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Transcript
```1
Technical Article
Recent Developments in Discontinuous Galerkin Methods for the Time–
Harmonic Maxwell’s Equations
P. Houston1 , I. Perugia2 and D. Schötzau3
ICS Newsletter, Vol. 11 (2004), pp. 11-17
discontinuous Galerkin (DG, for short) methods for the
discretization of the time-harmonic Maxwell’s equations,
based on the interior penalty discretization of the curl-curl
operator. Direct and mixed methods will be presented for
both the low- and high-frequency cases. The performance
of the proposed DG methods will be highlighted in a series
of numerical examples with known analytical solutions.
I. Introduction
In this article, we review recent developments in discontinuous Galerkin (DG, for short) methods with interior penalty for the discretization of the time-harmonic
Maxwell’s equations: find the complex-valued electric
field u such that
∇ × (µ−1 ∇ × u) − ω 2 (ε − iω −1 σ)u = j
n×u=0
in Ω, (1)
on Γ, (2)
where Γ = ∂Ω. Here, Ω is a simply-connected Lipschitz
polyhedron in R3 with connected boundary Γ = ∂Ω and
unit outward normal vector n. The function j is a given
(complex-valued) source term in L2 (Ω)3 . The temporal
frequency is denoted by ω > 0. The real-valued functions µ, ε and σ are the magnetic permeability, electric
permittivity, and electric conductivity, respectively.
The origins of DG methods can be traced back to the
seventies, where they were proposed for the numerical
solution of the neutron transport equation, as well as for
the weak enforcement of continuity in Galerkin methods
for elliptic and parabolic problems; see  for a historical
review. In the meantime, these methods have undergone
quite a remarkable development and are used in a wide
range of applications; see the recent survey articles ,
, , and the references cited therein. The main advantages of DG methods, and in particular the application
of DG methods for the numerical approximation of the
above problem are the following:
• DG methods are locally conservative. That is, since
the underlying partial differential equations are enforced
elementwise, local conservation properties can be easily
preserved on the discrete level, very much in the spirit of
finite volume methods. This is particularly advantageous
in time-domain computations (which are not considered
in this paper); see, e.g., ,  and the references therein.
• The numerical robustness of DG methods means that
they can easily treat a wide range of problems within the
same unified framework.
1 Department of Mathematics, University of Leicester, Leicester
LE1 7RH, UK. Email: [email protected] (Funded by
the EPSRC: Grant GR/R76615)
2 Dipartimento di Matematica, Università di Pavia, Via Ferrata
For mixed formulations, a wider choice of stable finite element spaces are available when DG methods are
employed in comparison to their conforming counterparts. Indeed, in Section II-C.2, we shall see that both
equal–order and mixed–order polynomial spaces may be
employed for the numerical approximation of the lowfrequency approximation to (1).
• Being based on discontinuous finite element spaces, DG
methods can easily handle meshes with hanging nodes
and local spaces of different orders. This renders DG
methods well-suited for adaptive mesh refinement, as well
• The implementation of discontinuous elements can be
based on standard shape functions; a convenience that
is particularly advantageous for high-order elements and
that is not straightforwardly shared by standard edge
elements commonly used in computational electromagnetics (see , ,  and the references cited therein for
Finally, we remark that while the total number of degrees
of freedom of a DG method is typically larger than that of
the corresponding conforming method on the same mesh,
this increase is not too dramatic for higher-order elements
since most of the degrees of freedom are in the interior
of the elements.
In this article, we focus on interior penalty DG discretizations for (1)–(2) in both the low- and highfrequency cases. We point out that many other DG approaches could be used instead of the interior penalty
method presented here, with no major changes in the
theoretical analysis; see, e.g., the discussion in .
In the low-frequency case, the term ω 2 ε is neglected
in (1). Problem (1)–(2) then has to be completed by
a divergence-free constraint in the subdomain Ω0 ⊆ Ω
covered by insulating materials where σ = 0 (additional
scalar constraints arise if ∂Ω0 is not connected, see,
e.g., ). This results in the following system:
•
∇ × (µ−1 ∇ × u) + iωσ u = j in Ω,
∇ · (εu) = 0 in Ω0 ,
n × u = 0 on Γ.
