Page 1
SIAM J. NUMER. ANAL.
Vol. 44, No. 6, pp. 2408–2431
c ? 2006 Society for Industrial and Applied Mathematics
DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR
THE WAVE EQUATION∗
MARCUS J. GROTE†, ANNA SCHNEEBELI†, AND DOMINIK SCH¨OTZAU‡
Abstract.
presented for the numerical discretization of the second-order wave equation. The resulting stiffness
matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method
is inherently parallel and leads to fully explicit time integration when coupled with an explicit time-
stepping scheme. Optimal a priori error bounds are derived in the energy norm and the L2-norm for
the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with
the optimal order O(hmin{s,?}) with respect to the mesh size h, the polynomial degree ?, and the
regularity exponent s of the continuous solution. Under additional regularity assumptions, the L2-
error is shown to converge with the optimal order O(h?+1). Numerical results confirm the expected
convergence rates and illustrate the versatility of the method.
The symmetric interior penalty discontinuous Galerkin finite element method is
Key words.
second-order hyperbolic problems, a priori error analysis, explicit time integration
discontinuous Galerkin finite element methods, wave equation, acoustic waves,
AMS subject classification. 65N30
DOI. 10.1137/05063194X
1. Introduction. The numerical solution of the wave equation is of fundamental
importance to the simulation of time dependent acoustic, electromagnetic, or elastic
waves. For such wave phenomena the scalar second-order wave equation often serves
as a model problem. Finite element methods (FEMs) can easily handle inhomoge-
nous media or complex geometry. However, if explicit time-stepping is subsequently
employed, the mass matrix arising from the spatial discretization by standard con-
tinuous finite elements must be inverted at each time step: a major drawback in
terms of efficiency. For low-order Lagrange (P1) elements, so-called mass lumping
overcomes this problem [6, 15], but for higher-order elements this procedure can lead
to unstable schemes unless particular finite elements and quadrature rules are used
[11]. In addition, continuous Galerkin methods impose significant restrictions on the
underlying mesh and discretization; in particular, they do not easily accommodate
hanging nodes.
To avoid these difficulties, we consider instead discontinuous Galerkin (DG) meth-
ods. Based on discontinuous finite element spaces, these methods easily handle el-
ements of various types and shapes, irregular nonmatching grids, and even locally
varying polynomial order; thus, they are ideally suited for hp-adaptivity. Here con-
tinuity is weakly enforced across mesh interfaces by adding suitable bilinear forms,
so-called numerical fluxes, to standard variational formulations.
easily included within an existing conforming finite element code.
These fluxes are
∗Received by the editors May 19, 2005; accepted for publication (in revised form) May 7, 2006;
published electronically December 1, 2006.
http://www.siam.org/journals/sinum/44-6/63194.html
†Department of Mathematics, University of Basel, Rheinsprung 21, 4051 Basel, Switzerland
(Marcus.Grote@unibas.ch, anna.schneebeli@unibas.ch). The second author was supported by the
Swiss National Science Foundation.
‡Mathematics Department, University of British Columbia, 121–1984 Mathematics Road, Van-
couver V6T 1Z2, BC, Canada (schoetzau@math.ubc.ca). This author was supported in part by the
Natural Sciences and Engineering Research Council of Canada (NSERC).
2408
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DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2409
Because individual elements decouple, DGFEMs are also inherently parallel; see
[8, 9, 10, 7] for further details and recent reviews. Moreover, the mass matrix aris-
ing from the spatial DG discretization is block-diagonal, with block size equal to the
number of degrees of freedom per element; it can therefore be inverted at very low
computational cost. In fact, for a judicious choice of (locally orthogonal) shape func-
tions, the mass matrix is diagonal. When combined with explicit time integration,
the resulting time marching scheme will be fully explicit.
The origins of DG methods can be traced back to the 1970s, when they were pro-
posed for the numerical solution of hyperbolic neutron transport equations, as well as
for the weak enforcement of continuity in Galerkin methods for elliptic and parabolic
problems; see Cockburn, Karniadakis, and Shu [8] for a review of the development of
DG methods. When applied to second-order hyperbolic problems, most DG methods
first require the problem to be reformulated as a first-order hyperbolic system, for
which various DG methods are available. In [9], for instance, Cockburn and Shu used
a DGFEM in space combined with a Runge–Kutta scheme in time to discretize hy-
perbolic conservation laws. Hesthaven and Warburton [13] used the same approach to
implement high-order methods for Maxwell’s equations in first-order hyperbolic form.
Space-time DG methods for linear symmetric first-order hyperbolic systems were pre-
sented by Falk and Richter in [12] and later generalized by Monk and Richter in [17]
and by Houston, Jensen, and S¨ uli in [14]. A first DG method for the acoustic wave
equation in its original second-order formulation was recently proposed by Rivi` ere and
Wheeler [21]; it is based on a nonsymmetric interior penalty formulation and requires
additional stabilization terms for optimal convergence in the L2-norm [20].
Here we propose and analyze the symmetric interior penalty DG method for the
spatial discretization of the second-order scalar wave equation.
shall derive optimal a priori error bounds in the energy norm and the L2-norm for
the semidiscrete formulation. Besides the well-known advantages of DG methods
mentioned above, a symmetric discretization of the wave equation in its second-order
form offers an additional advantage, which also pertains to the classical continuous
Galerkin formulation: since the stiffness matrix is positive definite, the semidiscrete
formulation conserves (a discrete version of) the energy for all time; thus, it is free
of any (unnecessary) damping. The dispersive properties of the symmetric interior
penalty DG method were recently analyzed by Ainsworth, Monk, and Muniz [1].
The outline of our paper is as follows. In section 2 we describe the setting of
our model problem. Next, we present in section 3 the symmetric interior penalty DG
method for the wave equation. Our two main results, optimal error bounds in the
energy norm and the L2-norm for the semidiscrete scheme, are stated at the beginning
of section 4 and proved subsequently. The analysis relies on an idea suggested by
Arnold et al. [2] together with the approach presented by Perugia and Sch¨ otzau in
[18] to extend the DG bilinear forms by suitable lifting operators. In section 5, we
demonstrate the sharpness of our theoretical error estimates by a series of numerical
experiments. By combining our DG method with the second-order Newmark scheme,
we obtain a fully discrete method. To illustrate the versatility of our method, we
also propagate a wave across an inhomogenous medium with discontinuity, where
the underlying finite element mesh contains hanging nodes. Finally, we conclude with
some remarks on possible extensions of our DG method to electromagnetic and elastic
waves.
In particular, we
2. Model problem. We consider the (second-order) scalar wave equation
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M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
utt− ∇ · (c∇u) = fin J × Ω,
on J × ∂Ω,
in Ω,
(2.1)
u = 0(2.2)
u|t=0= u0
ut|t=0= v0
(2.3)
in Ω, (2.4)
where J = (0,T) is a finite time interval and Ω is a bounded domain in Rd, d = 2,3.
For simplicity, we assume that Ω is a polygon (d = 2) or a polyhedron (d = 3). The
(known) source term f lies in L2(J;L2(Ω)), while u0∈ H1
prescribed initial conditions. We assume that the speed of propagation,
piecewise smooth and satisfies the bounds
0(Ω) and v0∈ L2(Ω) are
?c(x), is
0 < c?≤ c(x) ≤ c?< ∞,x ∈ Ω.(2.5)
The standard variational form of (2.1)–(2.4) is to find u ∈ L2(J;H1
ut∈ L2(J;L2(Ω)) and utt∈ L2(J;H−1(Ω)), such that u|t=0= u0, ut|t=0= v0, and
?utt,v? + a(u,v) = (f,v)
Here, the time derivatives are understood in a distributional sense, ?·,·? denotes the
duality pairing between H−1(Ω) and H1
a(·,·) is the elliptic bilinear form given by
a(u,v) = (c∇u,∇v).
