the forms by a variant of Strang’s lemma.
Introducing the global form Bh (u, p; v, q) defined by
Bh (u, p; v, q) :=alift (u, v) + c(u, v) + d(u, v) − k 2 (u, v)
+ blift (v, p) − blift (u, q) + e(p, q),
we can rewrite the DG method (9)–(10) in the following compact form: find
(uh , ph ) ∈ Vh × Qh such that
Bh (uh , ph ; v, q) = F (v),
(14)
for all (v, q) ∈ Vh × Qh .
5.2 Stability of the lifting operators
The following standard inequalities (see, e.g., [38]) will be useful in the rest of
the paper.
14
Lemma 11 For polynomials r ∈ P ` (K), we have
−1
|r|1,K ≤ Cinv h−1
K krk0,K ,
krk0,∂K ≤ Cinv hK 2 krk0,K
with a constant Cinv > 0 only depending on the shape regularity constant κ
and the polynomial degree `.
We start by establishing stability estimates for the lifting operators.
Proposition 12 Let L and M be the lifting operators defined above. We have
that, for all v ∈ V(h) and for all q ∈ Q(h),
1
1
kL([[q]]N )k0,Ω ≤ Clift kh− 2 [[q]]N k0,E
kL([[v]]T )k0,Ω ≤ Clift kh− 2 [[v]]T k0,E
1
1
kM([[v]]N )k0,Ω ≤ Clift kh− 2 [[v]]N k0,EI kN ([[v]]N )k0,Ω ≤ Clift kh− 2 [[v]]N k0,EI
with a constant Clift > 0 only depending on the shape regularity constant κ
and the polynomial degree `.
Proof We prove the first estimate. Given v = wh + w ∈ V(h), observe that
[[v]]T = [[wh ]]T on E. By the definition of the operator L and the CauchySchwarz inequality, we have
kL([[v]]T )k0,Ω = sup
z∈Vh
R
Ω
L([[v]]T ) · z dx
= sup
kzk0,Ω
z∈Vh
1
1
R
kh− 2 [[v]]T k0,E kh 2 {{z}}k0,E
.
≤ sup
kzk0,Ω
z∈Vh
· {{z}} ds
kzk0,Ω
E [[v]]T
By using the definitions of {{·}} and h, and the first inequality in Lemma 11,
we obtain
1
kh 2 {{z}}k20,E ≤ C
X
K∈Th
hK kzk20,∂K ≤ C
X
K∈Th
kzk20,K = Ckzk20,Ω .
This proves the first estimate. The other estimates are obtained similarly. 2
5.3 Continuity
We can state the following continuity properties.
Proposition 13 There exists a positive constant C only depending on κ and
` such that, for all (u, p), (v, q) ∈ V(h) × Q(h),
|alift (u, v)| ≤ C||| (u, 0) |||h||| (v, 0) |||h
|blift (u, p)| ≤ C||| (u, 0) |||h||| (0, p) |||h
15
|c(u, v)| ≤ C||| (u, 0) |||h||| (v, 0) |||h
|d(u, v)| ≤ C||| (u, 0) |||h ||| (v, 0) |||h
|e(p, q)| ≤ C||| (0, p) |||h||| (0, q) |||h.
Consequently, for all (u, p), (v, q) ∈ V(h) × Q(h),
Bh (u, p; v, q) ≤ Ccont ||| (u, p) |||h||| (v, q) |||h,
for a continuity constant Ccont > 0 only depending on the shape regularity
constant κ and the polynomial degree `.
Proof Using the first estimate in Proposition 12, we have
|alift (u, v)| ≤k∇h × uk0,Ω k∇h × vk0,Ω + k∇h × vk0,Ω kL([[u]]T )k0,Ω
+ k∇h × uk0,Ω kL([[v]]T )k0,Ω ≤ C||| (u, 0) |||h||| (v, 0) |||h.
Then, owing to the second estimate in Proposition 12,
|blift (u, p)| ≤ k 2 k∇h p − L([[p]]N )k0,Ω kuk0,Ω ≤ C||| (u, 0) |||h||| (0, p) |||h.
The estimates for c(·, ·), d(·, ·) and e(·, ·) are straightforward.