(3)
The main difficulty here is the incorporation of the divergence-free constraint in the DG framework. We will consider two approaches: in the first one, the constraint is
accounted for by introducing a regularization term as in
 (see ); in the second one, a mixed approach as
in  and  is adopted, and the constraint is imposed
by introducing a suitable Lagrange multiplier (see 
and ). As will be shown in our numerical tests, the
regularized DG approach suffers from the same drawback
as its conforming counterpart, namely, if the solution ex-
2
lized and non-stabilized formulations that were designed
and analyzed in  and , respectively. We discuss
the corresponding energy norm a priori error estimates,
as well as the energy norm a posteriori estimates obtained
in  for the non-stabilized formulation in . For further numerical tests, we also refer the reader to .
In the high-frequency case, renaming (ε − iω −1σ) by ε,
problem (1)–(2) becomes
∇×(µ−1 ∇×u)−ω 2 εu = j in Ω, n×u = 0 on Γ. (4)
Here, we assume that ω 2 is not an eigenvalue of the underlying Maxwell eigenproblem. While the design of interior penalty DG methods is straightforward, the key difficulty for (4) arises in the numerical analysis of the methods due to the indefiniteness caused by the zeroth order
term. The first DG method for problem (4), based on a
mixed formulation, was introduced and studied in .
The method there contains volume stabilization terms
which have been numerically observed to be unnecessary.
Here, we present the main results of a novel error analysis that was recently developed in  and , for a
direct and a mixed formulation, respectively, which do
not contain the volume stabilization terms of the method
in . These results show that DG methods for (4) yield
optimal rates of convergence in the energy norm and the
L2 (Ω)-norm.
that µ and ε are constants. However, the analysis in ,
 covers the case of piecewise smooth material coefficients, and the theoretical results in ,  can be
element spaces:
Vh := {v ∈ L2 (Ω)3 : v|K ∈ P ` (K)3 ∀K ∈ Th },
Qh := {q ∈ L2 (Ω) : q|K ∈ P m (K) ∀K ∈ Th },
where P k (K) denotes the space of (complex) polynomials
of total degree at most k on K.
For s ≥ 0 and D a bounded domain in R2 or R3 , we denote by k · ks,D the standard norm in the Sobolev space
H s (D)d , d ≥ 1. For s = 0, we write L2 (D) in lieu of
H 0 (D), and denote by (·, ·)D Rthe standard L2 (Ω)-inner
product defined by (v, w)D = D v w dx. We further define the broken Sobolev space
H s (Th ) = {v ∈ L2 (Ω) : v|K ∈ H s (K), ∀K ∈ Th },
endowed with the broken Sobolev norm denoted by k ·
ks,Th .
For the computational domain Ω ⊂ R3 , H(curl; Ω) is
the space of functions in L2 (Ω)3 with curl in L2 (Ω)3 , and
H(div; Ω) the space of functions in L2 (Ω)3 with divergence in L2 (Ω). We denote by H0 (curl; Ω) and H0 (div; Ω)
the subspaces of H(curl; Ω) and H(div; Ω), respectively,
with zero tangential and normal boundary trace, respectively.
Finally, we set V(h) = H0 (curl; Ω) + Vh and Q(h) =
H01 (Ω) + Qh and define the following DG norms with
which we will measure the approximation errors:
1
1
kvk2V(h) = kε 2 vk20,Ω + kµ− 2 ∇h × vk20,Ω
1
1
+ kµ− 2 h− 2 [[v]]T k20,Fh ,
1
II. Discontinuous Galerkin Discretizations
In this section, we introduce interior penalty DG methods for the two model problems in (3) and (4), and review
their theoretical properties.
A. Preliminaries
We consider conforming, shape-regular affine meshes
Th that partition the domain Ω into tetrahedra {K}; the
parameter h denotes the mesh size of Th given by h =
maxK∈Th hK , where hK is the diameter of the element
K ∈ Th . We denote by FhI the set of all interior faces of
elements in Th , by FhB the set of all boundary faces, and
set Fh := FhI ∪ FhB . We define the local meshsize h on Fh
by setting h(x) := max{hK + , hK − }, if x is in the interior
of ∂K + ∩ ∂K − , and by h(x) := hK if x ∈ ∂K is on the
boundary Γ.
