![Comparison between the spectrum of 2N n −1 C [2,3,2] N,n (K) and the canonical rearranged version κ of the symbol f [1,K] [2,3,2] (x, θ) for N = n = 30 and K(x) = 3 − cos(10x).](https://www.researchgate.net/profile/Carlo-Garoni/publication/322520269/figure/fig2/AS:614258014421007@1523461931885/Comparison-between-the-spectrum-of-2N-n-1-C-2-3-2-N-n-K-and-the-canonical_Q320.jpg)
![Comparison between the spectrum of 2N n −1 C [1,2,1] N,n (1) and the canonical rearranged version κ of the symbol f [1,1] [1,2,1] (θ) for N = 35 and n = 70.](https://www.researchgate.net/profile/Carlo-Garoni/publication/322520269/figure/fig1/AS:614258014437387@1523461931844/Comparison-between-the-spectrum-of-2N-n-1-C-1-2-1-N-n-1-and-the-canonical_Q320.jpg)
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DISCONTINUOUS GALERKIN DISCRETIZATION OF THE HEAT
EQUATION IN ANY DIMENSION: THE SPECTRAL SYMBOL∗
PIETRO BENEDUSI†, CARLO GARONI‡, ROLF KRAUSE§, XIAOZHOU LI¶,AND
STEFANO SERRA-CAPIZZANOk
Abstract. The multidimensional heat equation, along with its more general version involving
variable diffusion coefficients, is discretized by a discontinuous Galerkin (DG) method in time and a
finite element (FE) method of arbitrary regularity in space. We show that the resulting space-time
discretization matrices enjoy an asymptotic spectral distribution as the mesh fineness increases, and
we determine the associated spectral symbol, i.e., the function that carefully describes the spectral
distribution. The analysis of this paper is carried out in a stepwise fashion, without omitting details,
and it is supported by several numerical experiments. It is preparatory to the development of
specialized solvers for linear systems arising from the DG/FE approximation of the heat equation in
the case of both constant and variable diffusion coefficients.
Key words. Spectral distribution, symbol, discontinuous Galerkin method, finite element
method, B-splines, heat equation
AMS subject classifications. 15A18, 65M60, 41A15, 35K05, 15B05, 15A69
1. Introduction. Suppose a linear partial differential equation (PDE) is dis-
cretized by a linear numerical method characterized by a mesh fineness parameter n.
In this situation, the computation of the numerical solution reduces to solving a linear
system of the form Lnun=fn, where the size of the matrix Lnincreases with n. What
is often observed in practice is that Lnenjoys an asymptotic spectral distribution as
n→ ∞. More precisely, it often turns out that, for a large class of test functions F,
lim
n→∞
1
dn
dn
X
j=1
F(λj(Ln)) = 1
µ`(D)ZDPs
i=1 F(λi(f(y)))
sdy,
where dnis the size of Ln,λj(Ln), j= 1, . . . , dn, are the eigenvalues of Ln,µ`is the
Lebesgue measure in R`, and λi(f(y)), i= 1, . . . , s, are the eigenvalues of a certain
matrix-valued function
f:D⊂R`→Cs×s.
We refer to fas the spectral symbol of the sequence {Ln}n.
The spectral information carried by the symbol, which is detailed in Remark 2.2,
is not only interesting from a theoretical viewpoint, but can also be used for practical
∗Submitted to the editors June 18, 2017.
Funding: Carlo Garoni is a Marie-Curie fellow of the Italian INdAM (Istituto Nazionale di Alta
Matematica) under grant agreement PCOFUND-GA-2012-600198.
†University of Italian Switzerland, Institute of Computational Science, Lugano, Switzerland
(pietro.benedusi@usi.ch).
‡University of Italian Switzerland, Institute of Computational Science, Lugano, Switzerland, and
Insubria University, Department of Science and High Technology, Como, Italy (carlo.garoni@usi.ch,
carlo.garoni@uninsubria.it).
§University of Italian Switzerland, Institute of Computational Science, Lugano, Switzerland
(rolf.krause@usi.ch).
¶University of Italian Switzerland, Institute of Computational Science, Lugano, Switzerland
(xiaozhou.li@usi.ch).
kInsubria University, Department of Science and High Technology, Como, Italy, and Uppsala
University, Department of Information Technology, Division of Scientific Computing, Uppsala,
Sweden (stefano.serrac@uninsubria.it,stefano.serra@it.uu.se).
1
2P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
purposes. For example, it is known that the convergence properties of mainstream
iterative solvers, such as multigrid and preconditioned Krylov methods, strongly de-
pend on the spectral features of the matrices to which they are applied. The symbol
fcan then be exploited to design efficient solvers of this kind for the matrix Ln,
and to analyze/predict their performance. In this regard, we recall that noteworthy
estimates on the superlinear convergence of the conjugate gradient method obtained
by Beckermann and Kuijlaars in [1] are closely related to the asymptotic spectral
distribution of the considered matrices. Furthermore, in the context of Galerkin and
collocation isogeometric analysis (IgA) discretizations of elliptic boundary value prob-
lems, the symbol computed in a sequel of recent papers [7,10,11,12,13] was exploited
in [5,6,8] to devise and analyze optimal and robust multigrid solvers for IgA linear
systems.
In the present paper, we focus on the heat equation (3.1) defined over a rectan-
gular space-time domain in Rd+1, with d≥1 being an arbitrary positive integer. We
consider for this equation a discontinuous Galerkin (DG) discretization in time and
a finite element (FE) discretization of arbitrary regularity in space, as described in
section 3. It is worth recalling that DG methods for the time integration of (ordi-
nary) differential equations were proposed by Lesaint and Raviart [23] and applied to
parabolic equations by Jamet [22]. They enjoy several appealing properties, such as
the unconditional stability and the very high order of convergence [22,23]. Moreover,
when performing the time integration by a Galerkin method such as the DG method,
the error does not grow significantly over time, so that a long-time integration is
possible [9].
After proving in section 4some key properties of the space discretization matrices
arising from our DG/FE technique, in section 5we determine the spectral symbol for
the (normalized) space-time discretization matrices as a function of all the relevant
parameters of the considered DG/FE approximation. Our main results are Theo-
rems 5.1 and 5.2; note that in Theorem 5.2 we actually consider an even more general
version of the standard heat equation (3.1), namely the heat equation with variable
diffusion coefficients (5.8). Numerical experiments in support of the theoretical analy-
sis are provided in section 6. We draw conclusions in section 7, where we also outline
future lines of research. The study of this paper is motivated by our intention to
exploit the spectral analysis carried out herein to design/analyze appropriate solvers
for linear systems arising from the DG/FE discretization of both the heat equation
(3.1) and its more general version (5.8) involving variable diffusion coefficients.