2
5.4 Inf-sup condition
We show the stability of the form Bh in the following two steps: we start
by proving, in Lemma 14, a Gårding inequality for the form Bh in terms
of the seminorm | (·, ·) |h and then, in Lemma 16, a stability estimate for
Bh (u, p; −∇h p, −p); by combining these results, we obtain the inf-sup condition in Proposition 17.
Lemma 14 There exists a positive constant C independent of h and k such
that, for all (u, p) in Vh × Qh ,
Bh (u, p; u, p) ≥ C| (u, p) |2h − k 2 kuk20,Ω ,
2
provided that α > Clift
, where Clift is the constant in the estimates of Proposition 12.
Proof First, we prove the following coercivity property: for all u ∈ Vh ,
1
alift (u, u) + c(u, u) ≥ C(k∇h × uk20,Ω + α kh− 2 [[u]]T k20,E ).
16
(15)
Using the arithmetic geometric mean inequality |ab| ≤
first bound in Proposition 12, we have, for any δ > 0,
δ 2
a
2α
+
α 2
b ,
2δ
and the
alift (u, u) + c(u, u)
= k∇h × uk20,Ω − 2
Z
Ω
1
∇h × u · L([[u]]T ) dx + α kh− 2 [[u]]T k20,E
1
δ
α
≥ 1−
k∇h × uk20,Ω − kL([[u]]T )k20,Ω + α kh− 2 [[u]]T k20,E
α
δ
2 1
δ
Clift
2
≥ 1−
k∇h × uk0,Ω + α 1 −
kh− 2 [[u]]T k20,E .
α
δ
2
2
Owing to the assumption α > Clift
, we can take Clift
< δ < α and obtain (15).
Now, since Bh (u, p; u, p) := alift (u, u) + c(u, u) + d(u, u) − k 2 kuk20,Ω + e(p, p),
the result immediately follows from the coercivity property (15) and from the
definition of the seminorm | (·, ·) |h.
2
Remark 15 If we replace the form alift by its nonsymmetric variant derived
from (11), Lemma 14 holds true for any α > 0. For the symmetric method
2
in (9)–(10), we will assume throughout the text that α satisfies α > C lift
(the
2
constant α0 in Theorem 7 is actually Clift ).
Lemma 16 Let (u, p) ∈ Vh × Qh . Then there exist positive constants C1 , C2
and C3 independent of h and k such that
||| (∇h p, p) |||h ≤ C1 ||| (u, p) |||h
Bh (u, p; −∇h p, −p) ≥ C2 k 2 k∇h pk20,Ω − C3 | (u, p) |2h − C3 k 2 kuk20,Ω .
Proof Let us first prove the continuity property. From Lemma 11, the following
1
bounds hold: kh 2 [[∇h p]]N k0,EI ≤ Ck∇h pk0,Ω and k∆pk0,K ≤ Ch−1
K k∇pk0,K .
Then, from the definition of ||| (·, ·) |||h, we obtain
1
1
||| (∇h p, p) |||2h = 2 k 2 k∇h pk20,Ω + 2 α kh− 2 [[∇h p]]T k20,E + k 2 β kh 2 [[∇h p]]N k20,EI
+ k2 β
X
K∈Th
1
h2K k∆pk20,K + k 2 γ kh− 2 [[p]]N k20,E
1
2
1
≤ C k k∇h pk20,Ω + 2 α kh− 2 [[∇h p]]T k20,E + k 2 γ kh− 2 [[p]]N k20,E
≤ C1 ||| (u, p) |||2h,
for any u ∈ Vh , with C1 = C1 (Cinv , β).
In order to prove the bound for Bh (u, p; −∇h p, −p), we estimate separately
the bilinear forms that are involved. We consider first alift (u, −∇h p). From
17
Proposition 12:
alift (u, −∇h p) =
Z
Ω
≥−
L([[∇h p]]T ) · ∇h × u dx
2
1
1
Clift
kh− 2 [[∇h p]]T k20,E − k∇h × uk20,Ω .