For piecewise smooth vector-valued and scalar-valued
functions v and q, respectively, we introduce the following trace operators. On an interior face f ∈ FhI shared
by two neighboring elements K + and K − with unit outward normal vectors n± , respectively, denoting by v±
and q ± the traces of v and q taken from within K ± , respectively, we define the jumps and averages across f by
[[v]]T := n+ × v+ + n− × v− , [[v]]N = v+ · n+ + v− · n− ,
[[q]]N = q + n+ + q − n− , {{v}} := (v+ + v− )/2 and {{q}} :=
(q + + q − )/2, respectively. On a boundary face f ∈ FhB ,
we set [[v]] := n × v, {{v}} := v and [[q]] = qn.
(5)
1
1
kqk2Q(h) = kε 2 ∇h qk20,Ω + kε 2 h− 2 [[q]]N k20,Fh ,
where we have used ∇h to denote the elementwise application of the operator ∇ and have set kϕk20,Fh =
P
2
f ∈Fh kϕk0,f .
B. DG Discretization of the Curl-Curl Operator
The common ingredient to all the methods presented
below is the DG discretization of the curl-curl operator based on an interior penalty approach, for which the
discrete (complex-valued) sesquilinear form associated to
the term ∇ × (µ−1 ∇ × u) is defined by
ah (u, v) = (µ−1 ∇h × u, ∇h × v)Ω
Z
[[u]]T · {{µ−1 ∇h × v}} + [[v]]T · {{µ−1 ∇h × u}} ds
−
F
Z h
+
a µ−1 [[u]]T · [[v]]T ds.
Fh
Here,
and in
we use the convention that
R
R
Pthe following,
∞
f ∈Fh f ϕ ds. The function a in L (Fh ) is
Fh ϕ ds =
the usual interior penalty stabilization function defined
by
a := α h−1 ,
(6)
where α > 0 is a parameter independent of the mesh
3
C. DG Discretization of the Low-Frequency Problem (Insulating Materials)
We consider the problem (3) in the case of insulating
materials, i.e., Ω0 = Ω and σ = 0 in Ω, since all the key
difficulties in the numerical treatment of (3) are already
present in this particular case.
C.1 Regularized DG Method
Following , we consider the regularized DG method:
find uh in Vh such that
ah (uh , v) + rh (uh , v) = (j, v)Ω
(a)
(b)
(c)
(d)
(7)
for all v ∈ Vh , where ah (·, ·) is the curl-curl form defined
in Section II-B, and rh (·, ·) is the form defined by
rh (u, v)
= (µ
−
Z
−1
∇h · u, ∇h · v)Ω −
Z
FhI
[[εu]]N {{µ−1 ∇h · (εv)}} ds
Z
[[εv]]N {{µ−1 ∇h · (εu)}} ds + dµ−1 [[εu]]N · [[εv]]N ds.
FhI
FhI
It represents a divergence regularization, analogous to
the one introduced in  for continuous nodal elements.
The function d in L∞ (FhI ) is defined by
d := δ h−1 ,
(8)
where δ > 0 is a parameter independent of the mesh size.
Again, the form rh is positive semi-definite provided that
δ > δmin for a threshold value δmin . The formulation in
(7) is then well-posed and possesses a unique solution.
For this method, the error is measured in terms of the
following DG norm, which is naturally associated with
the formulation (7):
1
1
1
kvk2DG = kµ− 2 ∇h × vk20,Ω + kµ− 2 h− 2 [[v]]T k20,Fh
1
1
1
+ kµ− 2 ∇h · vk20,Ω + kµ− 2 h− 2 [[εv]]N k20,Fh .
With this notation, we have the following a-priori error
estimate taken from , .
Theorem 1: Assume that the analytical solution u
of (3) satisfies the smoothness assumption u ∈
H s+1 (Th )3 , for s > 1/2, and let uh be the DG approximation defined by (7). Then we have the a priori error
bound
ku − uh kDG ≤ C hmin{s,`} kuks+1,Th .
While Theorem 1 ensures optimal convergence for
smooth solutions, it does not cover the case of singular solutions with regularity below H 1 (Ω)3 . Indeed, numerical results in  have confirmed that the regularity assumptions in Theorem 1 are sharp and cannot be
weakened. Thus, as for their conforming counterparts,
regularized DG approaches cannot resolve the strongest
singularities.