2. Preliminaries.
2.1. Multi-index notation. Throughout this paper, we will systematically use
the multi-index notation. A multi-index m∈Zd, also called a d-index, is simply a
(row) vector in Zd; its components are denoted by m1, . . . , md. We denote by 0,1,2,
etc., the vectors consisting of all zeros, all ones, all twos, etc. (their size will be
clear from the context). For any d-index m, we set P(m) = Qd
i=1 miand we write
m→ ∞ to indicate that min(m)→ ∞. Inequalities between multi-indices must be
interpreted in the componentwise sense. For example, j≤kmeans that ji≤kifor
every i. If j,kare d-indices such that j≤k, the multi-index range j,...,kis the set
{i∈Zd:j≤i≤k}. We assume for this set the standard lexicographic ordering:
(2.1) . . . h[ (i1, . . . , id) ]id=jd,...,kdiid−1=jd−1,...,kd−1
. . . i1=j1,...,k1
.
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 3
For instance, in the case d= 2 this ordering is
(j1, j2),(j1, j2+ 1), . . . , (j1, k2),(j1+ 1, j2),(j1+ 1, j2+ 1), . . . , (j1+ 1, k2),
. . . , . . . , . . . , (k1, j2),(k1, j2+ 1), . . . , (k1, k2).
When a d-index ivaries in a multi-index range j,...,k(this is often written as
i=j,...,k), it is always assumed that ivaries from jto kfollowing the specific
ordering (2.1). In particular, if m∈Ndand x= [xi]m
i=1then xis a vector of length
P(m) whose components xi,i=1,...,m, are ordered in accordance with (2.1): the
first component is x1=x(1,...,1,1), the second component is x(1,...,1,2), and so on until
the last component, which is xm=x(m1,...,md). Similarly, if X= [xij]m
i,j=1then Xis
aP(m)×P(m) matrix whose entries are indexed by two d-indices i,j, both varying
from 1to maccording to the lexicographic ordering (2.1). The symbol Pk
i=jdenotes
the summation over all multi-indices i=j,...,k. Operations involving multi-indices
that do not have a meaning when considering multi-indices like usual vectors must
always be interpreted in the componentwise sense. For example, jk = (j1k1, . . . , jdkd),
j/k= (j1/k1, . . . , jd/kd), etc.
2.2. Matrix norms. For all X∈Cm×mthe eigenvalues and singular values
of Xare denoted by λj(X), j = 1, . . . , m, and σj(X), j = 1, . . . , m, respectively.
The conjugate transpose of Xis denoted by X∗. The identity matrix and the zero
matrix of order mare denoted by Imand Om, respectively. The ∞-norm and the
2-norm (spectral norm) of both vectors and matrices are denoted by k · k∞and k · k,
respectively. We recall that
(2.2) kXk ≤ qkXk∞kXTk∞,∀X∈Cm×m;
see, e.g., [20, section 2.3]. For X∈Cm×m, let kXk1be the trace-norm (or Schatten
1-norm) of X, i.e., the sum of all the singular values of X; see [2]. Since rank(X) is
the number of nonzero singular values of Xand kXkis the maximal singular value
of X, we have
(2.3) kXk1≤rank(X)kXk ≤ mkXk,∀X∈Cm×m.
2.3. Tensor products. If X, Y are matrices of any dimension, say X∈Cm1×m2
and Y∈C`1×`2, the tensor (Kronecker) product of Xand Yis the m1`1×m2`2matrix
defined by
X⊗Y=xij Yi=1,...,m1
j=1,...,m2
=
x11Y· ·· x1m2Y
.
.
..
.
.
xm11Y··· xm1m2Y
.
Tensor products possess a lot of nice algebraic properties. One of them is the associa-
tivity, which allows one to omit parentheses in expressions like X1⊗X2⊗ ·· · ⊗ Xd.
Another property is the bilinearity: for each matrix X, the application Y7→ X⊗Yis
linear on C`1×`2for all `1, `2∈N; and for each matrix Y, the application X7→ X⊗Y
is linear on Cm1×m2for all m1, m2∈N. If X1, X2can be multiplied and Y1, Y2can
be multiplied, then
(2.4) (X1⊗Y1)(X2⊗Y2)=(X1X2)⊗(Y1Y2).
4P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
For all matrices X, Y , we have (X⊗Y)∗=X∗⊗Y∗and (X⊗Y)T=XT⊗YT.
In particular, if X, Y are Hermitian (resp., symmetric) then X⊗Yis also Hermitian
(resp., symmetric). If X∈Cm×mand Y∈C`×`, the eigenvalues and singular values of
X⊗Yare, respectively, {λi(X)λj(Y) : i= 1, . . . , m, j = 1, . . . , `}and {σi(X)σj(Y) :
i= 1, . . . , m, j = 1, . . . , `}; see, e.g., [14, Exercise 2.5]. In particular, for all X∈
Cm×mand Y∈C`×`, we have
(2.5) kX⊗Yk=kXkkYk.
If X`∈Cm`×m`for `= 1, . . . , d, then
(2.6) (X1⊗X2⊗ · ·· ⊗ Xd)ij = (X1)i1j1(X2)i2j2···(Xd)idjd,i,j=1,...,m,
where m= (m1, m2, . . . , md). For every m= (m1, m2)∈N2there exists a permuta-
tion matrix Πmof size m1m2such that
(2.7) X2⊗X1= Πm(X1⊗X2)ΠT
m
for all matrices X1∈Cm1×m1and X2∈Cm2×m2; see, e.g., [17, Lemma 1].
2.4. Spectral distribution and spectral symbol. We say that a matrix-
valued function f:D→Cs×s, defined on a measurable set D⊆R`, is measurable
(resp., is continuous, belongs to Lp(D)) if its components fij :D→C, i, j = 1, . . . , s,
are measurable (resp., are continuous, belong to Lp(D)). Moreover, we say that fis
Hermitian (resp., symmetric) if f(y) is Hermitian (resp., symmetric) for all y∈D.
We denote by µ`the Lebesgue measure in R`and by Cc(R) (resp., Cc(C)) the set of
continuous complex-valued functions with compact support defined over R(resp., C).
Definition 2.1. Let {Xn}nbe a sequence of matrices, with Xnof size dntending
to infinity, and let f:D→Cs×sbe a measurable matrix-valued function defined on
a set D⊂R`with 0< µ`(D)<∞. We say that {Xn}nhas an asymptotic spectral
distribution described by f, and we write {Xn}n∼λf, if
lim
n→∞
1
dn
dn
X
j=1
F(λj(Xn)) = 1
µ`(D)ZDPs
i=1 F(λi(f(y)))
sdy,∀F∈Cc(C).
In this case, fis referred to as the spectral symbol of the sequence {Xn}n.
Whenever we write a spectral distribution relation such as {Xn}n∼λf, it is under-
stood that {Xn}nand fare as in Definition 2.1.