2
2
For c(·, ·) and e(·, ·) we have
Z
c(u, −∇h p) = −α h−1 [[u]]T · [[∇h p]]T ds
E
1
α
α −1
≥ − kh 2 [[u]]T k20,E − kh− 2 [[∇h p]]T k20,E
2
2
1
2
2
− 21
e(p, −p) = −k γkh [[p]]N k0,E − αkh− 2 [[∇h p]]T k20,E .
Let us consider now blift (·, ·). We have again from Proposition 12 with arithmetic geometric mean inequalities
blift (−∇h p, p) =k
≥
2
k∇h pk20,Ω
−k
2
Z
Ω
L([[p]]N ) · ∇h p ds
2
1
k2
k 2 Clift
k∇h pk20,Ω −
kh− 2 [[p]]N k20,E
2
2
and
−blift (u, −p) = − k 2
Z
Ω
u · ∇h p dx + k 2
Z
Ω
L([[p]]N ) · u ds
2
1
9
k2
k 2 Clift
≥ − k 2 kuk20,Ω − k∇h pk20,Ω −
kh− 2 [[p]]N k20,E .
2
16
2
For d(·, ·), we have
2
d(u, ∇h p) = − k β
Z
2
Ω
N (h[[u]]N ) · ∇h p dx − k β
X
h2K
K∈Th
Z
K
∇ · u ∆p dx
k2
k∇h pk20,Ω
16
X
k2
− 4 k2C 2β 2
h2K k∇ · uk20,K − k∇h pk20,Ω ,
16
K∈Th
1
2
≥ − 4 k 2 β 2 Clift
kh 2 [[u]]N k20,E −
where we used the estimates in Proposition 12, the inverse estimate k∆pk0,K ≤
Ch−1
K k∇pk0,K and arithmetic geometric mean inequalities with appropriate
weights. Finally, we note that
k 2 (u, ∇h p) ≥ −4k 2 kuk20,Ω −
k2
k∇h pk20,Ω .
16
Adding together all the contributions from the bilinear forms we obtain the
result (with our choice of the weights, C2 = 41 ).
2
We are now ready to prove the following inf-sup condition.
18
Proposition 17 There are positive constants C1 , C2 and C3 independent of
h and k such that, for any (v, q) ∈ Vh × Qh , there is (w, s) ∈ Vh × Qh such
that
||| (w, s) |||h ≤ C1 ||| (v, q) |||h
Bh (v, q; w, s) ≥ C2 (| (v, q) |2h + k 2 k∇h qk20,Ω ) − C3 k 2 kvk20,Ω .
(16)
Proof Set (w, s) = δ(v, q) − (∇h q, q), combine Lemma 14 and Lemma 16 and
choose δ large enough.
2
5.5 A variant of Strang’s lemma
We prove the following abstract error estimate involving the residual
Rh (u, p; v, q) = Bh (u, p; v, q) − F (v),
which takes into account the inconsistency of the formulation (14).
Theorem 18 There is a positive constant C independent of h and k such that
the error (u − uh , p − ph ) satisfies
||| (u − uh , p − ph ) |||h ≤C
+
inf
(v,q)∈Vh ×Qh
||| (u − v, p − q) |||h
sup
(0,0)6=(w,s)∈Vh ×Qh
|Rh (u, p; w, s)|
+ k ku − uh k0,Ω .
||| (w, s) |||h
Proof Fix (v, q) ∈ Vh × Qh . We split the error (u − uh , p − ph ) into
(u − uh , p − ph ) = (u − v, p − q) + (v − uh , q − ph ) =: (ϕu , ϕp ) + (ξu , ξp ).
We bound (ξ u , ξp ), which we may assume to be nonzero. By Proposition 17,
there exists a nonzero test function (w, s) ∈ Vh × Qh satisfying (16) with
(v, q) = (ξ u , ξp ). We obtain
||| (ξu , ξp ) |||2h = k 2 kξu k20,Ω + k 2 k∇h ξp k20,Ω + | (ξu , ξp ) |2h
≤ C |Bh (ξ u , ξp ; w, s)| + C k 2 kξ u k20,Ω
≤ C |Bh (ϕu , ϕp ; w, s)| + C|Rh (u, p; w, s)| + C k 2 kξu k20,Ω
|Rh (u, p; w, s)|2
≤ Cδ −1 ||| (ϕu , ϕp ) |||2h + Cδ −1
||| (w, s) |||2h
+ Cδ||| (w, s) |||2h + C k 2 kξu k20,Ω
|Rh (u, p; w, s)|2
≤ Cδ −1 ||| (ϕu , ϕp ) |||2h + Cδ −1
||| (w, s) |||2h
+ Cδ||| (ξu , ξp ) |||2h + C k 2 kξ u k20,Ω ,
19
for any δ > 0, where we used the definition of the residual Rh , the continuity
of Bh , arithmetic geometric mean inequalities and ||| (w, s) |||h ≤ C ||| (ξu , ξp ) |||h .