As a numerical illustration of this behavior, we consider the following example of a real–valued model problem with a singular solution and constant material coefficients µ ≡ ε ≡ 1. To this end, for simplicity, we restrict
Fig. 1. Regularized DG Method: (a) & (b) First and second components of the analytical solution, respectively; (c) & (d) First
and second components of the regularized DG approximation,
respectively.
conditions for u) so that the analytical solution u of (3)
is given, in terms of the polar coordinates (r, ϑ), by
u(x, y) = ∇S(r, ϑ), where S(r, ϑ) = r 2n/3 sin(2nϑ/3),
(9)
where n ≥ 1 is an integer parameter. Here, the boundary conditions are enforced in the usual DG manner by
adding boundary terms in the formulation (7); see ,
 for details. The analytical solution given by (9) then
contains a singularity at the re-entrant corner located at
the origin of Ω; in particular, we note that u lies in the
Sobolev space H 2n/3−ε (Ω)2 , ε > 0. In particular, for
n = 1, u has a regularity below H 1 (Ω)2 .
For n > 1, the numerical experiments presented in the
article  confirm the optimality of the a priori error
bound stated in Theorem 1. However, in the case of the
strongest singularity when n = 1, the regularized DG
method (7) no longer converges to the correct analytical
solution given by (9). Indeed, in Fig. 1 we show the first
and second components of the analytical solution and
their respective regularized DG approximation; here, we
set the polynomial degree ` = 1 and the stabilization parameters α = δ = 10. Furthermore, the underlying computational mesh consists of uniform quadrilaterals with
3072 elements. Here, we clearly observe that when the
regularity assumptions of Theorem 1 are violated, then
the regularized DG method no longer converges to the
correct analytical solution; indeed, under further mesh
refinement, the numerical approximation converges to
the solution of a ‘nearby’ problem; cf. . We remark
that analogous behavior is also observed for regularized
conforming methods; cf. , . This drawback may
be overcome by employing the weighted regularization
4
C.2 Mixed DG Methods
Mixed DG methods for the discretization of the problem (3) with Ω0 = Ω are based on the following mixed
formulation of the problem:
∇ × (µ−1 ∇ × u) − ε∇p = j
in Ω,
∇ · (εu) = 0
in Ω,
n × u = 0,
p=0
(10)
on Γ.
Here, p is the Lagrange multiplier related to the
divergence-free constraint. The standard variational formulation of (10) is well-posed in H0 (curl; Ω) × H01 (Ω);
see, e.g., , .
Mixed DG methods for (10) are given by: find (uh , ph )
in Vh × Qh such that
ah (uh , v) + sh (uh , v) + bh (v, ph ) = (j, v)Ω ,
bh (uh , q) − ch (ph , q) = 0
(11)
for all (v, q) ∈ Vh × Qh , where ah (·, ·) is defined in Section II-B, and sh (·, ·), bh (·, ·) and ch (·, ·) are defined, respectively, by
Z
b [[u]]N [[v]]N ds,
sh (u, v) =
FhI
Z
{{εv}} · [[p]]N ds,
= −(εv, ∇h p) +
Fh
Z
c ε[[p]]N · [[q]]N ds.
ch (p, q) =
bh (v, p)
Fh
The form sh (·, ·) is a normal-jump stabilization form, the
form bh (·, ·) discretizes the divergence operator in a DG
fashion, and the form ch (·, ·) is the interior penalty form
that weakly enforces the continuity of ph .
The functions b and c in L∞ (Fh ) are taken to be
b := β h,
c := γ h−1 ,
(12)
where β ≥ 0 and γ > 0 are parameters independent of
the mesh size.
In particular, we consider the following two methods:
• Method I (stabilized): we take β > 0, and m = `
in (5) (equal-order polynomial spaces).
• Method II (non-stabilized): we take β = 0, and
m = ` + 1 in (5) (mixed-order polynomial spaces).
It can be shown that both methods are well-posed and
possess unique solutions provided that α > αmin .
The error estimates contained in the following theorem
have been proved and numerically validated in  for
Method I, and in  for Method II.