Remark 2.2. The informal meaning behind Definition 2.1 is the following: as-
suming that fpossesses sRiemann-integrable eigenvalue functions λi(f(y)),i=
1, . . . , s, the eigenvalues of Xn, except possibly for o(dn)outliers, can be subdivided
into sdifferent subsets of approximately the same cardinality; and the eigenvalues
belonging to the ith subset are approximately equal to the samples of the ith eigen-
value function λi(f(y)) over a uniform grid in the domain D. For instance, if `= 1,
dn=ns, and D= [a, b], then, assuming we have no outliers, the eigenvalues of Xn
are approximately equal to
λifa+jb−a
n, j = 1, . . . , n, i = 1, . . . , s,
for nlarge enough; similarly, if `= 2,dn=n2s, and D= [a1, b1]×[a2, b2], then,
assuming we have no outliers, the eigenvalues of Xnare approximately equal to
λifa1+j1
b1−a1
n, a2+j2
b2−a2
n, j1, j2= 1, . . . , n, i = 1, . . . , s,
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 5
for nlarge enough; and so on for `≥3.
Remark 2.3. Let D= [a1, b1]× ·· · × [a`, b`]⊂R`and let f:D→Cs×sbe a
measurable function possessing sreal-valued Riemann-integrable eigenvalue functions
λi(f(y)),i= 1, . . . , s. Compute for each r∈Nthe uniform samples
λifa1+j1
b1−a1
r, . . . , a`+j`
b`−a`
r, j1, . . . , j`= 1, . . . , r, i = 1, . . . , s,
sort them in non-decreasing order and put them in a vector (ς1, ς2, . . . , ςsr`). Let
κr: [0,1] →Rbe the piecewise linear non-decreasing function that interpolates the
samples (ς0=ς1, ς1, ς2, . . . , ςsr`)over the nodes (0,1
sr`,2
sr`,...,1), i.e.,
κri
sr`=ςi, i = 0, . . . , sr`,
κrlinear on i
sr`,i+ 1
sr`for i= 0, . . . , sr`−1.
Suppose κrconverges in measure over [0,1] to some function κas r→ ∞ (this is
always the case in real-world applications). Then,
(2.8) Z1
0
F(κ(y))dy=1
µ`(D)ZDPs
i=1 F(λi(f(y)))
sdy,∀F∈Cc(C).
This result can be proved by adapting the argument used in [14,solution of Exer-
cise 3.1]. The function κis referred to as the canonical rearranged version of f. What
is interesting about κis that, by (2.8), if {Xn}n∼λfthen {Xn}n∼λκ, i.e., if fis a
spectral symbol of {Xn}nthen κis a spectral symbol of {Xn}nas well. Moreover, κ
is a univariate scalar function and hence it is much easier to handle than f.
Two very useful tools for determining spectral distributions are the following; see
[16, Theorem 3.3] for the first one and [24, Theorem 4.3] for the second one.
Theorem 2.4. Let {Xn}n,{Yn}nbe sequences of matrices, with Xn, Yn∈Cdn×dn
and dntending to infinity as n→ ∞, and assume the following.
1. Every Xnis Hermitian and {Xn}n∼λf.
2. kXnk,kYnk ≤ Cfor all n, with Ca constant independent of n.
3. kYnk1=o(dn)as n→ ∞.
Then {Xn+Yn}n∼λf.
Theorem 2.5. Let {Xn}nbe a sequence of Hermitian matrices, with Xn∈Cdn×dn
and dntending to infinity as n→ ∞, and let {Pn}nbe a sequence of matrices, with
Pn∈Cdn×δnsuch that P∗
nPn=Iδnand δn≤dnsuch that δn/dn→1as n→ ∞.
Then,
{Xn}n∼λf⇐⇒ {P∗
nXnPn}n∼λf.
Another result of interest herein is stated and proved in the next lemma. Through-
out this paper, for any s∈Ndand any functions fi:Di→Csi×si,i= 1, . . . , d, the
tensor-product function f1⊗ · ·· ⊗ fd:D1× · · · × Dd→CP(s)×P(s)is defined as
(f1⊗ · ·· ⊗ fd)(ζ1, . . . , ζd) = f1(ζ1)⊗ · ·· ⊗ fd(ζd),(ζ1, . . . , ζd)∈D1× · · · × Dd.
Lemma 2.6. Let {Xn}n,{Yn}nbe sequences of Hermitian matrices, with Xn∈
Cdn×dn,Yn∈Cδn×δn, and both dnand δntending to infinity as n→ ∞. Assume
kXnk,kYnk ≤ Cfor all nand for some constant Cindependent of n. Let f:D⊆
6P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
R`→Cr×rand g:E⊆R`→Cs×sbe measurable Hermitian matrix-valued functions,
with 0< µ`(D)<∞and 0< µ`(E)<∞. Then,
{Xn}n∼λf,{Yn}n∼λg=⇒ {Xn⊗Yn}n∼λf⊗g.
Proof. We have to show that, for all F∈Cc(C),
lim
n→∞
1
dnδn
dn
X
i=1
δn
X
j=1
F(λi(Xn)λj(Yn))
=1
µ`(D)µ`(E)ZDZEPr
i=1 Ps
j=1 F(λi(f(x))λj(g(y)))
rs dxdy.(2.9)
Actually, since all the eigenvalues λi(Xn), λj(Yn), λi(f(x)), λj(g(y)) are real, it
suffices to prove (2.9) for all real-valued functions F∈Cc(R). Throughout the proof,
the letter Cwill denote a generic constant independent of n. Since kXnk,kYnk ≤ C,
we have
λ1(Xn), . . . , λdn(Xn), λ1(Yn), . . . , λδn(Yn)∈[−C, C],
and consequently, by [19, Theorem 4.2], we also have
λ1(f(x)), . . . , λr(f(x)), λ1(g(y)), . . . , λs(g(y)) ∈[−C, C] almost everywhere.
We start with proving (2.9) in the case where F(y) = yNis a monomial over [−C, C].
In this case the proof can be done by direct computation, due to the separability
property F(xy) = F(x)F(y) and the hypotheses {Xn}n∼λfand {Yn}n∼λg:
lim
n→∞
1
dnδn
dn
X
i=1
δn
X
j=1
F(λi(Xn)λj(Yn)) = lim
n→∞
1
dn
dn
X
i=1
F(λi(Xn)) 1
δn
δn
X
j=1
F(λj(Yn))
=1
µ`(D)ZDPr
i=1 F(λi(f(x)))
rdx1
µ`(E)ZEPs
j=1 F(λj(g(y)))
sdy
=1
µ`(D)µ`(E)ZDZEPr
i=1 Ps
j=1 F(λi(f(x))λj(g(y)))
rs dxdy.
By linearity, (2.9) holds for all functions Fsuch that F(y) is a polynomial over
[−C, C]. Thus, (2.9) holds for all real-valued F∈Cc(R) because, by the Weierstrass
approximation theorem, for any such Fand any ε > 0 we can find a polynomial pε
such that kF−pεk∞≤ε.
2.5. Multilevel block Toeplitz matrices. Given m∈Nd, a matrix of the
form
(2.10) [Ai−j]m
i,j=1∈CP(m)s×P(m)s,
with blocks Ak∈Cs×s,k=−(m−1),...,m−1,is called a multilevel block
Toeplitz matrix, or, more precisely, a d-level block Toeplitz matrix. Given a function
f: [−π, π]d→Cs×sin L1([−π , π]d), we denote its Fourier coefficients by
(2.11) fk=1
(2π)dZ[−π,π]d
f(θ)e−ik·θdθ∈Cs×s,k∈Zd,
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 7
where the integrals are computed componentwise and k·θ=k1θ1+. . . +kdθd. For
every m∈Nd, the mth Toeplitz matrix associated with fis defined as
(2.12) Tm(f)=[fi−j]m
i,j=1.