Hence, the parameter δ can be chosen such that
||| (ξu , ξp ) |||2h ≤ C||| (ϕu , ϕp ) |||2h + C
|Rh (u, p; w, s)|2
+ C k 2 kξu k20,Ω .
||| (w, s) |||2h
Since k 2 kξ u k20,Ω ≤ k 2 ku − uh k20,Ω + k 2 kϕu k20,Ω , we have
||| (ξu , ξp ) |||2h
≤ C ||| (ϕu , ϕp ) |||2h
+
sup
(0,0)6=(w,s)∈Vh ×Qh
|Rh (u, p; w, s)|2
+ k 2 ku − uh k20,Ω .
||| (w, s) |||2h
The assertion now follows by applying the triangle inequality and taking the
infimum over all (v, q) ∈ Vh × Qh .
2
Remark 19 The result of Theorem 18 holds true also in the case β = 0.
The positivity of β and the stability induced by the corresponding forms will
be invoked in the duality argument of the next section.
6
Error estimates
In this section, we make explicit the abstract error estimate in Theorem 18.
6.1 Approximation properties
First, we review the approximation results for the L2 -projection, for standard
H 1 -conforming and for curl-conforming Nédéléc operators.
Lemma 20 Let w ∈ H t (K), t ≥ 0. Let Π be the L2 -projection from H t (K)
onto P ` (K). Then for m integer, 0 ≤ m ≤ t, we have
min{`+1,t}−m
|w − Πw|m,K ≤ ChK
kwkt,K .
Moreover, if t > 21 ,
min {`+1,t}− 21
kw − Πwk0,∂K ≤ ChK
kwkt,K .
The constant C > 0 only depends on the shape regularity constant κ, the
polynomial degree ` and the smoothness parameter t.
20
Proof For natural numbers t, the first estimate follows from the classical
Bramble-Hilbert theory (see, e.g., [38]); for t = 0, it is a consequence of the
stability of the L2 -projection. For non-integer t, it can be obtained by interpolation. The second estimate follows from the trace theorem from L2 (∂K) to
H t (K), from the first estimate, and scaling arguments.
2
We also need a standard H 1 -conforming approximant, see [38], and a Clément
operator, as constructed in [44].
Lemma 21 The standard nodal H 1 -conforming interpolant ΠH 1 : [H t+1 (Ω) ∩
H01 (Ω)] → [Qh ∩ H01 (Ω)] satisfies
kw − ΠH 1 wkm,Ω ≤ C hmin{`+1,t+1}−m kwkt+1,Ω
m = 0, 1,
for t > 21 . The constant C > 0 only depends on the shape regularity constant κ,
the polynomial degree ` and and the smoothness parameter t.
This lemma is proved in [38] for integer t, and can be proved for non-integer
t by using the arguments in [45].
For the Clément operator ΠCl : H01 (Ω) → [Qh ∩ H01 (Ω)], we recall the following
result (see, e.g., [44, pp. 109–111]).
Lemma 22 There exists a constant C > 0 only depending on the shape regularity constant κ and the polynomial degree ` such that
X K∈T
2
−1
2
2
kw − ΠCl wk21,K + h−2
K kw − ΠCl wk0,K + hK kw − ΠCl wk0,∂K ) ≤ Ckwk1,Ω .
Finally, we establish the next lemma for curl-conforming Nédéléc operators.
Lemma 23 Let Πcurl be the curl-conforming Nédéléc operator (either of the
first type [6] or of the second type [7]) into Vh ∩H0 (curl; Ω). Then, there exists
a positive constant C = C(κ, `, t) such that, for any w ∈ H0 (curl; Ω) ∩ H t (Ω)3
with ∇ × w ∈ H t (Ω)3 , t > 12 ,
kw − Πcurl wkcurl,Ω ≤ C hmin{`,t} (kwkt,Ω + k∇ × wkt,Ω ).