It is well known that problem (2.6) is well posed [16]. Moreover, the weak solution u
can be shown to be continuous in time; that is,
0(Ω)), with
∀v ∈ H1
0(Ω)a.e. in J.(2.6)
0(Ω), (·,·) is the inner product in L2(Ω), and
(2.7)
u ∈ C0(J;H1
0(Ω)),ut∈ C0(J;L2(Ω));(2.8)
see [16, Chapter III, Theorems 8.1 and 8.2] for details. In particular, this result implies
that the initial conditions in (2.3) and (2.4) are well defined.
3. Discontinuous Galerkin discretization. We shall now discretize the wave
equation (2.1)–(2.4) by using the interior penalty discontinuous Galerkin finite element
method in space, while leaving the time dependence continuous.
3.1. Preliminaries. We consider shape-regular meshes Th that partition the
domain Ω into disjoint elements {K} such that Ω = ∪K∈ThK. For simplicity, we
assume that the elements are triangles or parallelograms in two space dimensions,
and tetrahedra or parallelepipeds in three dimensions, respectively. The diameter of
element K is denoted by hK, and the mesh size h is given by h = maxK∈ThhK. We
assume that the partition is aligned with the discontinuities of the wave speed√c.
Generally, we allow for irregular meshes with hanging nodes. However, we assume that
the local mesh sizes are of bounded variation; that is, there is a positive constant κ,
depending only on the shape-regularity of the mesh, such that
κhK≤ hK? ≤ κ−1hK
(3.1)
for all neighboring elements K and K?.
An interior face of This the (nonempty) interior of ∂K+∩∂K−, where K+and K−
are two adjacent elements of Th. Similarly, a boundary face of This the (nonempty)
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DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2411
interior of ∂K ∩∂Ω, which consists of entire faces of ∂K. We denote by FI
all interior faces of Thand by FB
Here we generically refer to any element of Fh as a “face,” in both two and three
dimensions.
For any piecewise smooth function v we now introduce the following trace oper-
ators. Let F ∈ FI
K−and let x ∈ F; we write n±to denote the unit outward normal vectors on the
boundaries ∂K±. Denoting by v±the trace of v taken from within K±, we define the
jump and average of v at x ∈ F by
[[v]] := v+n++ v−n−,
hthe set of
h∪FB
hthe set of all boundary faces and set Fh= FI
h.
hbe an interior face shared by two neighboring elements K+and
{ {v} } := (v++ v−)/2,
respectively. On every boundary face F ∈ FB
n is the unit outward normal vector on ∂Ω.
For a piecewise smooth vector-valued function q, we analogously define the av-
erage across interior faces by { {q} } := (q++ q−)/2, and on boundary faces we set
{ {q} } := q. The jump of a vector-valued function will not be used. For a vector-valued
function q with continuous normal components across a face f, the trace identity
h, we set [[v]] := vn and { {v} } := v. Here,
v+(n+· q+) + v−(n−· q−) = [[v]] · { {q} }
immediately follows from the definitions.
on f
3.2. Discretization in space. For a given partition Thof Ω and an approxi-
mation order ? ≥ 1, we wish to approximate the solution u(t,·) of (2.1)–(2.4) in the
finite element space
Vh:= {v ∈ L2(Ω) : v|K∈ S?(K) ∀K ∈ Th},
where S?(K) is the space P?(K) of polynomials of total degree at most ? on K if K
is a triangle or a tetrahedra, or the space Q?(K) of polynomials of degree at most ?
in each variable on K if K is a parallelogram or a parallelepiped.
Then, we consider the following (semidiscrete) DG approximation of (2.1)–(2.4):
find uh: J × Vh→ R such that
(uh
(3.3)
(3.2)
tt,v) + ah(uh,v) = (f,v)
∀v ∈ Vh,t ∈ J,
uh|t=0= Πhu0,
uh
t|t=0= Πhv0.
(3.4)
(3.5)
Here, Πh denotes the L2-projection onto Vh, and the discrete bilinear form ah on
Vh× Vhis given by
?
−
F∈Fh
ah(u,v) :=
K∈Th
?
?
K
?
c∇u · ∇v dx −
?
F∈Fh
?
F
[[u]] · { {c∇v} }dA
?
F
[[v]] · { {c∇u} }dA +
?
F∈Fh
F
a[[u]] · [[v]]dA.
(3.6)
The last three terms in (3.6) correspond to jump and flux terms at element boundaries;
they vanish when u,v ∈ H1
DG formulation (3.3) is consistent with the original continuous problem (2.6).
0(Ω) ∩ H1+σ(Ω) for σ >1
2. Hence the above semidiscrete
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M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
In (3.6) the function a penalizes the jumps of u and v over the faces of Th. It is
referred to as the interior penalty stabilization function and is defined as follows. We
first introduce the function h by
?
hK,
h|F=
min{hK,hK?},F ∈ FI
F ∈ FB
h, F = ∂K ∩ ∂K?,
h, F = ∂K ∩ ∂Ω.
For x ∈ F, we further define c by
c|F(x) =
?
max{c|K(x),c|K?(x)},
c|K(x),
F ∈ FI
F ∈ FB
h, F = ∂K ∩ ∂K?,
h, F = ∂K ∩ ∂Ω.
Then, on each F ∈ Fh, we set
a|F:= αch−1,(3.7)
where α is a positive parameter independent of the local mesh sizes and the coeffi-
cient c.
To conclude this section we recall the following stability result for the DG form ah.
Lemma 3.1. There exists a threshold value αmin> 0 which depends only on the
shape-regularity of the mesh, the approximation order ?, the dimension d, and the
bounds in (2.5), such that for α ≥ αmin
??
ah(v,v) ≥ Ccoer
K∈Th
?c
1
2∇v?2
0,K+
?
F∈Fh
?a
1
2[[v]]?2
0,F
?
,v ∈ Vh,
where the constant Ccoeris independent of c and h.
The proof of this lemma follows readily from the arguments in [2]. However, to
make explicit the dependence of αminon the bounds in (2.5), we present the proof
of a slightly more general stability result in Lemma 4.4 below. Throughout the rest
of the paper we shall assume that α ≥ αmin, so that by Lemma 3.1 the semidiscrete
problem (3.3)–(3.5) has a unique solution.
We remark that the condition α ≥ αmincan be omitted by using other symmetric
DG discretizations of the div-grad operator, such as the local discontinuous Galerkin
(LDG) method; see, e.g., [2] for details. It can also be avoided by using the nonsym-
metric interior penalty method proposed in [20]. However, since the symmetry of ah
is crucial in the analysis below, our error estimates (section 4) do not hold for the
nonsymmetric DG method in [20].
Remark 3.2. Because the bilinear form ah is symmetric and coercive, for α ≥
αmin, the semidiscrete DG formulation (3.3)–(3.5) with f = 0 conserves the (discrete)
energy
Eh(t) :=1
2?uh
t(t)?2
0+1
2ah(uh(t),uh(t)).
4. A priori error estimates. We shall now derive optimal a priori error bounds
for the DG method (3.3)–(3.5), first with respect to the DG energy norm and then
with respect to the L2-norm. These two key results are stated immediately below,
while their proofs are postponed to subsequent sections.
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DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2413
4.1. Main results. To state our a priori error bounds, we define the space
V (h) = H1
0(Ω) + Vh.
On V (h), we define the DG energy norm
?v?2
h:=
?
K∈Th
?c
1
2∇v?2
0,K+
?
F∈Fh
?a
1
2[[v]]?2
0,F.
Furthermore, for 1 ≤ p ≤ ∞ we will make use of the Bochner space Lp(J;V (h)),
endowed with the norm
???
esssupt∈J?v?h,
Our first main result establishes an optimal error estimate of the energy norm ? · ?h
of the error. It also gives a bound in the L2(Ω)-norm on the error in the first time
derivative.