Theorem 2: Assume that the analytical solution (u, p)
of (10) satisfies the smoothness assumptions u ∈
H s (Th )3 , ∇ × u ∈ H s (Th )3 and p ∈ H s+1 (Th ), for
s > 1/2, and let (uh , ph ) be the DG approximation obtained by (11) with Method I or Method II. Then we
have the optimal a priori error bound
ku − uh kV(h) + kp − ph kQ(h)
≤ C hmin{s,`} kεuks,Th + kµ−1 ∇ × uks,Th + kpks+1,Th .
Unlike Theorem 1, the result in Theorem 2 is valid for
Section III below. There, we also compare the practical
performance of Method I and Method II.
Although the result in Theorem 2 ensures convergence
for highly singular solutions, the overall accuracy of the
numerical simulations can be highly improved using local
mesh refinement. Such refinement strategies are typically
based on suitable a posteriori error estimates. Here, we
present a simple energy norm a posteriori error estimate
for Method II. It has been established and tested in .
Theorem 3: Assume that ∇ · j = 0. Let (uh , ph ) be
the DG approximation obtained by (11) with Method II.
Then there is a parameter σ ∈ (1/2, 1] only depending
on Ω and a constant C > 0 independent of the mesh size,
such that
X
1/2
2
ku − uh kV(h) + kp − ph kQ(h) ≤ C
ηK
,
K∈Th
where the elemental error indicator ηK is given by
2
−1
= h2σ
∇ × uh ) + ε∇ph k20,K
ηK
K kj − ∇ × (µ
1
2σ−1
2
2
+ hK
kb
τK (uh ) − τK (uh )k20,∂K + h−1
K kµ [[uh ]]T k0,∂K
+ hK k[[εuh ]]N k20,∂K\Γ + h2K k∇ · (εuh )k20,K
1
1
2
2
+ kε 2 ∇ph k20,K + h−1
K kε [[ph ]]N k0,∂K ,
and τbK (v) is the numerical flux defined by
nK × ({{µ−1 ∇ × v}} − µ−1 a [[v]]T ) on ∂K \ Γ,
τbK (v) =
nK × (µ−1 ∇ × v − µ−1 a (nK × v)) on ∂K ∩ Γ.
The parameter σ depends on the opening angles of
Ω and can be determined easily for the domain under
consideration. In particular, if Ω is convex, we can choose
σ = 1; cf. .
D. DG Discretization of the High-Frequency Problem
Next, we introduce two methods for the DG approximation of the indefinite problem (4).
D.1 Direct DG Method
For problem (4), a direct interior penalty DG method
is given by: find uh ∈ Vh such that
ah (uh , v) − ω 2 (εuh , v)Ω = (j, v)Ω
(13)
for all v ∈ Vh , where the discrete form ah (·, ·) is the
curl-curl discretization from Section II-B. The interior
penalty stabilization function a ∈ L∞ (Fh ) is defined
again by (6), with α chosen independently of the mesh
size and the frequency.
The following a priori error estimates in the energy
norm and the L2 (Ω)-norm have been proved in .
Their proofs are based on techniques similar to those
of  and [25, Section 7.2], for the energy error bound,
and of [26, Theorem 3.2], for the L2 (Ω)-error bound, combined with novel results that allow for the approximation
of a discontinuous function by a conforming one. This result is instrumental in controlling the non-conformity of
the DG method.
Theorem 4: Assume that the analytical solution u
of (4) satisfies the regularity assumptions u ∈ H s (Th )3
and ∇×u ∈ H s (Th )3 , for s > 12 , and let uh be the DG approximation defined by (13). Then, for sufficiently small
5
1
kε 2 (u − uh )k0,Ω ≤ C h`+1 kuk`+1,Th .
The analysis of  is based on duality arguments;
thus, the result of Theorem 4 can easily be extended to
smooth material coefficients µ and ε. However, the extension to piecewise smooth coefficients cannot be based
on duality and is the subject of ongoing research.