We call {Tm(f)}m∈Ndthe family of (multilevel block) Toeplitz matrices associated
with f, which in turn is called the generating function of {Tm(f)}m∈Nd.
Let L1([−π, π]d,Cs×s) be the space consisting of all matrix-valued functions f:
[−π, π]d→Cs×sbelonging to L1([−π, π ]d). For every s≥1 and m∈Nd, the map
Tm(·) : L1([−π, π]d,Cs×s)→CP(m)s×P(m)sis linear, i.e.,
Tm(αf+βg) = αTm(f) + βTm(g)
for all α, β ∈Cand f,g∈L1([−π, π]d,Cs×s). Moreover, if f∈L1([−π, π]d,Cs×s) is
a Hermitian matrix-valued function, then all the matrices Tm(f) are Hermitian.
Theorem 2.7 is a fundamental result concerning multilevel block Toeplitz matrices
generated by Hermitian matrix-valued functions; its proof can be found in [31].
Theorem 2.7. Let f: [−π, π]d→Cs×sbe a Hermitian matrix-valued function in
L1([−π, π]d). Then {Tm(f)}n∼λffor all sequences of multi-indices {m=m(n)}n
such that m→ ∞ as n→ ∞.
Besides Theorem 2.7, we shall need the following lemma [17, Lemma 4]. Any
matrix-valued function of the form p(θ) = PM
r=−MPreirθ , with Pr∈Cs×sfor all r,
is referred to as a matrix-valued trigonometric polynomial.
Lemma 2.8. For every m,s∈Ndthere exists a permutation matrix Γm,sof size
P(ms)such that
Tm1(p1)⊗ · ·· ⊗ Tmd(pd)=Γm,s(Tm(p1⊗ · ·· ⊗ pd))ΓT
m,s
for all matrix-valued trigonometric polynomials pj: [−π, π]→Csj×sj,j= 1, . . . , d.
3. Problem setting and discretization. Consider the heat equation
(3.1)
∂tu(t, x)−∆u(t, x) = f(t, x),(t, x)∈(0,T) ×(0,1)d,
u(t, x) = 0,(t, x)∈(0,T) ×∂((0,1)d),
u(0,x)=0,x∈(0,1)d.
We are imposing homogeneous Dirichlet initial/boundary conditions both for sim-
plicity and because the case of inhomogeneous Dirichlet initial/boundary conditions
reduces to the homogeneous case by considering a lifting of the boundary data; see [25]
for more on this subject. We stress that the spatial domain (0,1)dmay be replaced
by any other rectangular domain in Rdwithout affecting the essence of this paper.
To approximate the solution u(t, x) of the differential problem (3.1), we use a q-
degree DG discretization in time and a p-degree CkFE discretization in space, with
0≤ki≤pi−1 for all i= 1, . . . , d. For the sake of completeness, this numerical
technique is described here in some detail. For more on DG methods we refer the
reader to [4,21,22,23,26].
3.1. Weak form. Consider a partition in time 0 = t0< t1<··· < tN= T
and define the mth space-time slab Em= [tm, tm+1]×[0,1]dfor m= 0, . . . , N −1.
Assuming the solution u(t, x) is sufficiently regular over [0,T] ×[0,1]d, we multiply
the PDE in (3.1) by a sufficiently regular test function v(t, x) satisfying the same
8P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
boundary conditions as u(t, x), i.e., v(t, x) = 0 for (t, x)∈(0,T) ×∂((0,1)d), and we
integrate over Em:
ZEm∂tu(t, x)−∆u(t, x)v(t, x)dtdx=ZEm
f(t, x)v(t, x)dtdx
⇐⇒ Z[0,1]d
dxZtm+1
tm
∂tu(t, x)v(t, x)dt−Ztm+1
tm
dtZ[0,1]d
∆u(t, x)v(t, x)dx
=ZEm
f(t, x)v(t, x)dtdx
⇐⇒ Z[0,1]d
dxu(t, x)v(t, x)tm+1
tm−Ztm+1
tm
u(t, x)∂tv(t, x)dt
−Ztm+1
tm
dtZ∂([0,1]d)
v(t, x)∇u(t, x)·n(x)dσ(x)
| {z }
=0
−Z[0,1]d∇u(t, x)·∇v(t, x)dx
=ZEm
f(t, x)v(t, x)dtdx.
This means that, for every m= 0, . . . , N −1 and every sufficiently regular test function
v(t, x) satisfying v(t, x) = 0 for (t, x)∈(0,T) ×∂((0,1)d), the solution u(t, x) satisfies
(3.2) am(u, v) = Fm(v),
where
am(u, v) = −ZEm
u(t, x)∂tv(t, x)dtdx+ZEm∇u(t, x)· ∇v(t, x)dtdx
+Z[0,1]du(t−
m+1,x)v(t−
m+1,x)−u(t+
m,x)v(t+
m,x)dx,(3.3)
Fm(v) = ZEm
f(t, x)v(t, x)dtdx.(3.4)
Here, the symbols w(τ−,x) and w(τ+,x) stand for the limits limt→τ−w(t, x) and
limt→τ+w(t, x), respectively. For m= 0 it is assumed that u(t+
0,x) = u(t0,x) = 0
according to the initial condition in (3.1).
3.2. Space-time discretization. Let N∈Nand n∈Nd, and consider uniform
partitions in time and space:
ti=i∆t, i = 0, . . . , N, ∆t= T/N,
xi=i∆x= (i1∆x1, . . . , id∆xd),i=0,...,n,∆x= (∆x1,...,∆xd)
= (1/n1,...,1/nd) = 1/n.
Define the q-degree DG approximation space and the p-degree CkFE approximation
space as follows:
WN,[q]=w:w|[tm,tm+1 ]∈Pqfor all m= 0, . . . , N −1,
Wn,[p,k]=Wn1,[p1,k1]⊗ · ·· ⊗ Wnd,[pd,kd]
= span(w1⊗ · ·· ⊗ wd:wi∈Wni,[pi,ki]for all i= 1, . . . , d),
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 9
where Pqis the space of polynomials of degree less than or equal to qand, for all
p, n ∈Nand 0 ≤k≤p−1, the space Wn,[p,k]is defined as
Wn,[p,k]=w∈Ck([0,1]) :
w|[i
n,i+1
n]∈Ppfor all i= 0, . . . , n −1, w(0) = w(1) = 0.(3.5)
Note that the generic element w∈WN,[q]is not a function from [0,T] to Rin the true
sense of this word, because it takes two values at the points tm,m= 1, . . . , N −1.