Theorem 4.1. Let the analytical solution u of (2.1)–(2.4) satisfy
?v?Lp(J;V (h))=
J?v?p
hdt?1/p,1 ≤ p < ∞,
p = ∞.
u ∈ L∞(J;H1+σ(Ω)),
for a regularity exponent σ >1
approximation obtained by (3.3)–(3.5), with α ≥ αmin. Then, the error e = u − uh
satisfies the estimate
?
?u?L∞(J;H1+σ(Ω))+ T?ut?L∞(J;H1+σ(Ω))+ ?utt?L1(J;Hσ(Ω))
ut∈ L∞(J;H1+σ(Ω)),utt∈ L1(J;Hσ(Ω))
2, and let uhbe the semidiscrete discontinuous Galerkin
?et?L∞(J;L2(Ω))+ ?e?L∞(J;V (h))≤ C
+ Chmin{σ,?}?
with a constant C that is independent of T and h.
We remark that the fact that ut∈ L∞(J;H1+σ(Ω)) implies that u is continuous
in time on J with values in H1+σ(Ω). Similarly, utt ∈ L1(J;Hσ(Ω)) implies the
continuity of ut on J with values in Hσ(Ω).
assume that the initial conditions satisfy u0 ∈ H1+σ(Ω) and v0 ∈ Hσ(Ω). Hence,
standard approximation properties imply that
?et(0)?0+ ?e(0)?h
?
?
,
In Theorem 4.1 we thus implicitly
?et(0)?0= ?v0− Πhv0?0≤ C hmin{σ,?+1}?v0?σ,
?e(0)?h= ?u0− Πhu0?h≤ C hmin{σ,?}?u0?1+σ;
see also Lemma 4.6 below. As a consequence, Theorem 4.1 yields optimal convergence
in the (DG) energy norm
?et?L∞(J;L2(Ω))+ ?e?L∞(J;V (h))≤ Chmin{σ,?},
with a constant C = C(T) that is independent of h.
Next, we state an optimal error estimate with respect to the L2-norm (in space).
To do so, we need to assume elliptic regularity; that is, we assume that there is a
stability constant CSsuch that for any λ ∈ L2(Ω) the solution of the problem
−∇ · (c∇z) = λin Ω,z = 0on Γ,(4.1)
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M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
belongs to H2(Ω) and satisfies the stability bound
?z?2≤ CS?λ?0. (4.2)
This condition is certainly satisfied for convex domains and smooth coefficients. Then,
the following L2-error bound holds.
Theorem 4.2. Assume elliptic regularity as in (4.1)–(4.2), and let the analytical
solution u of (2.1)–(2.4) satisfy
u ∈ L∞(J;H1+σ(Ω)),ut∈ L∞(J;H1+σ(Ω)),utt∈ L1(J;Hσ(Ω))
for a regularity exponent σ >
tained by (3.3)–(3.5) with α ≥ αmin. Then, the error e = u−uhsatisfies the estimate
?e?L∞(J;L2(Ω))≤ Chmin{σ,?}+1??u0?1+σ+ ?u?L∞(J;H1+σ(Ω))+ T?ut?L∞(J;H1+σ(Ω))
with a constant C that is independent of T and the mesh size.
For smooth solutions, Theorem 4.2 thus yields optimal convergence rates in the
L2-norm:
1
2. Let uhbe the semidiscrete DG approximation ob-
?,
?e?L∞(J;L2(Ω))≤ Ch?+1,
with a constant C that is independent of h.
The rest of this section is devoted to the proofs of Theorems 4.1 and 4.2. We
shall first collect preliminary results in section 4.2. In section 4.3, we present the proof
of Theorem 4.1. Following an argument by Baker [3] for conforming finite element
approximations, we shall then derive the estimate of Theorem 4.2 in section 4.4.
4.2. Preliminaries.
Extension of the DG form ah. The DG form ahin (3.6) does not extend in
a standard way to a continuous form on the (larger) space V (h) × V (h). Indeed the
average { {c∇v} } on a face F ∈ Fhis not well defined in general for v ∈ H1(Ω). To
circumvent this difficulty, we shall extend the form ahin a nonstandard and noncon-
sistent way to the space V (h) × V (h) by using the lifting operators from [2] and the
approach in [18]. Thus, for v ∈ V (h) we define the lifted function, Lc(v) ∈?Vh?d,
?
F∈Fh
d = 2,3, by requiring that
Ω
Lc(v) · wdx =
?
?
F
[[v]] · { {cw} }dA,w ∈?Vh?d,
where c is the material coefficient from (2.1). We shall now show that the lifting
operator Lc is stable in the DG norm; see [18] for a similar result for the LDG
method.
Lemma 4.3.There exists a constant Cinv which depends only on the shape-
regularity of the mesh, the approximation order ?, and the dimension d such that
?Lc(v)?2
0≤ α−1c?C2
inv
?
F∈Fh
?a
1
2[[v]]?2
0,F
for any v ∈ V (h).
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DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2415
Moreover, if the speed of propagation c
aligned with the finite element mesh Th, then
?c−1
1
2 is piecewise constant, with discontinuities
2Lc(v)?2
0≤ α−1C2
inv
?
F∈Fh
?a
1
2[[v]]?2
0,F.
Proof. We have
?Lc(v)?0=max
w∈(Vh)d
?
??
F∈Fh
?
F[[v]] · { {cw} }dA
?w?0
?
?
??
≤
max
w∈(Vh)d
F∈Fh
Fa[[v]]2dA?1
Fa[[v]]2dA?1
2??
F∈Fh
?
Fa−1|{ {cw} }|2dA?1
?
?w?0
Fa[[v]]2dA?1
?w?0
2
?w?0
≤ α−1
2
max
w∈(Vh)d
??
F∈Fh
2??
F∈Fh
Fhc−1|{ {cw} }|2dA?1
?
2
≤ α−1
2(c?)
1
2
max
w∈(Vh)d
F∈Fh
?
2??
K∈ThhK
∂K|w|2dA?1
2
.
Here, we have used the Cauchy–Schwarz inequality, the definition of a in (3.7), and
the upper bound for c in (2.5). We recall the inverse inequality
?w?2
0,∂K≤ C2
invh−1
K?w?2
0,K,w ∈?S?(K)?d,(4.3)
with a constant Cinv that depends only on the shape-regularity of the mesh, the
approximation order ?, and the dimension d. Using this bound, we obtain
??
which shows the first statement.
With c
obtain as before
?
≤ α−1C2
F∈Fh
which completes the proof.
Next, we introduce the auxiliary bilinear form
?
−
K∈Th
The following result establishes that ? ahis continuous and coercive on the entire space
? ah= ah
K∈Th
hK
?
∂K
|w|2dA
?1
2
≤ Cinv?w?0,
1
2 piecewise constant, we have c−1
2z ∈?Vh?dfor all z ∈?Vh?d. Hence, we
?
?w?0
?
?c−1
2Lc(v)?0= max
w∈(Vh)d
F∈Fh
F[[v]] · { {c
1
2w} }dA
inv
?a
1
2[[v]]?2
0,F,
? ah(u,v) :=
K∈Th
?
?
K
?
c∇u · ∇v dx −
?
K∈Th
?
K
?
Lc(u) · ∇v dx
K
Lc(v) · ∇udx +
?
F∈Fh
F
a[[u]] · [[v]]dA.
(4.4)
V (h) × V (h); hence it is well defined. Furthermore, since
on Vh× Vh,
? ah= aon H1
0(Ω) × H1
0(Ω),(4.5)
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M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
the form ? ahcan be viewed as an extension of the two forms ahand a to the space
Lemma 4.4. Let the interior penalty parameter a be defined as in (3.7), and set
V (h) × V (h).
αmin= 4c−1
?c?C2
inv
for a general piecewise smooth c, and
αmin= 4C2
inv
for a piecewise constant c, with discontinuities aligned with the finite element mesh
Th. Cinvis the constant from Lemma 4.3.
Setting Ccont= 2 and Ccoer= 1/2, we have for α ≥ αmin
|? ah(u,v)| ≤ Ccont?u?h?v?h,
In particular, the coercivity bound implies the result in Lemma 3.1.