(17)
Moreover, if w also belongs to H(div0 ; Ω),
1
kh 2 [[w − Πcurl w]]N k0,EI ≤ C hmin{`,t} kwkt,Ω + k∇ × wkt,Ω
X
K∈Th
h2K k∇ · (w − Πcurl w)k20,K
1
2
≤ C hmin{`,t} kwkt,Ω + k∇ × wkt,Ω .
21
Proof The first part is proved in [45, Section 5]. For the second part, denote by
ΠVh the L2 -projection onto Vh . By using the triangle inequality, the approximation results in Lemma 20, the first estimate in Lemma 11, the L2 -stability
of ΠVh and the first part of this lemma, we obtain
1
kh 2 [[w − Πcurl w]]N k20,EI
≤C
X
K∈Th
hK kw − ΠVh wk20,∂K + kΠVh (w − Πcurl w)k20,∂K
≤C
X
hK
2 min{`+1,t}
K∈Th
2 min{`+1,t}
≤ Ch
kwk2t,K + C
kwk2t,Ω
X
K∈Th
kΠVh (w − Πcurl w)k20,K
+ Ckw − Πcurl wk20,Ω
2
≤ Ch2 min{`,t} kwkt,Ω + k∇ × wkt,Ω .
To prove the last estimate, we integrate by parts and obtain
X
K∈Th
=−
+
h2K k∇ · (w − Πcurl w)k20,K
X
h2K
Z
X
h2K
Z
K∈Th
K∈Th
K
∇∇ · (w − Πcurl w) · (w − Πcurl w)dx
∂K
∇ · (w − Πcurl w)(w − Πcurl w) · nK ds =: T1 + T2 .
Let us first consider the volume term T1 . Using the fact that ∇ · w = 0,
Cauchy-Schwarz inequalities, the second estimate in Lemma 11 and the first
part of this lemma, we get
T1 ≤
X
K∈Th
h4K k∇∇ · Πcurl wk20,K
1
kw − Πcurl wk0,Ω
1
kw − Πcurl wk0,Ω
2
≤C
X
h2K k∇ · Πcurl wk20,K
=C
X
h2K k∇ · (w − Πcurl w)k20,K
K∈Th
K∈Th
≤ Chmin{`,t}
X
K∈Th
2
1
2
kw − Πcurl wk0,Ω
h2K k∇ · (w − Πcurl w)k20,K
22
1 2
kwkt,Ω + k∇ × wkt,Ω .
Similarly, we can bound the term T2 by
T2 ≤
X
≤
X
K∈Th
K∈Th
h3K k∇ · Πcurl wk20,∂K
h2K k∇ · Πcurl wk20,K
X
K∈Th
≤ Chmin{`,t}
1 X
2
K∈Th
hK kw − Πcurl wk20,∂K
1
2
1
2
hK kw − ΠVh wk20,∂K + kΠVh (w − Πcurl w k20,∂K
X
K∈Th
h2K k∇ · (w − Πcurl w)k20,K
1 2
1
2
kwkt,Ω + k∇ × wkt,Ω ,
where we used the first estimate in Lemma 11 for the term containing the sum
1
and proceeded as in the estimate of kh 2 [[w−Πcurl w]]N k0,EI for the second term.
This completes the proof of the last estimate.
2
6.2 Error in the principal part
We have the following estimate of the residual in Theorem 18.
Lemma 24 Let (u, p) be the exact solution and assume that ∇ × u ∈ H s (Ω)3
and p ∈ H s+1 (Ω), for s > 21 . Then, for all (v, q) ∈ V(h) × Q(h),
Rh (u, p; v, q) =
Z
E
[[v]]T · {{∇ × u − ΠVh (∇ × u)}} ds
+k
2
Z
EI
[[v]]N {{p − ΠQh p}} ds,
where ΠVh and ΠQh denote the L2 -projections onto Vh and Qh , respectively.