However, for simplicity we will refer to each w∈WN,[q]as a function without further
specifications. It can be shown that
dim(Wn,[p,k]) = n(p−k) + k−1
and
N= dim(WN,[q]) = N(q+ 1),
n= dim(Wn,[p,k]) =
d
Y
i=1
dim(Wni,[pi,ki]) = P(n(p−k) + k−1).
Let {φ1, . . . , φN}be a basis for WN,[q], let {ϕ1, . . . , ϕn}be a basis for Wn,[p,k], and set
W=WN,[q]⊗Wn,[p,k]= span(ψj=φj1⊗ϕj2:j=1,...,N),N= (N , n).
We look for an approximation uW(t, x) of the solution u(t, x) by solving the following
discrete problem: find uW∈Wsuch that, for all m= 0, . . . , N −1 and all v∈W,
(3.6) am(uW, v) = Fm(v),
where am(u, v) and Fm(v) are given by (3.3) and (3.4), respectively.
It should be noted, however, that, due to the structure of the DG approximation
space WN,[q], the solution of (3.6) for m= 1 is completely independent of the solution
of (3.6) for m= 0. Similarly, the solution of (3.6) for m= 2 is completely independent
of the solution of (3.6) for m= 1, and so on. In particular, the information provided
by the initial condition u(0,x) is present only until t=t−
1and it is lost for t>t1. To
avoid this decoupling of the various problems (3.6) corresponding to different indices
m, as well as to avoid the loss of information carried by the initial condition, we
impose that the initial condition of problem (3.6) for m= 1 is given by uW(t−
1,x),
which is obtained by solving (3.6) for m= 0. More generally, we impose that the
initial condition of problem (3.6) for m= 1, . . . , N −1 is given by uW(t−
m,x), which is
obtained by solving (3.6) for the previous index m−1. Of course, the initial condition
for m= 0 is u(0,x). In conclusion, we replace am(uW, v) in (3.6) with ˆam(uW, v),
where
ˆa0(uW, v) = a0(uW, v) = −ZE0
uW(t, x)∂tv(t, x)dtdx+ZE0∇uW(t, x)· ∇v(t, x)dtdx
+Z[0,1]duW(t−
1,x)v(t−
1,x)−u(0,x)
|{z }
=0
v(t+
0,x)dx(3.7)
and
ˆam(uW, v) = −ZEm
uW(t, x)∂tv(t, x)dtdx+ZEm∇uW(t, x)· ∇v(t, x)dtdx
+Z[0,1]duW(t−
m+1,x)v(t−
m+1,x)−uW(t−
m,x)v(t+
m,x)dx(3.8)
10 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
for m= 1, . . . , N −1. Then, we look for an approximation uW(t, x) of the solution
u(t, x) by solving the following discrete problem: find uW∈Wsuch that, for all
m= 0, . . . , N −1 and all v∈W,
(3.9) ˆam(uW, v) = Fm(v),
where ˆam(u, v) and Fm(v) are given by (3.7)–(3.8) and (3.4), respectively.
Considering that {ψj:j=1,...,N}is a basis for W, we have uW=PN
j=1ujψj
for a unique vector u= [uj]N
j=1and, by linearity, the computation of uWreduces to
finding usuch that, for all m= 0, . . . , N −1,
(3.10) Amu=Fm,
where
Fm= [Fm(ψi)]N
i=1,(3.11)
Am= [ˆam(ψj, ψi)]N
i,j=1.(3.12)
3.3. Choice of the bases in time and space. Fix a basis {`1,[q], . . . , `q+1,[q]}
for the polynomial space Pq, and let
(3.13) ˆ
φs,[q](τ) = `s,[q](τ),if τ∈[−1,1],
0,otherwise,s= 1, . . . , q + 1.
We refer to {ˆ
φ1,[q],..., ˆ
φq+1,[q]}and [−1,1] as the reference basis in time and the
reference interval in time, respectively.
The basis {φ1, . . . , φN}for WN,[q]is defined as follows:
φ(q+1)(r−1)+s(t) = ˆ
φs,[q]2t−(tr+tr−1)
tr−tr−1
=ˆ
φs,[q]2t−(2r−1)∆t
∆t, r = 1, . . . , N, s = 1, . . . , q + 1.(3.14)
Note that φ(q+1)(r−1)+sis identically zero outside [tr−1, tr]. In the context of (nodal)
DG methods, `1,[q], . . . , `q+1,[q]are often chosen as the Lagrange polynomials asso-
ciated with q+ 1 fixed points {τ1, . . . , τq+1} ⊆ [−1,1], such as, for example, the
Gauss–Lobatto or the right Gauss–Radau nodes in [−1,1]; see, e.g., [21]. Neverthe-
less, other choices are also allowed, and since the analysis of this paper is not affected
by the specific choice of the reference basis, we will not make any specific assumptions
on `1,[q], . . . , `q+1,[q].
For p, n ∈Nand 0 ≤k≤p−1, let B1,[p,k], . . . , Bn(p−k)+k+1,[p,k]be the B-splines
of degree pand smoothness Ckdefined on the knot sequence
{ξ1, . . . , ξn(p−k)+p+k+2}
=0,...,0
| {z }
p+1
,1
n,..., 1
n
| {z }
p−k
,2
n,..., 2
n
| {z }
p−k
, . . . , n−1
n,...,n−1
n
| {z }
p−k
,1,...,1
| {z }
p+1 .(3.15)
A few properties of the functions B1,[p,k], . . . , Bn(p−k)+k+1,[p,k]that we shall use in
this paper are listed below.
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 11
•Local support property: the support of the ith B-spline is given by
(3.16) supp(Bi,[p,k])=[ξi, ξi+p+1], i = 1, . . . , n(p−k) + k+ 1.
•Vanishment on the boundary: except for the first and the last one, all the other
B-splines vanish on the boundary of [0,1], i.e.,
(3.17) Bi,[p,k](0) = Bi,[p,k](1) = 0, i = 2, . . . , n(p−k) + k.
•Basis property: {B1,[p,k], . . . , Bn(p−k)+k+1,[p,k]}is a basis for the space of piecewise
polynomial functions of degree pand smoothness Ck, that is,
Vn,[p,k]=v∈Ck([0,1]) : v|[i
n,i+1
n]∈Ppfor all i= 0, . . . , n −1;
and {B2,[p,k], . . . , Bn(p−k)+k,[p,k]}is a basis for the space
Wn,[p,k]={w∈Vn,[p,k]:w(0) = w(1) = 0},
which has already been introduced in (3.5).
•Non-negativity and partition of unity:
Bi,[p,k]≥0 over R, i = 1, . . . , n(p−k) + k+ 1,(3.18)
n(p−k)+k+1
X
i=1
Bi,[p,k]= 1 over [0,1].(3.19)
•Bounds for derivatives:
(3.20)
n(p−k)+k+1
X
i=1 |B0
i,[p,k]| ≤ cpnover [0,1],
for some constant cpdepending only on p. Note that the derivatives B0
i,[p,k]may not
be defined at some of the grid points 0,1
n,...,n−1
n,1 in the case of C0smoothness
(k= 0). In (3.20) it is assumed that the undefined values are excluded from the
summation.