Proof. By taking into account the bounds in (2.5) and Lemma 4.3, application of
the Cauchy–Schwarz inequality readily gives in the general case
u,v ∈ V (h),
u ∈ V (h).
? ah(u,u) ≥ Ccoer?u?2
h,
|? ah(u,v)| ≤ max{2,α−1c−1
?c?C2
inv+ 1}?u?h?v?h.
For α ≥ αmin, the continuity of ? ahimmediately follows. The case of piecewise constant
To show the coercivity of the form ? ah, we note that
? ah(u,u) =
By using the weighted Cauchy–Schwarz inequality, the geometric-arithmetic inequal-
ity ab ≤εa2
for the lifting operator in Lemma 4.3, we obtain for general c
?
K∈Th
≤ 2
K∈Th
≤ ε
K∈Th
≤ ε
K∈Th
for a parameter ε > 0 still at our disposal. We conclude that
?
For ε =1
2and α ≥ αmin, we obtain the desired coercivity bound.
For a piecewise constant c we use the bound for ?c−1
and proceed analogously.
c follows analogously.
?
K∈Th
?c
1
2∇u?2
0,K− 2
?
K∈Th
?
K
Lc(u) · ∇udx +
?
F∈Fh
?a
1
2[[u]]?2
0,F.
2+b2
2ε, valid for any ε > 0, the bounds in (2.5), and the stability bound
2
?
K∈Th
K
Lc(u) · ∇udx = 2
?
?
?
?
?
?c−1
K
c−1
2Lc(u) · c
1
2∇udx
2Lc(u)?0,K?c
1
2∇u?0,K
?c
1
2∇u?2
0,K+ ε−1c−1
?
?
?c?C2
K∈Th
?Lc(u)?2
0,K
?c
1
2∇u?2
0,K+ ε−1α−1c−1
inv
?
F∈Fh
?a
1
2[[u]]?2
0,F
? ah(u,u) ≥ (1 − ε)
K∈Th
?c
1
2∇u?2
0,K+?1 − ε−1α−1c−1
?c?C2
inv
? ?
F∈Fh
?a
1
2[[u]]?2
0,F.
2Lc(u)?2
0from Lemma 4.4
Page 10
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2417
Error equation. Because ? ah coincides with ah on Vh× Vh, the semidiscrete
Find uh: J × Vh→ R such that uh|t=0= Πhu0, uh
(uh
(4.6)
scheme in (3.3)–(3.5) is equivalent to the following:
t|t=0= Πhv0, and
∀v ∈ Vh.
tt,v) + ? ah(uh,v) = (f,v)
?
We shall use the formulation in (4.6) as the basis of our error analysis.
To derive an error equation, we first define for u ∈ H1+σ(Ω) with σ > 1/2
?
Here Πhdenotes the L2-projection onto (Vh)d. The assumption u ∈ H1+σ(Ω) ensures
that rh(u;v) is well defined. From the definition in (4.7) it is immediate that rh(u;v) =
0 when v ∈ H1
Lemma 4.5. Let the analytical solution u of (2.1)–(2.4) satisfy
rh(u;v) =
F∈Fh
F
[[v]] · { {c∇u − cΠh(∇u)} }dA,v ∈ V (h). (4.7)
0(Ω).
u ∈ L∞(J;H1+σ(Ω)),utt∈ L1(J;L2(Ω)).
Let uhbe the semidiscrete DG approximation obtained by (4.6). Then, the error e =
u − uhsatisfies
(ett,v) + ? ah(e,v) = rh(u;v)
Proof. Let v ∈ Vh. Since utt∈ L1(J;L2(Ω)), we have ?utt,v? = (utt,v) almost
everywhere in J. Hence, using the discrete formulation in (3.3)–(3.5), we obtain that
∀v ∈ Vha.e. in J,
with rh(u;v) given in (4.7).
(ett,v) + ? ah(e,v) = (utt,v) + ? ah(u,v) − (f,v)
the defining properties of the L2-projection Πh, and the definition of the lifted ele-
ment Lc(v), we obtain
?
Since utt ∈ L1(J;L2(Ω)) and f ∈ L2(J;L2(Ω)), we have that ∇ · (c∇u) ∈ L2(Ω)
almost everywhere in J, which implies that c∇u has continuous normal components
across all interior faces. Therefore, elementwise integration by parts combined with
the trace operators defined in section 3.1 yields
?
−
F∈Fh
From the definition of rh(u,v) in (4.7), we therefore conclude that
a.e. in J.
Now, by definition of ? ah, the fact that Lc(u) = 0 and that [[u]] = 0 on all faces,
?
? ah(u,v) =
K∈Th
K
c∇u · ∇v dx −
?
F∈Fh
?
F
[[v]] · { {cΠh(∇u)} }dA.
? ah(u,v) = −
K∈Th
?
?
?
K
∇ · (c∇u)v dx +
?
F∈Fh
?
F
[[v]] · { {c∇u} }dA
F
[[v]] · { {cΠh(∇u)} }dA.
(utt,v) + ? ah(u,v) = (utt− ∇ · (c∇u),v) + rh(u;v)
(ett,v) + ? ah(e,v) = (utt− ∇ · (c∇u) − f,v) + rh(u;v) = rh(u;v),
and obtain
where we have used the differential equation (2.1).
Page 11
2418
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
Approximation properties. Let Πhand Πhdenote the L2-projections onto Vh
and (Vh)d, respectively. We recall the following approximation properties; see [6].
Lemma 4.6. Let K ∈ Th. Then the following hold:
(i) For v ∈ Ht(K), t ≥ 0, we have
?v − Πhv?0,K ≤ Chmin{t,?+1}
K
?v?t,K,
with a constant C that is independent of the local mesh size hK and depends
only on the shape-regularity of the mesh, the approximation order ?, the di-
mension d, and the regularity exponent t.
(ii) For v ∈ H1+σ(K), σ >1
2, we have
?∇v − ∇(Πhv)?0,K ≤ Chmin{σ,?}
?v − Πhv?0,∂K ≤ Chmin{σ,?}+1
?∇v − Πh(∇v)?0,∂K ≤ Chmin{σ,?+1}−1
K
?v?1+σ,K,
2
?v?1+σ,K,
2
?v?1+σ,K,
K
K
with a constant C that is independent of the local mesh size hK and depends
only on the shape-regularity of the mesh, the approximation order ?, the di-
mension d, and the regularity exponent σ.
As a consequence of the approximation properties in Lemma 4.6, we obtain the
following results.
Lemma 4.7. Let u ∈ H1+σ(Ω), σ >1
(i) We have
2. Then the following hold:
?u − Πhu?h≤ CAhmin{σ,?}?u?1+σ,
with a constant CA that is independent of the mesh size and depends only
on α, the constant κ in (3.1), the bounds in (2.5), and the constants in
Lemma 4.6.
(ii) For v ∈ V (h), the form rh(u;v) in (4.7) can be bounded by
??
|rh(u;v)| ≤ CRhmin{σ,?}
F∈Fh
?a
1
2[[v]]?2
0,F
?1
2
?u?1+σ,
with a constant CR independent of h, which depends only on α, the bounds
in (2.5), and the constants in Lemma 4.6.
Proof. The estimate in (i) is an immediate consequence of Lemma 4.6, the defi-
nition of a, and the bounded variation property (3.1). To show the bound in (ii), we
apply the Cauchy–Schwarz inequality and obtain
|rh(u;v)| ≤
??
F∈Fh
?
F
a[[v]]2ds
?1
2??
F∈Fh
?
F
a−1|{ {c∇u − cΠh(∇u)} }|2ds
?1
2
≤ α−1
2c−1
2
? c?
??
F∈Fh
?a
1
2[[v]]?2
0,F
?1
2??
K∈Th
hK?∇u − Πh(∇u)?2
0,∂K
?1
2
.