Moreover, there exists C > 0 independent of h and k such that
|Rh (u, p; v, q)| ≤ C hmin{`,s} ||| (v, 0) |||h k∇ × uk2s,Ω + kpk2s+1,Ω
1
2
.
Proof By straightforward calculations involving integration by parts and taking into account the definition of the lifting operators L and M, we have that,
for any (v, p) ∈ V(h) × Q(h),
Rh (u, p; v, q) =
Z
E
{{∇ × u}} · [[v]]T ds −
+ k2
Z
EI
Z
{{p}} [[v]]N ds − k 2
23
∇ × u · L([[v]]T ) dx
ΩZ
Ω
p M([[v]]N ) dx.
Since
Z
ZΩ
Ω
∇ × u · L([[v]]T ) dx =
p M([[v]]N ) dx =
Z
Ω
Z
Ω
ΠVh (∇ × u) · L([[v]]T ) dx
(18)
ΠQh p M([[v]]N ) dx,
we obtain the desired expression for Rh (u, p; v, q).
For the estimates of the residual, let us write Rh (u, p; v, q) =: T1 + T2 , where
T1 =
Z
E
T2 = k 2
[[v]]T · {{∇ × u − ΠVh (∇ × v)}} ds
Z
EI
[[v]]N {{p − ΠQh p}} ds.
By the Cauchy-Schwarz inequality, the definition of the norm ||| (·, ·) |||h and
the second estimate of Lemma 20, we obtain the following bound:
T1 ≤C ||| (v, 0) |||h
X
hK k∇ × u − ΠVh (∇ ×
≤C ||| (v, 0) |||h
X
2 min{`+1,s}
hK
k∇
×
T2 ≤ C ||| (v, 0) |||h
X
h−1
K kp
ΠQh pk20,∂K
≤ C ||| (v, 0) |||h
X
2 min{`,s}
hK
kpk2s+1,K
K∈Th
K∈Th
uk2s,K
u)k20,∂K
1
2
1
2
.
Similarly,
K∈Th
K∈Th
−
The estimate for Rh then follows.
1
2
1
2
.
2
We are now ready to prove the following error estimate.
Corollary 25 Under the assumptions of Theorem 7, there exists a constant
C > 0 independent of the meshsize h such that
||| (u−uh , p−ph ) |||h ≤ C hmin{`,s} (kuks,Ω +k∇×uks,Ω +kpks+1,Ω )+Ckku−uh k0,Ω .
Proof Consider the abstract estimate of Theorem 18 and bound the infimum
by ||| (u − Πcurl u, p − ΠH 1 p) |||h , where Πcurl is the curl-conforming Nédéléc
operator and ΠH 1 the standard H 1 -conforming interpolant from Lemma 21.
Taking into account the approximation properties in Lemma 21 and Lemma 23
and the estimate of the residual in Lemma 24, we obtain the result.
2
24
6.3 Error in the L2 -norm
In order to complete our error analysis, we need to estimate the term k ku −
uh k0,Ω . This is done in the next proposition by a duality approach. The main
difficulty in this argument is that we can not assume any smoothness for the
scalar potential of the dual solution. To overcome these difficulties we will
have to make the stabilization constant β large enough.
Proposition 26 Let σ > 21 be the regularity exponent from Proposition 2.
Furthermore, we assume that the exact solution satisfies the smoothness assumptions in (13) with s > 21 . Then we have
kku − uh k0,Ω ≤C1 hmin{σ,1} ||| (u − uh , p − ph ) |||h + C1 hmin{`,s} kpks+1,Ω
1
+ C2 β − 2 ||| (u − uh , p − ph ) |||h ,
with positive constants C1 = C1 (κ, `, k, β, Creg , σ, s) and C2 = C2 (κ, `, Cell ).
Consequently, there exists a positive constant β0 = β0 (κ, `, Cell ) such that for
β > β0
kku − uh k0,Ω ≤
1
||| (u − uh , p − ph ) |||h + C hmin{`,s} kpks+1,Ω ,
2
provided that h ≤ h0 for a positive constant h0 = h0 (κ, `, k, Creg , α, β, γ, σ, s).