D.2 Mixed DG Method
A mixed DG method, which can be used for the full
Maxwell problem (1) with σ = 0 and real-valued j,
irrespective of whether the problem is in the low- or
high-frequency regime, is obtained by performing the
Helmholtz decomposition of the unknown field u as
w + ∇ϕ, with ϕ ∈ H01 (Ω) and w ∈ H0 (curl; Ω) with
zero divergence. By setting p := ω 2 ϕ and renaming w
by u, the problem can be naturally written in the following mixed form:
∇ × (µ−1 ∇ × u) − ω 2 εu − ε∇p = j
in Ω,
∇ · (εu) = 0
in Ω,
n × u = 0,
p=0
(14)
on Γ.
The mixed DG method for the numerical approximation
of (14) studied in  is defined as follows: find (uh , ph )
in Vh × Qh , with m = ` + 1, such that
ah (uh , v) − ω 2 (εuh , v)Ω + bh (v, ph ) = (j, v)Ω ,
bh (uh , q) − ch (ph , q) = 0
(15)
for all (v, q) ∈ Vh × Qh , where ah (·, ·) is defined in Section II-B, and bh (·, ·) and ch (·, ·) are defined as in Section II-C.2.
Optimal a priori energy-norm and L2 (Ω)-norm error
estimates for the method in (15) can be readily obtained
by by combining the techniques of  for the analysis
of the method in (13) with the ones of  for the mixed
Method II of Section II-C.2; cf. .
III. Numerical Results
In this section we present a series of numerical experiments for model problems in two dimensions with known
analytical solutions in order to both confirm the optimality of our a priori error bounds, as well as to highlight
the practical performance of the DG methods reviewed
in this article. Throughout this section we select the interior penalty parameter α in (6) as follows: α = 10 `2 .
A. Example 1
sin(π(x − 1)/2) sin(π(y − 1)/2). Here, we investigate the
asymptotic convergence of both methods on a sequence
of successively finer uniform square and quasi-uniform
unstructured triangular meshes for ` = 1, 2, 3. For both
methods we set γ = 1; for Method I (stabilized method),
we set β = 1, while β = 0 for Method II (non-stabilized
method), cf. (12).
In Fig. 2 we first present a comparison of the sum
of the DG–norms ku − uh kV(h) and kp − ph kQ(h) with
respect to the square root of the number of degrees of
freedom in the finite element space Vh × Qh for the first
(stabilized) method; for brevity, we have not shown these
quantities individually, since for this equal-order method,
each of these norms of the error converges to zero at the
same rate. Indeed, here we observe convergence, for each
fixed `, at the optimal rate O(h` ) as the mesh is refined,
thereby confirming Theorem 2. We remark that here we
are only interested in demonstrating the general performance of the underlying method on meshes comprising of
either square or triangular elements, but not in making
a formal comparison of the accuracy of the scheme with
respect to each element type. Indeed, although for each
fixed ` we observe that the error on the uniform square
meshes is smaller than the corresponding quantity measured on triangular meshes, the former meshes consist of
uniform structured meshes, where we anticipate that local error cancellation will lead to a reduction in the size
of the global error. On the other hand, the triangular
meshes are completely unstructured, so we do not expect
the same level of local error cancellation.
In Figs. 3 & 4 we plot the DG–norms k · kV(h) and
k · kQ(h) of the errors u − uh and p − ph , respectively, as
the mesh size tends to zero for Method II. As for Method
I, we again observe that ku − uh kV(h) converges to zero,
for each fixed `, at the optimal rate O(h` ), as the mesh
is refined, in accordance with Theorem 2. On the other
hand, for this mixed-order method, kp − ph kQ(h) converges to zero at the rate O(h`+1 ), for each `, as h tends
to zero; this rate is indeed optimal, though this is not
reflected by Theorem 2.
Finally, we highlight the optimality of Method II when
the error in the computed vector field uh is measure in
terms of the L2 (Ω)-norm. As noted in , ku − uh k0,Ω
converges at the suboptimal rate O(h` ), for each `, as h
tends to zero when Method I is employed, cf. Fig. 5.
ku − uh kV(h) + kp − ph kQ(h)
Moreover, assume that the analytical solution u of (4)
satisfies u ∈ H `+1 (Th )3 and the domain Ω is convex.