Applying the approximation properties in Lemma 4.6 completes the proof.
Page 12
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2419
4.3. Proof of Theorem 4.1. We are now ready to complete the proof of The-
orem 4.1. We begin by proving the following auxiliary result.
Lemma 4.8. Let the analytical solution u of (2.1)–(2.4) satisfy
u ∈ L∞(J;H1+σ(Ω)),
2. Let v ∈ C0(J;V (h)) and vt∈ L1(J;V (h)). Then we have
?
·
where CRis the constant from the bound (ii) in Lemma 4.7.
Proof. From the definition of rhin (4.7) and integration by parts, we obtain
?
F∈Fh
?
F∈Fh
??
= −
J
Lemma 4.7 then implies the two estimates
????
????
To complete the proof of Theorem 4.1, we now set e = u−uhand recall that Πh
is the L2-projection onto Vh. Because of (2.8), we have
ut∈ L∞(J;H1+σ(Ω))
for σ >1
J
|rh(u;vt)| dt ≤ CRhmin{σ,?}?v?L∞(J;V (h))
?
2?u?L∞(J;H1+σ(Ω))+ T ?ut?L∞(J;H1+σ(Ω))
?
,
J
rh(u;vt)dt =
?
J
?
?
F
[[vt]] · { {c∇u − cΠh(∇u)} }dAdt
?
= −
J
?
F
[[v]] · { {c∇ut− cΠh(∇ut)} }dAdt
+
F∈Fh
?
?
F
[[v]] · { {c∇u − cΠh(∇u)} }dA
?
?t=T
t=0
rh(ut;v)dt +rh(u;v)
?t=T
t=0.
?
J
rh(ut;v)dt
????≤ CRhmin{σ,?}T ?v?L∞(J;V (h))?ut?L∞(J;H1+σ(Ω))
????≤ 2CRhmin{σ,?}?v?L∞(J;V (h))?u?L∞(J;H1+σ(Ω)),
and
?
rh(u;v)
?t=T
t=0
which concludes the proof of the lemma.
e ∈ C0(J;V (h)) ∩ C1(J;L2(Ω)).
Next, we use the symmetry of ? ahand the error equation in Lemma 4.5 to obtain
2 dt
= (ett,(u − Πhu)t) + ? ah(e,(u − Πhu)t)
We fix s ∈ J and integrate (4.8) over the time interval (0,s). This yields
1
2?et(s)?2
?s
+
0
1
d
??et?2
0+ ? ah(e,e)?= (ett,et) + ? ah(e,et)
+ rh(u;(Πhu − uh)t).
(4.8)
0+1
2? ah(e(s),e(s)) =1
2?et(0)?2
0+1
2? ah(e(0),e(0))
(ett,(u − Πhu)t)dt +
+
0
?s
0
? ah(e,(u − Πhu)t)dt
?s
rh(u;(Πhu − uh)t)dt.
Page 13
2420
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
Integration by parts of the third term on the right-hand side yields
?s
From the stability properties of ? ahin Lemma 4.4 and standard H¨ older’s inequalities,
1
2?et(s)?2
?
+ CcontT?e?L∞(J;V (h))?(u − Πhu)t?L∞(J;V (h))
+
J
0
(ett,(u − Πhu)t)dt = −
?s
0
(et,(u − Πhu)tt)dt +
?
(et,(u − Πhu)t)
?t=s
t=0.
we conclude that
0+1
2Ccoer?e(s)?2
h≤1
2?et(0)?2
0+1
2Ccont?e(0)?2
h
+ ?et?L∞(J;L2(Ω))
?(u − Πhu)tt?L1(J;L2(Ω))+ 2?(u − Πhu)t?L∞(J;L2(Ω))
?
????
?
rh(u;(Πhu − uh)t)dt
????.
Since this inequality holds for any s ∈ J, it also holds for the maximum over J, that
is
?et?2
L∞(J;L2(Ω))+ Ccoer?e?2
L∞(J;V (h))≤ ?et(0)?2
0+ Ccont?e(0)?2
h+ T1+ T2+ T3,
with
T1= 2?et?L∞(J;L2(Ω))
T2= 2CcontT?e?L∞(J;V (h))?(u − Πhu)t?L∞(J;V (h)),
T3= 2
J
?
?(u − Πhu)tt?L1(J;L2(Ω))+ 2?(u − Πhu)t?L∞(J;L2(Ω))
?
,
????
?
rh(u;(Πhu − uh)t)dt
????.
Using the geometric-arithmetic mean inequality |ab| ≤
and the approximation results in Lemma 4.6, we conclude that
?
≤1
≤1
1
2εa2+ε
2b2, valid for any ε > 0,
T1≤1
2?et?2
L∞(J;L2(Ω))+ 2
?(u − Πhu)tt?L1(J;L2(Ω))+ 2?(u − Πhu)t?L∞(J;L2(Ω))
?2
2?et?2
L∞(J;L2(Ω))+ 4?(u − Πhu)tt?2
L∞(J;L2(Ω))+ Ch2min{σ,?}?
with a constant C that depends only on the constants in Lemma 4.6. Similarly,
L1(J;L2(Ω))+ 16?(u − Πhu)t?2
L∞(J;L2(Ω)),
2?et?2
?utt?2
L1(J;Hσ(Ω))+ h2?ut?2
L∞(J;H1+σ(Ω))
?
,
T2≤1
≤1
4Ccoer?e?2
L∞(J;V (h))+ 4C2
cont
CcoerT2?(u − Πhu)t?2
L∞(J;V (h))
4Ccoer?e?2
L∞(J;V (h))+ T2Ch2min{σ,?}?ut?2
L∞(J;H1+σ(Ω)),
where the constant C depends on Ccoer, Ccont, and the constant CAin Lemma 4.7.
It remains to bound the term T3. To do so, we use Lemma 4.8 to obtain
T3≤ 2CRRhmin{σ,?}?Πhu − uh?L∞(J;V (h)),
with
R :=
?
2?u?L∞(J;H1+σ(Ω))+ T?ut?L∞(J;H1+σ(Ω))
?
.
Page 14
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2421
The triangle inequality, the geometric-arithmetic mean, and the approximation prop-
erties of Πhin Lemma 4.7 then yield
T3≤ 2CRRhmin{σ,?}?
≤1
?e?L∞(J;V (h))+ ?u − Πhu?L∞(J;V (h))
L∞(J;V (h))+ Ch2min{σ,?}?
?
4Ccoer?e?2
?u?2
L∞(J;H1+σ(Ω))+ R2?
,
with a constant C that depends only on Ccoer, CR, and CA. Combining the above
estimates for T1, T2, and T3then shows that
1
2?et?2
L∞(J;L2(Ω))+1
+ Ch2min{σ,?}??utt?2
with a constant that is independent of T and the mesh size. This concludes the proof
of Theorem 4.1.
2Ccoer?e?2
L1(J;Hσ(Ω))+ T2?ut?2
L∞(J;V (h))≤ ?et(0)?2
0+ Ccont?e(0)?2
L∞(J;H1+σ(Ω))+ ?u?2
h
L∞(J;H1+σ(Ω))
?,
4.4. Proof of Theorem 4.2. To prove the error estimate in Theorem 4.2, we
first establish the following variant of [3, Lemma 2.1].
Lemma 4.9. For u ∈ H1+σ(Ω) with σ >1
? ah(wh,v) = ? ah(u,v) − rh(u;v)
?u − wh?h≤ CEhmin{σ,?}?u?1+σ,
with a constant CE that is independent of h and depends only on Ccoer, Ccont in
Lemma 4.4 and CA, CRin Lemma 4.7.
Moreover, if the elliptic regularity defined in (4.1) and (4.2) holds, we have the
L2-bound
2, let wh∈ Vhbe the solution of
∀v ∈ Vh.