Proof The proof is given in several steps. We start by introducing a suitable
adjoint problem with right-hand side k 2 (u−uh ), denoting by (z, ψ) its solution,
and we express k 2 ku − uh k20,Ω as the sum of Bh (u − uh , p − ph ; z − Πcurl z, ψ −
ΠCl ψ) plus residual terms, with Πcurl the curl-conforming Nédéléc operator
into Vh ∩ H0 (curl; Ω) and ΠCl the Clément operator into Qh ∩ H01 (Ω) (step
1). Then, estimates of the residuals (step 2) and of the bilinear forms in the
definition of Bh give the result (steps 3 and 4).
Step 1: A dual problem. Let (z, ψ) be the solution of the dual problem
∇ × ∇ × z − k 2 z + k 2 ∇ψ = k 2 (u − uh )
∇·z =0
n×z=0
ψ=0
in Ω
in Ω
on ∂Ω
on ∂Ω.
(19)
By Proposition 1 and Proposition 2, we have
kzkσ,Ω + k∇ × zkσ,Ω ≤ Creg k 2 ku − uh k0,Ω
kψk1,Ω ≤ Cell ku − uh k0,Ω
(20)
for a regularity exponent σ = σ(Ω) > 21 . We may also assume that σ ≤ 1. The
main problem in the following arguments is that we can not assume ψ to be
25
smoother than belonging to H01 (Ω).
The solution (z, ψ) of problem (19) satisfies
Bh (z, −ψ; v, q) − Rh (z, −ψ; v, q) = k 2 (u − uh , v)
for all (v, q) ∈ H s (Th )3 × H s (Th ), s > 21 . Taking (v, q) = (u − uh , −(p − ph )),
observing (13), we obtain
Bh (z, −ψ; u − uh , −(p − ph )) − Rh (z, −ψ; u − uh , −(p − ph )) = k 2 ku − uh k20,Ω .
Since Bh (u − uh , p − ph ; zh , ψh ) = Rh (u, p; zh , ψh ) = −Rh (u, p; z − zh , ψ − ψh ),
for all (zh , ψh ) ∈ Vh × Qh , and using the skew-symmetry properties of Bh , we
can write
k 2 ku − uh k20,Ω = Bh (u − uh , p − ph ; z − zh , ψ − ψh ) + R1 + R2 + R3 + R4 , (21)
with residual terms
R1 = −
R2 = k 2
R3 = −
Z
ZE
[[u − uh ]]T · {{∇ × z − ΠVh (∇ × z)}} ds,
Z EI
[[z − zh ]]T · {{∇ × u − ΠVh (∇ × u)}} ds,
EZ
R4 = −k 2
[[u − uh ]]N {{ψ − ΠQh ψ}} ds,
EI
[[z − zh ]]N {{p − ΠQh p}} ds.
We define zh = Πcurl z, so that R3 = 0, and ψh = ΠCl ψ, with Πcurl the curlconforming Nédéléc operator from Lemma 23 and ΠCl the standard Clément
operator which satisfy the approximation property of Lemma 22.
Step 2: The residuals. We estimate the residual expressions R1 , R2 and
R4 in (21) (recall that R3 = 0). Let us start with R1 . The Cauchy-Schwarz
inequality and the approximation properties in Lemma 20 yield
R1 ≤ C||| (u − uh , 0) |||h
X
K∈Th
hK k∇ × z − ΠVh (∇ × z)k20,∂K
1
2
σ
≤ Ch ||| (u − uh , 0) |||hk∇ × zkσ,Ω .
Thus, from (20),
R1 ≤ C hσ k 2 ||| (u − uh , 0) |||h ku − uh k0,Ω .
The crucial term is R2 . Again from the Cauchy-Schwarz inequality, using the
second estimate of Lemma 20 and the stability estimates (20) for the dual
26
solution, we obtain
1
R2 ≤ C k 2 β kh 2 [[u − uh ]]N k0,EI
≤ Ckβ
≤ Ckβ
− 21
− 21
1 2
k 2 β −1
X
K∈Th
2
h−1
K kψ − ΠQh ψk0,∂K
1
2
||| (u − uh , 0) |||h kψk1,Ω
||| (u − uh , 0) |||h ku − uh k0,Ω ,
with C = C(κ, `, Cell ). Finally, for R4 we have
R4 ≤ C||| (z − Πcurl z, 0) |||h k 2 β −1
≤Ch
min{`,s}
X
K∈Th
2
h−1
K kp − ΠQh pk0,∂K
1
2
k ||| (z − Πcurl z, 0) |||hkpks+1,Ω .