Then, for sufficiently small mesh sizes, we have the optimal L2 -error bound
In this first example we consider the numerical perPSfrag
replacements
formance of the stabilized and non-stabilized mixed
DG
methods for the numerical approximation of the lowfrequency problem (10) in the case when µ ≡ ε ≡ 1
and the underlying analytical solution is smooth. To
this end, we set Ω = (−1, 1)2 and select j and suitable non-homogeneous boundary conditions for u, i.e.,
0
10
`=1
1
1
`=2
−1
10
1
`=3
−2
10
2
−3
10
1
−4
10
3
Square Elements
Triangular Elements
1
10
√
2
10
Degrees of Freedom
6
0
10
−1
`=1
10
`=1
1
ku − uh kV(h)
ku − uh k0,Ω
`=2
−2
10
1
−3
10
2
`=3
1
−4
g replacements
PSfrag replacements
3
1
−6
√
−5
10
1
3
10
Square Elements
Triangular Elements
10
2
−4
−6
10
1
1
`=3
10
10
Square Elements
Triangular Elements
−7
10
2
1
10
√
10
Degrees of Freedom
Fig. 3. Example 1. Method II: Convergence of ku − uh kV (h) .
2
10
Degrees of Freedom
Fig. 5. Example 1. Method I: Convergence of ku − uh k0,Ω .
0
−1
10
10
`=1
`=1
−1
1
−2
10
10
1
`=2
2
−2
10
`=2
2
−3
10
`=3
−3
10
ku − uh k0,Ω
kp − ph kQ(h)
1
`=2
−2
10
1
1
−4
10
g replacements
1
−5
10
PSfrag replacements
1
`=3
−4
10
3
3
−5
10
1
−6
10
4
4
−7
−6
10
10
Square Elements
Triangular Elements
−7
10
1
10
√
Square Elements
Triangular Elements
−8
10
2
10
Degrees of Freedom
1
10
√
2
10
Degrees of Freedom
Fig. 4. Example 1. Method II: Convergence of kp − ph kQ(h) .
Fig. 6. Example 1. Method II: Convergence of ku − uh k0,Ω .
On the other hand, Fig. 6 demonstrates that the mixedorder method (Method II) yields an optimal convergence
rate for the above quantity as the mesh is refined.
mixed form (14). Here, we take µ = µ0 and ε = ε0 ,
the permeability and permittivity of the free space, respectively, and, as is standard in electromagnetic computations, we re-scale each problem by µ0 in order to
√
introduce the wave number k = ω µ0 ε0 . Thereby, the
mixed problem (14) becomes
B. Example 2
In this section we now consider the application of
the stabilized and non-stabilized mixed DG methods to
the numerical approximation of the low-frequency model
problem considered in Section II-C.1, cf. (9). We recall that in the case of the strongest singularity when
n = 1, the regularized method (7) fails to converge to
the correct analytical solution defined in (9). In contrast, both the stabilized and non-stabilized mixed methods proposed in Section II-C.2 converge to the analytical
solution (9) at the optimal rate predicted in Theorem 2,
cf. Tables I and II, respectively. Here, r denotes the computed convergence rate; additionally, we have employed
the notation |||e|||DG to denote the sum of the DG–norms
ku−uh kV(h) and kp−ph kQ(h) . For the stabilized method,
Table I demonstrates the optimal convergence of the DG
scheme on a sequence of uniform square meshes consisting of 12 elements on the coarsest mesh and 3072 on
the finest; Table II shows analogous results for the nonstabilized method on uniform triangular meshes consisting of 24 elements on coarsest mesh and 6144 on the
finest.
C. Example 3
∇ × ∇ × u − k 2 u − ∇p = j
in Ω,
∇·u=0
in Ω,
n × u = 0,
p=0
(16)
on Γ,
with a rescaled Lagrange multiplier and right-hand side
(again denoted by p := k 2 ϕ and j, respectively).
We select Ω ⊂ R2 to be the square domain (−1, 1)2 .