Then, we have
?u − wh?0≤ CLhmin{σ,?}+1?u?1+σ,
with a constant CLthat is independent of h and depends only on the stability constant
CSin (4.2); Ccoer, Ccontin Lemma 4.4; and CA, CRin Lemma 4.7.
Proof. We first remark that the approximation whis well defined because of the
stability properties in Lemma 4.4 and the estimates in Lemma 4.7. To prove the
estimate for ?u − wh?h, we first use the triangle inequality,
?u − wh?h≤ ?u − Πhu?h+ ?Πhu − wh?h.
From the approximation properties of Πhin Lemma 4.7, we immediately infer that
(4.9)
?u − Πhu?h≤ CAhmin{σ,?}?u?1+σ.
From the coercivity and continuity of ? ah in
h≤ ? ah(Πhu − wh,Πhu − wh)
= ? ah(Πhu − u,Πhu − wh) + rh(u;Πhu − wh)
It remains to bound ?Πhu − wh?h.
Lemma 4.4, the definition of wh, and the bound in Lemma 4.7, we conclude that
Ccoer?Πhu − wh?2
= ? ah(Πhu − u,Πhu − wh) + ? ah(u − wh,Πhu − wh)
≤ Ccont?Πhu − u?h?Πhu − wh?h+ CRhmin{σ,?}?u?1+σ?Πhu − wh?h.
Page 15
2422
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
Thus,
?Πhu − wh?h≤
?CcontCA+ CR
Ccoer
?
hmin{σ,?}?u?1+σ,
which proves the bound for ?u − wh?h.
We shall now prove the L2-bound. To do so, let z ∈ H1
−∇ · (c∇z) = u − wh
0(Ω) be the solution of
in Ω,z = 0on Γ. (4.10)
Then, the elliptic regularity assumption in (4.1) and (4.2) implies that
z ∈ H2(Ω),
?z?2≤ CS?u − wh?0. (4.11)
Next, we multiply (4.10) by u − whand integrate the resulting expression by parts.
Since c∇z has continuous normal components across all interior faces, we have
?
=
K
?u − wh?2
0=
K∈Th
?
??
?
K
c∇z · ∇(u − wh)dx −
?
∂K
c∇z · nK(u − wh)dA
?
?
K∈Th
c∇z · ∇(u − wh)dx −
?
F∈Fh
F
{ {c∇z} } · [[u − wh]]dA,
with nK denoting the unit outward normal on ∂K. By definition of ? ahand rh, we
?u − wh?2
From the symmetry of ? ah, the definition of wh, and the fact that [[z]] = 0 on all faces,
?u − wh?2
=: T1+ T2+ T3.
immediately find that
0= ? ah(z,u − wh) − rh(z;u − wh).
we conclude that
0= ? ah(u − wh,z − Πhz) − rh(u;z − Πhz) − rh(z;u − wh)
We shall now derive upper bounds for each individual term T1, T2, and T3in (4.12).
To estimate the term T1, we use the continuity of ? ah, the approximation result in
T1≤ Ccont?u − wh?h?z − Πhz?h
≤ CcontCAh?u − wh?h?z?2
≤ CcontCACSh?u − wh?h?u − wh?0.
(4.12)
Lemma 4.7 with σ = 1, and the bound in (4.11). Thus,
By using Lemma 4.7 and the stability bound in (4.11), we can estimate T2by
T2≤ CRhmin{σ,?}?z − Πhz?h?u?1+σ
≤ CRCAhmin{σ,?}+1?z?2?u?1+σ
≤ CRCACShmin{σ,?}+1?u − wh?0?u?1+σ.
Similarly,
T3≤ CRh?z?2?u − wh?h≤ CRCSh?u − wh?0?u − wh?h.
Page 16
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2423
The use of these bounds for T1, T2, and T3in (4.12) then leads to
?u − wh?0≤ Ch?u − wh?h+ Chmin{σ,?}+1?u?1+σ,
which completes the proof of the lemma, since ?u−wh?h≤ Chmin{σ,?}?u?1+σ.
Now, let u be defined by the exact solution of (2.1)–(2.4). We may define wh(t,·) ∈
Vhalmost everywhere in J by
? ah(wh(t,·),v) = ? ah(u(t,·),v) − rh(u(t,·);v)
we also have ut∈ L∞(J;H1+σ(Ω)), then wh
? ah(wh
as well as
∀v ∈ Vh. (4.13)
If u ∈ L∞(J;H1+σ(Ω)), it can be readily seen that wh∈ L∞(J;V (h)). Moreover, if
t∈ L∞(J;V (h)) and
v ∈ Vha.e. in J,
t,v) = ? ah(ut,v) − rh(ut;v),
? ah(wh(0),v) = ? ah(u0,v) − rh(u0;v),
Therefore Lemma 4.9 immediately implies the following estimates.
Lemma 4.10. Let whbe defined by (4.13). Under the regularity assumptions of
Theorem 4.2, we have
v ∈ Vh.
?(u − wh)t?L∞(J;V (h))≤ CEhmin{σ,?}?ut?L∞(J;H1+σ(Ω)),
?(u − wh)(0)?h
Moreover, if elliptic regularity as defined in (4.1) and (4.2) holds, we have the L2-
bounds
≤ CEhmin{σ,?}?u0?1+σ.
?(u − wh)t?L∞(J;L2(Ω))≤ CLhmin{σ,?}+1?ut?L∞(J;H1+σ(Ω)),
?(u − wh)(0)?0
≤ CLhmin{σ,?}+1?u0?1+σ.
The constants CEand CLare as in Lemma 4.9.
To complete the proof of Theorem 4.2, let wh∈ L∞(J;V (h)) be defined by (4.13)
and consider
?e?2
L∞(J;L2(Ω))≤ 2?u − wh?2
L∞(J;L2(Ω))+ 2?wh− uh?2
L∞(J;L2(Ω)).(4.14)
The first term can be estimated from the L2-bounds in Lemma 4.9. We shall now
derive an estimate for the second term. First, we fix v ∈ L∞(J;Vh) and assume
that vt∈ L∞(J;Vh). From the definition of whin (4.13) and the error equation in
Lemma 4.5, we have
((uh− wh)tt,v) + ? ah(uh− wh,v) = (uh
tt,v) + ? ah(uh,v) − ? ah(wh,v) − (wh
= (utt,v) − (wh
tt,v)
= (uh
tt,v) + ? ah(uh− u,v) + rh(u;v) − (wh
tt,v)
tt,v).
We rewrite this identity as
d
dt((uh−wh)t,v)−((uh−wh)t,vt)+? ah(uh−wh,v) =d
dt((u−wh)t,v)−((u−wh)t,vt),
Page 17
2424
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
which yields
−((uh− wh)t,vt) + ? ah(uh− wh,v) =d
Let τ ∈ (0,T] be fixed, and consider the function
dt((u − uh)t,v) − ((u − wh)t,vt). (4.15)
? v(t,·) =
?τ
t
(uh− wh)(s,·)ds,t ∈ J.
Note that
? v(τ,·) = 0,
? vt(t,·) = −(uh− wh)(t,·) a.e. t ∈ J.
Next, choose v = ? v in (4.15) which yields
((uh− wh)t,uh− wh) − ? ah(? vt,? v) =d
Since the DG form ? ah(·,·) is symmetric, we obtain
22
dt((u − uh)t,? v) + ((u − wh)t,uh− wh).
1
d
dt?uh− wh?2
0−1
d
dt? ah(? v,? v) =d
dt((u − uh)t,? v) + ((u − wh)t,uh− wh).
Integration over (0,τ) and using that ? v(τ,·) = 0 then yield
= −2((u − uh)t(0),? v(0)) + 2
Since ut(0) = v0, uh
?(uh− wh)(τ)?2
0− ?(uh− wh)(0)?2
0+ ? ah(? v(0),? v(0))
0
?τ
((u − wh)t,uh− wh)dt.