•All the B-splines Bi,[p,k], except for the first k+ 1 and the last k+ 1, are uni-
formly shifted-scaled versions of p−kfixed reference functions ˆϕ1,[p,k],..., ˆϕp−k,[p,k],
namely the first p−kB-splines defined on the reference knot sequence
(3.21) 0,...,0
| {z }
p−k
,1,...,1
| {z }
p−k
, . . . , η, . . . , η
| {z }
p−k
, η =p+ 1
p−k.
In formulas, setting
(3.22) ν=k+ 1
p−k,
for the B-splines Bk+2,[p,k], . . . , Bk+1+(n−ν)(p−k),[p,k]we have
Bk+1+(p−k)(r−1)+s,[p,k](x) = ˆϕs,[p,k](nx −r+ 1), r = 1, . . . , n −ν,(3.23)
s= 1, . . . , p −k.
We point out that the supports of the reference B-splines ˆϕs,[p,k ]satisfy
(3.24) supp( ˆϕ1,[p,k])⊆supp( ˆϕ2,[p,k])⊆. . . ⊆supp( ˆϕp−k,[p,k]) = [0, η].
12 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
Fig. 3.1.B-splines B1,[p,k],...,Bn(p−k)+k+1,[p,k]for p= 3 and k= 1, with n= 10.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
Fig. 3.2.Reference B-splines ˆϕ1,[p,k],ˆϕ2,[p,k]for p= 3 and k= 1.
Figs. 3.1–3.2 show the graphs of the B-splines B1,[p,k], . . . , Bn(p−k)+k+1,[p,k]for the
degree p= 3 and the smoothness k= 1, and the graphs of the associated reference
B-splines ˆϕ1,[p,k],ˆϕ2,[p,k]. For the formal definition of the B-splines, as well as for the
proof of the properties mentioned above, we refer the reader to [3,27].
The basis {ϕ1, . . . , ϕn}={ϕ1, . . . , ϕn(p−k)+k−1}for Wn,[p,k]is defined as follows:
ϕs=Bs+1,[p,k]
=Bs1+1,[p1,k1]⊗ · ·· ⊗ Bsd+1,[pd,kd],s=1,...,n(p−k) + k−1.(3.25)
3.4. Space-time matrix assembly. For every i,j=1,...,N, the (i,j) entry
of the matrix Amappearing in (3.12) is given by
(Am)ij = ˆam(ψj, ψi)
=−ZEm
ψj(t, x)∂tψi(t, x)dtdx+ZEm∇ψj(t, x)· ∇ψi(t, x)dtdx
+Z[0,1]dψj(t−
m+1,x)ψi(t−
m+1,x)−ψj(t−
m,x)ψi(t+
m,x)dx
=−Ztm+1
tm
φj1(t)φ0
i1(t)dtZ[0,1]d
ϕj2(x)ϕi2(x)dx
+Ztm+1
tm
φj1(t)φi1(t)dtZ[0,1]d∇ϕj2(x)· ∇ϕi2(x)dx
+φj1(t−
m+1)φi1(t−
m+1)−φj1(t−
m)φi1(t+
m)Z[0,1]d
ϕj2(x)ϕi2(x)dx(3.26)
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 13
for m= 1, . . . , N −1, and similarly
(A0)ij =−Zt1
t0
φj1(t)φ0
i1(t)dtZ[0,1]d
ϕj2(x)ϕi2(x)dx
+Zt1
t0
φj1(t)φi1(t)dtZ[0,1]d∇ϕj2(x)· ∇ϕi2(x)dx
+φj1(t−
1)φi1(t−
1)Z[0,1]d∇ϕj2(x)· ∇ϕi2(x)dx(3.27)
Since the values φj(t−
0) are not defined, we set for convenience φj(t−
0) = 0 for all
j= 1, . . . , N , so that formula (3.26) is true also for m= 0. For all m= 0, . . . , N −1,
define the matrices
M[m]
N,[q]=Ztm+1
tm
φj(t)φi(t)dtN
i,j=1
,(3.28)
H[m]
N,[q]=−Ztm+1
tm
φj(t)φ0
i(t)dtN
i,j=1
,(3.29)
C[m]
N,[q]=φj(t−
m+1)φi(t−
m+1)N
i,j=1 ,(3.30)
J[m]
N,[q]=φj(t−
m)φi(t+
m)N
i,j=1 ,(3.31)
Mn,[p,k]="Z[0,1]d
ϕj(x)ϕi(x)dx#n
i,j=1
="Z[0,1]d
Bj+1,[p,k](x)Bi+1,[p,k](x)dx#n(p−k)+k−1
i,j=1
,(3.32)
Kn,[p,k]="Z[0,1]d∇ϕj(x)· ∇ϕi(x)dx#n
i,j=1
="Z[0,1]d∇Bj+1,[p,k](x)· ∇Bi+1,[p,k](x)dx#n(p−k)+k−1
i,j=1
.(3.33)
From (3.26) we obtain
(Am)ij = (H[m]
N,[q])i1j1(Mn,[p,k])i2j2+ (M[m]
N,[q])i1j1(Kn,[p,k])i2j2
+(C[m]
N,[q])i1j1−(J[m]
N,[q])i1j1(Mn,[p,k])i2j2.
Hence, by (2.6),
(3.34) Am= (H[m]
N,[q]+C[m]
N,[q]−J[m]
N,[q])⊗Mn,[p,k]+M[m]
N,[q]⊗Kn,[p,k].
In view of the choice (3.14) for the basis in time, we can further simplify the expressions
of the matrices (3.28)–(3.31). Concerning the matrices (3.28)–(3.30), the components
(M[m]
N,[q])ij , (H[m]
N,[q])ij , (C[m]
N,[q])ij are nonzero at most for the pairs (i, j) such that
14 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
i= (q+ 1)m+sand j= (q+ 1)m+s0with 1 ≤s, s0≤q+ 1. Moreover, for such
pairs we have
(M[m]
N,[q])ij =Ztm+1
tm
φ(q+1)m+s0(t)φ(q+1)m+s(t)dt
=Ztm+1
tm
ˆ
φs0,[q]2t−(2m+ 1)∆t
∆tˆ
φs,[q]2t−(2m+ 1)∆t
∆tdt
=∆t
2Z1
−1
`s0,[q](τ)`s,[q](τ)dτ,
(H[m]
N,[q])ij =−Ztm+1
tm
φ(q+1)m+s0(t)φ0
(q+1)m+s(t)dt
=−Ztm+1
tm
ˆ
φs0,[q]2t−(2m+ 1)∆t
∆t2
∆tˆ
φ0
s,[q]2t−(2m+ 1)∆t
∆tdt
=−Z1
−1
`s0,[q](τ)`0
s,[q](τ)dτ,
(C[m]
N,[q])ij =φ(q+1)m+s0(t−
m+1)φ(q+1)m+s(t−
m+1) = `s0,[q](1)`s,[q](1).