Step 3: The term ||| (z − Πcurl z, 0) |||h. We claim that
||| (z − Πcurl z, 0) |||h ≤ C hσ k ku − uh k0,Ω ,
(22)
with C = C(κ, `, k, Creg , β, σ).
To see (22), we first note that, since the Nédéléc projection is curl-conforming,
1
kh− 2 [[z − Πcurl z]]T k0,E = 0. Furthermore, by (17) and (20),
k∇ × (z − Πcurl z)k20,Ω + k 2 kz − Πcurl zk20,Ω ≤ Ch2σ (kzkσ,Ω + k∇ × zkσ,Ω )2
≤ C h2σ k 4 ku − uh k20,Ω .
From the second and third estimates of Lemma 23 and (20), we have
1
k 2 β kh 2 [[z − Πcurl z]]N k20,EI ≤ C h2σ k 2 ku − uh k20,Ω
k2 β
X
K∈T
h2K k∇ · (z − Πcurl z)k20,K ≤ C h2σ k 2 ku − uh k20,Ω .
The proof of estimate (22) now follows from the definition of ||| (z−Πcurl z, 0) |||h .
Step 4: The assertion. We are now able to complete the proof Proposition 26. Define ξ z = z − Πcurl z and ξψ = ψ − ΠCl ψ. From (21), taking
into account that, due to the conformity of the projectors Πcurl and ΠCl ,
c(u − uh , ξz ) = 0 and e(p − ph , ξψ ) = 0, we have that
k 2 ku − uh k20,Ω =alift (u − uh , ξz ) + d(u − uh , ξz ) − k 2 (u − uh , ξz )
+ blift (ξ z , p − ph ) − blift (u − uh , ξψ ) + R1 + R2 + R3 .
From the continuity properties of Proposition 13, we obtain
k 2 ku − uh k20,Ω ≤C ||| (u − uh , p − ph ) |||h ||| (ξz , 0) |||h
+ |blift (u − uh , ξψ )| + R1 + R2 + R3 ,
27
(23)
where we isolated the term blift (u − uh , ξψ ) that needs to be treated separately.
By the Cauchy-Schwarz inequality we have
|blift (u − uh , ξψ )|2
=
Z
2
k
Ω
≤C
2
ξ ψ ∇h
2
k β
X
2
· (u − uh ) dx
K∈Th
h2K k∇h
· (u −
Z
+ k 2
Ω
uh )k20,K
2 −1
· k β
X
K∈Th
2
M([[u − uh ]]N ) ξψ dx
2
+ k βkM(h[[u −
2
h−2
K kξψ k0,K
uh ]]N )k20,Ω
,
with C = C(κ). Now, using the approximation property of Lemma 22 of the
Clément operator, the third estimate in Proposition 12, the second estimate
in (20) and the definition of ||| (u − uh ) |||h , we get
|blift (u − uh , ξψ )|2
≤ C 2 k 2 β −1 k 2 β
2
2 −1
≤C k β
X
K∈Th
1
h2K k∇h · (u − uh )k20,K + k 2 βkh 2 [[u − uh ]]N k20,EI kψk21,Ω
||| (u − uh , 0) |||2h ku − uh k20,Ω ,
where C = C(κ, `, Cell ). Inserting this, (22) and the estimates for the residuals
obtained in Step 2 in (23) completes the proof.
2
The proof of Theorem 7 follows now from Corollary 25 and Proposition 26.
7
Conclusions
In this paper, we have carried out an error analysis for stabilized interior
penalty discontinuous Galerkin methods for the discretization of the indefinite
time-harmonic Maxwell equations. We have derived error estimates that are
optimal, provided that the stabilization parameters are large enough and the
meshsize is small enough. A numerical study of the proposed methods is the
subject of ongoing work.
Acknowledgment and disclaimer
The effort of Peter Monk was sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number
F49620-96-1-0039. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright no28
tation thereon. The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the Air Force Office
of Scientific Research or the US Government.
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