(4.16)
t(0) = Πhv0, and ? v(0) belongs to Vh, we conclude that
((u − uh)t(0),? v(0)) = (v0− Πhv0,? v(0)) = 0.
civity of the form ? ahin Lemma 4.4 ensures that ? ah(? v(0),? v(0)) ≥ 0. This leads to the
?(uh− wh)(τ)?2
Hence, the first term on the right-hand side of (4.16) vanishes. Moreover, the coer-
inequality
0≤ ?(uh− wh)(0)?2
0+ 2
?τ
0
?(u − wh)t?0?uh− wh?0dt. (4.17)
By using the Cauchy–Schwarz inequality and the geometric-arithmetic mean inequal-
ity, we obtain
?τ
≤1
2
0
?(u − wh)t?0?uh− wh?0dt ≤ 2T ?(u − wh)t?L∞(J;L2(Ω))?uh− wh?L∞(J;L2(Ω))
2?uh− wh?2
L∞(J;L2(Ω))+ 2T2?(u − wh)t?2
L∞(J;L2(Ω)).
Because this upper bound is independent of τ, it also holds for the supremum over
τ ∈ J, which yields the estimate
1
2?uh− wh?2
≤ 2?(uh− u)(0)?2
L∞(J;L2(Ω))≤ ?(uh− wh)(0)?2
0+ 2?(u − wh)(0)?2
0+ 2T2?(u − wh)t?2
0+ 2T2?(u − wh)t?2
L∞(J;L2(Ω))
L∞(J;L2(Ω)).
Page 18
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2425
Next, we use this estimate in (4.14) to obtain
?e?2
L∞(J;L2(Ω))≤ 2?u − wh?2
+8?u0− Πhu0?2
From the L2-approximation properties in Lemma 4.6, Lemma 4.9, and Lemma 4.10,
we finally conclude that
L∞(J;L2(Ω))≤ h2min{σ,?}+2?
+2C2
L∞(J;L2(Ω))
0+ 8?u0− wh(0)?2
0+ 8T2?(u − wh)t?2
L∞(J;L2(Ω)).
?e?2
max{8C,8C2
L?u?2
L}?u0?2
L∞(J;H1+σ(Ω))+ 8CLT2?ut?2
1+σ
L∞(J;H1+σ(Ω))
?
.
Here, C is the constant from Lemma 4.6. This completes the proof of Theorem 4.2.
5. Numerical results. We shall now present a series of numerical experiments
which verify the sharpness of the theoretical error bounds stated in Theorems 4.1
and 4.2. Furthermore, we shall demonstrate the robustness and flexibility of our DG
method by propagating a pulse through an inhomogeneous medium with discontinuity
on a finite element mesh with hanging nodes.
To obtain a fully discrete discretization of the wave equation, we choose to aug-
ment our DG spatial discretization with the second-order Newmark scheme in time;
see, e.g., [19, sections 8.5–8.7]. The resulting scheme has been implemented using the
general purpose finite element library deal.II,1which provides powerful C++classes
to handle the finite element mesh and the degrees of freedom, and to solve the linear
system of equations; see [5, 4]. In all our examples, the DG stabilization parameter
is set to α = 20.
5.1. Time discretization. The discretization of (2.1)–(2.4) in space by the DG
method (3.3)–(3.5) leads to the linear second-order system of ordinary differential
equations
M¨ uh(t) + Auh(t) = fh(t),t ∈ J,(5.1)
with initial conditions
Muh(0) = uh
0,M˙ uh(0) = vh
0. (5.2)
Here, M denotes the mass matrix and A the stiffness matrix. To discretize (5.1) in
time, we employ the Newmark time-stepping scheme; see, e.g., [19]. We let k denote
the time step and set tn = n · k. Then the Newmark method consists in finding
approximations {uh
(5.3)
n}nto uh(tn) such that
(M + k2βA)uh
1=
?
M − k2
?1
2− β
?
A
?
uh
0+ kMvh
0+ k2
?
βfh
1+
?1
2− β
?
fh
0
?
and
(M + k2βA)uh
n+1=
?
?
2M − k2
?1
?1
2− 2β + γ
?
?
?1
A
?
uh
n−
?
?
M + k2
?1
2+ β − γ
?
A
?
uh
n−1
+k2
βfh
n+1+
2− 2β + γfn+
2+ β − γfn−1
?
(5.4)
1See www.dealii.org.
Page 19
2426
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
for n = 1,...,N − 1. Here, fn:= f(tn), while β ≥ 0 and γ ≥ 1/2 are free parameters
that still can be chosen. We recall that for γ = 1/2 the Newmark scheme is second-
order accurate in time, whereas it is only first-order accurate for γ > 1/2. For β = 0,
the Newmark scheme (5.3)–(5.4) requires at each time step the solution of a linear
system with the mass matrix M. However, because individual elements decouple,
M is block-diagonal with a block size equal to the number of degrees of freedom
per element. It can be inverted at very low computational cost, and the scheme is
essentially fully explicit. In fact, if the basis functions are chosen mutually orthogonal,
M reduces to the identity; see [8] and the references therein. Then, with γ = 1/2, the
explicit Newmark method corresponds to the standard leap-frog scheme.
For β > 0, the resulting scheme is implicit and involves the solution of a linear
system with the symmetric positive definite stiffness matrix A at each time step. We
finally note that the second-order Newmark scheme with γ = 1/2 is unconditionally
stable for β ≥ 1/4, whereas for 1/4 > β ≥ 0 the time step k has to be restricted by a
CFL condition. In the case β = 0, that condition is k2λmax(A) ≤ 4(1 − ε), ε ∈ (0,1),
where λmax(A) is the maximal eigenvalue of the DG stiffness matrix A (which is of
the order O(h−2) and also depends on α).
In all our tests, we will employ the explicit second-order Newmark scheme, setting
γ = 1/2 and β = 0 in (5.3)–(5.4).
5.2. Example 1: Smooth solution. First, we consider the two-dimensional
wave equation (2.1)–(2.4) in J × Ω = (0,1) × (0,1)2, with c ≡ 1 and data f,u0, and
v0chosen such that the analytical solution is given by
u(x1,x2,t) = t2sin(πx1) sin(πx2). (5.5)
This solution is arbitrarily smooth so that all our theoretical regularity assumptions
are satisfied. We discretize this problem using the polynomial spaces Q?(K), ? =
1,2,3, on a sequence {Th}i≥1 of square meshes of size hi = 2−i. With increasing
polynomial degree ? and decreasing mesh size hi, smaller time steps kiare necessary
to ensure stability. We found that the choice ki = hi/20 provides a stable time
discretization on every mesh. Because our numerical scheme is second-order accurate
in time, the time integration of (5.5) is exact so that the spatial error is the only error
component in the discrete solution.
In Figure 5.1 we show the relative errors at time T = 1 in the energy norm and
in the L2-norm, as we decrease the mesh size hi. The numerical results corroborate
with the expected theoretical rates of O(h?) for the energy norm and of O(h?+1) for
the L2-norm; see Theorems 4.1 and 4.2.
Next, we modify the data so that the analytical solution u is given by
u(x1,x2,t) = sin(t2) sin(πx1) sin(πx2). (5.6)
Although u remains arbitrarily smooth, it is no longer integrated exactly in time by
(5.3)–(5.4). Since the Newmark scheme is only second-order accurate, we repeat the
above experiment only for the lowest order spatial discretization, ? = 1. Again, we
set ki= hi/20. In Figure 5.2, the relative errors for the fully discrete approximation
of (5.6) show convergence rates of order h in the energy norm and order h2in the
L2-norm, thereby confirming the theoretical estimates of Theorems 4.1 and 4.2.
5.3. Example 2: Singular solution. Here, we consider the two-dimensional
wave equation (2.1)–(2.4) on the L-shaped domain Ω = (−1,1)2\[0,1)2. We set c = 1
Page 20
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2427
10
−2
10
−1
10
0
10
−8
10
−6
10
−4
10
−2
10
0
1
1
1
1
2
3
l=1
l=2
l=3
?u − uh?DG
mesh size h
Convergence of Energy Error
10
−2
10
−1
10
0
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
1
1
1
2
3
4
l=1
l=2
l=3
?u − uh?0
mesh size h
Convergence of L2-Error
Fig. 5.1. Example 1: Convergence of the error at time T = 1 in the energy norm and the
L2-norm for ? = 1,2,3.