Thus, if Eij is the N×Nmatrix having 1 in position (i, j) and 0 elsewhere, the
matrices M[m]
N,[q],H[m]
N,[q],C[m]
N,[q]can be written as
M[m]
N,[q]=Em+1,m+1 ⊗∆t
2M[q], M[q]=Z1
−1
`s0,[q](τ)`s,[q](τ)dτq+1
s,s0=1
,(3.35)
H[m]
N,[q]=Em+1,m+1 ⊗H[q], H[q]=−Z1
−1
`s0,[q](τ)`0
s,[q](τ)dτq+1
s,s0=1
,(3.36)
C[m]
N,[q]=Em+1,m+1 ⊗C[q], C[q]=`s0,[q](1)`s,[q](1)q+1
s,s0=1 ,(3.37)
for all m= 0, . . . , N −1. Concerning the matrix (3.31), the component (J[m]
N,[q])ij is
nonzero at most for the pairs (i, j) such that i= (q+1)m+sand j= (q+1)(m−1) +s0
with 1 ≤s, s0≤q+ 1. Moreover, for such pairs we have
(J[m]
N,[q])ij =φ(q+1)(m−1)+s0(t−
m)φ(q+1)m+s(t+
m) = `s0,[q](1)`s,[q](−1).
Thus,
J[m]
N,[q]=Em+1,m ⊗J[q], J[q]=`s0,[q](1)`s,[q](−1)q+1
s,s0=1 ,(3.38)
for all m= 0, . . . , N −1; note that E1,0is the N×Nzero matrix. In conclusion, the
matrix (3.34) can be expressed as
(3.39) Am=Em+1,m ⊗B[q,p,k]
n+Em+1,m+1 ⊗A[q,p,k]
n
for all m= 0, . . . , N −1, where
A[q,p,k]
n= (H[q]+C[q])⊗Mn,[p,k]+∆t
2M[q]⊗Kn,[p,k],(3.40)
B[q,p,k]
n=−J[q]⊗Mn,[p,k].(3.41)
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 15
Now we recall that the computation of the numerical solution is equivalent to
finding the vector usatisfying the linear system (3.10) for all m= 0, . . . , N −1. In
view of the definition (3.11) of Fmand the choice (3.14) for the basis in time, the
component (Fm)iis nonzero at most for the pairs i= (i1, i2) such that i1= (q+1)m+s
with 1 ≤s≤q+ 1. Partition the vectors uand Fmas follows:
u=
u1
u2
.
.
.
uN
,Fm=
(Fm)1
(Fm)2
.
.
.
(Fm)N
,
where each ujand (Fm)jis a vector of length n(q+ 1). Then, (Fm)j=0for all
j6=m+ 1, and in view of (3.39) the requirement that usatisfies the linear system
(3.10) for all m= 0, . . . , N −1 is equivalent to the requirement that usatisfies
A[q,p,k]
n
B[q,p,k]
nA[q,p,k]
n
......
B[q,p,k]
nA[q,p,k]
n
u1
u2
.
.
.
uN
=
f1
f2
.
.
.
fN
⇐⇒ C[q,p,k]
N,nu=f,
where f= [f1,f2,...,fN]T,fm+1 = (Fm)m+1 for all m= 0, . . . , N −1, and
(3.42) C[q,p,k]
N,n=
A[q,p,k]
n
B[q,p,k]
nA[q,p,k]
n
......
B[q,p,k]
nA[q,p,k]
n
.
4. Properties of the space matrices. We investigate here some properties
of the space matrices Mn,[p,k]and Kn,[p,k]. Using the tensor-product structure of
the B-spline basis functions B2,[p,k], . . . , Bn(p−k)+k−1,[p,k]and the rectangularity of
the domain (0,1)d, in the next lemma we show that Mn,[p,k]and Kn,[p,k]possess a
tensor-product structure.
Lemma 4.1. Let p,n∈Ndand 0≤k≤p−1. Then,
Kn,[p,k]=
d
X
s=1 s−1
O
r=1
Mnr,[pr,kr]!⊗Kns,[ps,ks]⊗ d
O
r=s+1
Mnr,[pr,kr]!,(4.1)
Mn,[p,k]=
d
O
r=1
Mnr,[pr,kr],(4.2)
where, for p, n ∈Nand 0≤k≤p−1, the matrices Mn,[p,k]and Kn,[p,k]are defined by
Kn,[p,k]=Z1
0
B0
j+1,[p,k](x)B0
i+1,[p,k](x)dxn(p−k)+k−1
i,j=1
,(4.3)
Mn,[p,k]=Z1
0
Bj+1,[p,k](x)Bi+1,[p,k](x)dxn(p−k)+k−1
i,j=1
.(4.4)
16 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
Proof. We only prove (4.1) as (4.2) is proved in the same way. For every i,j=
1,...,n(p−k) + k−1, we have
(Kn,[p,k])ij =Z[0,1]d∇Bj+1,[p,k](x)· ∇Bi+1,[p,k](x)dx
=
d
X
s=1 Z1
0
B0
js+1,[ps,ks](xs)B0
is+1,[ps,ks](xs)dxs×
×
d
Y
r=1
r6=sZ1
0
Bjr+1,[pr,kr](xr)Bir+1,[pr,kr](xr)dxr
=
d
X
s=1 s−1
O
r=1
Mnr,[pr,kr]!⊗Kns,[ps,ks]⊗ d
O
r=s+1
Mnr,[pr,kr]!!ij
= d
X
s=1 s−1
O
r=1
Mnr,[pr,kr]!⊗Kns,[ps,ks]⊗ d
O
r=s+1
Mnr,[pr,kr]!!ij
,
where the third equality holds by (2.6) and by definition of Kn,[p,k]and Mn,[p,k].
For p, n ∈Nand 0 ≤k≤p−1, let
K[`]
[p,k]=ZR
ˆϕ0
s0,[p,k](x) ˆϕ0
s,[p,k](x−`)dxp−k
s,s0=1
, ` ∈Z,(4.5)
M[`]
[p,k]=ZR
ˆϕs0,[p,k](x) ˆϕs,[p,k](x−`)dxp−k
s,s0=1
. ` ∈Z,(4.6)
Due to (3.24), the integrals over Rappearing in (4.5)–(4.6) actually reduce to integrals
over [0, η]. For the same reason, the blocks (4.5)–(4.6) corresponding to indices `6∈
{−η+ 1, . . . , η −1}reduce to the zero block:
K[`]
[p,k]=M[`]
[p,k]=Op−k,|`| ≥ η.