Furthermore, we set j = 0 and select suitable nonhomogeneous boundary conditions for u, so that the analytical solution to the two-dimensional analogue of (16) is
T
given by the smooth field u(x, y) = (sin(ky), sin(kx)) ,
p = 0. We investigate the asymptotic convergence of
the mixed DG method (15) on a sequence of successively
finer (quasi-uniform) unstructured triangular meshes for
` = 1, 2, 3 as the wave number k increases. To this end,
in Fig. 7 we first present a comparison of the DG–norm
ku − uhkV(h) with respect to the square root of the number of degrees of freedom in the finite element space Vh
for k = 1, 2. Here, we observe that (asymptotically)
ku−uh kV(h) converges to zero at the optimal rate O(h` ),
7
|||e|||DG
1.473
1.245
0.905
0.607
0.393
`=2
r
0.24
0.46
0.58
0.63
|||e|||DG
1.876
1.381
0.930
0.602
0.384
r
0.44
0.57
0.63
0.65
`=3
|||e|||DG
2.083
1.485
0.988
0.637
0.405
0
10
r
0.49
0.59
0.63
0.65
−2
10
ku − uh k0,Ω
`=1
`=1
−4
10
PSfrag replacements
TABLE I
Example 2. Method I: Convergence of |||e|||DG on uniform
square meshes with h–refinement.
`=2
−6
10
`=3
−8
10
k=1
k=2
1
10
`=1
|||e|||DG
2.677
2.439
1.799
1.196
0.765
`=2
r
0.13
0.44
0.59
0.65
|||e|||DG
3.704
2.907
2.002
1.300
0.826
r
0.35
0.54
0.62
0.65
`=3
|||e|||DG
4.348
3.254
2.196
1.417
0.8989
r
0.42
0.57
0.63
0.66
TABLE II
Example 2. Method II: Convergence of |||e|||DG on uniform
triangular meshes with h–refinement.
√
2
10
Degrees of Freedom
Fig. 8. Example 3. Convergence of ku − uh k0,Ω .
agreement with the optimal rate predicted in . Numerical experiments also indicate that the L2 (Ω)-norm
of the error in the approximation to p converges to zero
at the optimal rate O(h`+2 ), for each fixed ` and each k,
as h tends to zero; for brevity, these results have been
omitted.
IV. Conclusions
`+1
O(h ), for each ` and k, as h tends to zero; for brevity,
these results have been omitted. We now make two key
observations: firstly, we note that for a given fixed mesh
and fixed polynomial degree, an increase in the wave
number k leads to an increase in the DG-norm of the error in the approximation to u. Indeed, as pointed out in
 and , where interior penalty and curl-conforming
finite element methods, respectively, were employed for
the numerical approximation of (16), the pre-asymptotic
region increases as k increases. Secondly, we observe that
the DG-norm of the error decreases when either the mesh
is refined, or the polynomial degree is increased as we
would expect for this smooth problem.
Finally, in Fig. 8 we present a comparison of the L2 (Ω)norm of the error in the approximation to u, with the
square root of the number of degrees of freedom in the finite element space Vh . Here, we observe that (asymptotically) ku−uh k0 converges to zero at the rate O(h`+1 ), for
each fixed ` and each k, as h tends to zero. This is in full
0
10
−1
ku − uh kV(h)
10
g replacements
`=1
−2
10
−3
methods for the numerical approximation of the timeharmonic Maxwell’s equations in both the low- and highfrequency regimes. For the low-frequency problem, both
regularized and mixed DG formulations have been proposed; the latter are advantageous in the sense that convergence of the underlying schemes is guaranteed even
in the case when only minimal regularity assumptions
on the analytical solution hold. In the mixed setting,
two schemes, a stabilized and a non-stabilized variant,
have been proposed. While both schemes deliver optimal rates of convergence when the error is measured in
terms of a discrete energy norm, only the non-stabilized
scheme ensures optimality of the error in the electric
field in the L2 (Ω)–norm. For this latter scheme, optimal a posteriori error bounds have also been deduced.
Finally, for the high-frequency problem, both direct and
mixed methods have been proposed. The key advantage of the mixed approach is that it allows for a unified
approximation of the time-harmonic Maxwell system in
both the low- and high-frequency regimes; indeed, in the
low-frequency case, this method corresponds to the nonstabilized mixed method studied in . Our current and
future research is devoted to the extension of the analysis of the high-frequency problem to piecewise smooth
coefficients, as well as the application of these methods
to practical problems of industrial interest.
References
`=2
10

−4
10
−5
10
`=3
k=1
k=2
−6
10
1
10

√
2
10
Degrees of Freedom
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