10
−2
10
−1
10
0
10
−4
10
−3
10
−2
10
−1
10
0
1
1
1
2
energy−error
L2−error
errors
mesh size h
Fig. 5.2. Example 1: Convergence of the error at time T = 1 in the energy norm and the
L2-norm for ? = 1.
everywhere and choose the data f,u0, and v0such that the analytical solution u is
given by
u(r,φ,t) = t2r2/3sin(2/3φ) (5.7)
in polar coordinates (r,φ). Although u is smooth in time (and can even be inte-
grated exactly in time), it has a spatial singularity at the origin, such that u ∈
C∞(J;H5/3(Ω)). Hence, this example is well suited to establishing the sharpness of
the regularity assumptions in our theoretical results. Since u is inhomogeneous at
the boundary of Ω, we need to impose inhomogeneous Dirichlet conditions within our
DG discretization. We do so in straightforward fashion by modifying the semidiscrete
formulation as follows: find uh(t,·) : J → Vhsuch that
(uh
tt,v) + ah(uh,v) = (f,v) +
?
F∈FB
h
?
F
g (av − c∇v · n)dA.(5.8)
Here, g is the boundary data and n is the outward unit normal vector on ∂Ω.
We discretize (5.8) by using bilinear polynomials (? = 1) on the same sequence of
meshes as before. Again, we set ki= hi/20 and integrate the problem up to T = 1.
For the analytical solution u in (5.7), the regularity assumptions in Theorem 4.1 hold
Page 21
2428
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
Table 5.1
Example 2: Relative errors at time T = 1 in the energy norm and L2-norm, and corresponding
numerical convergence rates.
iCellsEnergy-error
1.11e-01
7.18e-02
4.61e-02
2.94e-02
1.87e-02
1.18e-02
L2-error
1.61e-02
5.96e-03
2.27e-03
8.72e-04
3.38e-04
1.32e-04
1
2
3
4
5
6
12
48
--
0.62
0.64
0.65
0.66
0.66
1.43
1.40
1.38
1.37
1.36
192
768
3072
12288
with σ = 2/3. Thus, Theorem 4.1 predicts numerical convergence rates of 2/3 in the
energy norm, as confirmed by our numerical results in Table 5.1.
As the elliptic regularity assumptions (4.1)–(4.2) from Theorem 4.2 are violated,
we do not expect L2-error rates of the order 1+σ for this problem. Indeed, in Table 5.1
we observe convergence rates close to 4/3. To explain this behavior, let us consider
the following weaker elliptic regularity assumption: for any λ ∈ L2(Ω) we assume that
the solution of the problem
−∇ · (c∇z) = λ in Ω,z = 0 on ∂Ω, (5.9)
belongs to H1+s(Ω) for a parameter s ∈ (1/2,1] and satisfies the stability bound
?z?1+s≤ CS?λ?0
(5.10)
for a stability constant CS. The results from Lemmas 4.9 and 4.10 can be easily
adapted to this case. As a consequence, the L2-bound for e = u−uhfrom Theorem 4.2
can then be generalized to this weaker setting as
?e?L∞(J;L2(Ω))≤ Chmin{σ,?}+s??u0?1+σ
+ ?u?L∞(J;H1+σ(Ω))+ T?ut?L∞(J;H1+σ(Ω))
For the L-shaped domain Ω and c ≡ 1, the (weaker) regularity assumption in (5.9)–
(5.10) holds with s = 2/3, which underpins the rate σ+s = 4/3 observed in Table 5.1.
?.
5.4. Example 3: Inhomogeneous medium. Finally, we consider (2.1)–(2.4)
on the rectangular domain Ω = (−1,2)×(−1,1), with homogeneous initial and bound-
ary conditions and the piecewise constant material coefficient
?
The wave is locally excited until t = 0.2 by the source term
?
We discretize the problem by the DG method (3.3)–(3.5) on a fixed mesh Th
that consists of nonmatching components, which are adapted to the discontinuity c;
see Figure 5.3. The mesh Th is composed of 9312 nonuniform squares, where the
smallest local mesh size is given by hmin≈ 0.016. The hanging nodes are naturally
incorporated in the DG method without any difficulty. Here, the time step k = 0.002,
c(x1,x2) =
0.1,
1,
x1≤ 0,
else.
f(x1,x2,t) =
1,
0,
0.2 < x1< 0.4 and t < 0.2,
else.
Page 22
DISCONTINUOUS GALERKIN FEM FOR THE WAVE EQUATION
2429
Fig. 5.3. Example 3: Domain Ω with a finite element mesh Thadapted to the values of the
piecewise constant wave speed√c = 0.1 (left),√c = 1 (right).
Fig. 5.4. Example 3: The approximate DG solutions uh
display the behavior of a wave propagating through an inhomogeneous medium with homogeneous
Dirichlet boundary.
nshown at times tn = 0.2,0.6,1.8,4
that is, k ≈ hmin/8, proved to be sufficiently small to ensure the stability of the
explicit Newmark method (β = 0).
In Figure 5.4, the numerical solution is shown after n = 100,300,900, and 2000
time steps. The initial pulse splits into two planar wave fronts propagating in opposite
directions to either side of the domain. After n = 300 time steps, the left-moving wave
hits the much slower medium in the region x1≤ 0, resulting in a much steeper and
narrower wave front. Meanwhile, the right-moving wave rapidly reaches the boundary
at x1 = 2, where it is reflected and eventually enters the slow medium, too. The
discontinuous interface at x1= 0 generates multiple reflections, which interact with
each other at later times.
Page 23
2430
M. J. GROTE, A. SCHNEEBELI, AND D. SCH¨OTZAU
6. Conclusion. We have presented and analyzed the symmetric interior penalty
discontinuous Galerkin finite element method (DGFEM) for the numerical solution
of the (second-order) scalar wave equation. Taking advantage of the symmetry of the
method, we have carried out an a priori error analysis of the semidiscrete method and
derived optimal error bounds in the energy norm and, under additional regularity
assumptions, optimal error bounds in the L2-norm. Our numerical results confirm
the expected convergence rates and demonstrate the versatility of the method. The
error analysis of the fully discrete scheme is the subject of ongoing work.
Based on discontinuous finite element spaces, the proposed DG method easily
handles elements of various types and shapes, irregular nonmatching grids, and even
locally varying polynomial order. As continuity is only weakly enforced across mesh
interfaces, domain decomposition techniques immediately apply. Since the resulting
mass matrix is essentially diagonal, the method is inherently parallel and leads to
fully explicit time integration schemes. Moreover, as the stiffness matrix is symmetric
positive definite, the DG method shares the following two important properties with
the classical continuous Galerkin approach. First, the semidiscrete formulation con-
serves (a discrete version of) the energy for all time and therefore is nondissipative.
Second, if implicit time integration is used to overcome CFL constraints, the resulting
linear system to be solved at each time step will also be symmetric positive definite.
The symmetric interior penalty DGFEM, applied here to the scalar wave equation,
can also be utilized for other second-order hyperbolic equations, such as in electro-
magnetics or elasticity. In fact, our error analysis for the semidiscrete (scalar) case
readily extends to the second-order (vector) wave equation for time dependent elastic
waves. The same DG approach also extends to Maxwell’s equations in second-order
form:
εutt+ σut+ ∇ × (μ−1∇ × u) = f,σ ≥ 0.
Here, DG discretizations with standard discontinuous piecewise polynomials indeed
offer an alternative to edge elements, as typically used in conforming discretizations
of Maxwell’s equations. The corresponding error analysis, however, is more involved
and will be reported elsewhere in the near future.
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