By direct computation one can show that M[−`]
[p,k]= (M[`]
[p,k])Tand K[−`]
[p,k]= (K[`]
[p,k])T
for all `∈Z. Define the following (p−k)×(p−k) Hermitian matrix-valued functions:
f[p,k]: [−π, π]→C(p−k)×(p−k),
f[p,k](θ) = X
`∈Z
K[`]
[p,k]ei`θ =K[0]
[p,k]+
η−1
X
`=1K[`]
[p,k]ei`θ + (K[`]
[p,k])Te−i`θ ,(4.7)
h[p,k]: [−π, π]→C(p−k)×(p−k),
h[p,k](θ) = X
`∈Z
M[`]
[p,k]ei`θ =M[0]
[p,k]+
η−1
X
`=1M[`]
[p,k]ei`θ + (M[`]
[p,k])Te−i`θ .(4.8)
Lemma 4.2. Let p, n ∈Nand 0≤k≤p−1. Let ˜
Mn,[p,k](resp., ˜
Kn,[p,k]) be the
principal submatrix of Mn,[p,k](resp., Kn,[p,k]) of size (n−ν)(p−k)corresponding to
the indices k+ 1, . . . , k + (n−ν)(p−k), where νis defined in (3.22). Then,
(4.9) ˜
Mn,[p,k]=n−1Tn−ν(h[p,k]),˜
Kn,[p,k]=n Tn−ν(f[p,k]).
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 17
Proof. We only prove the left equation in (4.9) as the proof of the right equation
is completely analogous. For convenience, we index the B-splines (3.23), as well as
the entries of the matrices ˜
Mn,[p,k]and Tn−ν(h[p,k]), by a bi-index (r, s) such that
1≤r≤n−νand 1 ≤s≤p−k. Of course, it is understood that (r, s) varies in
the bi-index range (1,1),...,(n−ν, p −k) according to the standard lexicographic
ordering (2.1). In this notation, we can rewrite (3.23) as follows:
B(r,s),[p,k](x) = Bk+1+(p−k)(r−1)+s,[p,k](x)
= ˆϕs,[p,k](nx −r+ 1),(r, s) = (1,1),...,(n−ν, p −k).(4.10)
For all (r, s),(r0, s0) = (1,1),...,(n−ν, p −k), we have
(˜
Mn,[p,k])(r,s),(r0,s0)=Z1
0
B(r0,s0),[p,k](x)B(r,s),[p,k](x)dx
=Z1
0
ˆϕs0,[p,k](nx −r0+ 1) ˆϕs,[p,k](nx −r+ 1)dx
=n−1Zn−r0+1
−r0+1
ˆϕs0,[p,k](y) ˆϕs,[p,k](y−r+r0)dy
and
(Tn−ν(h[p,k]))(r,s),(r0,s0)= (M[r−r0]
[p,k])s,s0=ZR
ˆϕs0,[p,k](y) ˆϕs,[p,k](y−r+r0)dy.
Since, by (3.24), supp( ˆϕs0,[p,k ])⊆[0, η]⊆[−r0+ 1, n −r0+ 1] for all r0= 1, . . . , n −ν
and s0= 1, . . . , p −k, we have
Zn−r0+1
−r0+1
ˆϕs0,[p,k](y) ˆϕs,[p,k](y−r+r0)dy=ZR
ˆϕs0,[p,k](y) ˆϕs,[p,k](y−r+r0)dy
for all (r, s),(r0, s0) = (1,1),...,(n−ν, p −k). Hence,
(˜
Mn,[p,k])(r,s),(r0,s0)= (n−1Tn−ν(h[p,k]))(r,s),(r0,s0)
for all (r, s),(r0, s0) = (1,1),...,(n−ν, p −k), i.e., ˜
Mn,[p,k]=n−1Tn−ν(h[p,k]).
Lemma 4.3. For all p, n ∈Nand 0≤k≤p−1we have
knMn,[p,k]k∞,kn−1Kn,[p,k]k∞≤Cp
for some constant Cpdepending only on p.
Proof. By the local support property, non-negativity and partition of unity prop-
erty of B-splines (see (3.16), (3.18) and (3.19)), we have
knMn,[p,k]k∞= max
i=1,...,n(p−k)+k−1
n(p−k)+k−1
X
j=1 |(nMn,[p,k])ij |
= max
i=1,...,n(p−k)+k−1
n(p−k)+k−1
X
j=1
nZ1
0
Bj+1,[p,k](x)Bi+1,[p,k](x)dx
≤nmax
i=1,...,n(p−k)+k−1Z1
0
Bi+1,[p,k](x)dx
18 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
=nmax
i=1,...,n(p−k)+k−1Z[ξi+1,ξi+p+2 ]
Bi+1,[p,k](x)dx
≤nmax
i=1,...,n(p−k)+k−1(ξi+p+2 −ξi+1)≤p+ 1.
By the local support property and the bounds for the derivatives of B-splines (see
(3.16) and (3.20)), we have
kn−1Kn,[p,k]k∞= max
i=1,...,n(p−k)+k−1
n(p−k)+k−1
X
j=1 |(n−1Kn,[p,k])ij |
≤max
i=1,...,n(p−k)+k−1
n(p−k)+k−1
X
j=1
n−1Z1
0|B0
j+1,[p,k](x)||B0
i+1,[p,k](x)|dx
≤cpmax
i=1,...,n(p−k)+k−1Z1
0|B0
i+1,[p,k](x)|dx
=cpmax
i=1,...,n(p−k)+k−1Z[ξi+1,ξi+p+2 ]|B0
i+1,[p,k](x)|dx
≤c2
pnmax
i=1,...,n(p−k)+k−1(ξi+p+2 −ξi+1)≤c2
p(p+ 1).
5. Spectral symbol for the sequence of normalized space-time matrices.
This section is devoted to the proof of the first main result of this paper, which
gives the spectral distribution and the spectral symbol of the normalized space-time
matrices C[q,p,k]
N,nunder suitable assumptions on the discretization parameters Nand
n. Drawing inspiration from the multi-index notation, for any vector α= (α1, . . . , αd)
we set P(α) = Qd
i=1 αi.
Theorem 5.1. Let q∈N,p∈Ndand 0≤k≤p−1. Suppose that the following
conditions are met:
•n=αn, where α= (α1, . . . , αd)is a vector with positive components in Qdand n
varies in some infinite subset of Nsuch that n=αn∈Nd;
•N=N(n)is such that N→ ∞ and N/n2→0as n→ ∞.
Then, for the sequence of normalized space-time matrices {2Nnd−2C[q,p,k]
N,n}nwe have
{2Nnd−2C[q,p,k]
N,n}n∼λf[α]
[q,p,k],
where:
•f[α]
[q,p,k]is defined as
f[α]
[q,p,k]: [−π, π ]d→C(q+1)P(p−k)×(q+1)P(p−k),
f[α]
[q,p,k](θ) = f[α]
[p,k](θ)⊗TM[q];(5.1)
•f[α]
[p,k]is defined as
f[α]
[p,k]: [−π, π]d→CP(p−k)×P(p−k),
f[α]
[p,k](θ) = 1
P(α)
d
X
s=1
α2
s s−1
O
r=1
h[pr,kr](θr)!⊗f[ps,ks](θs)⊗ d
O
r=s+1
h[pr,kr](θr)
!;(5.2)
DG DISCRETIZATION OF THE HEAT EQUATION: THE SYMBOL 19
•f[p,k]and h[p,k]are given by (4.7)and (4.8)for all p∈Nand 0≤k≤p−1;
•Tis the final time in (3.1)and M[q]is given in (3.35).
Proof. The proof consists of two steps. The letter Cwill be used to denote a
generic constant independent of n.
Step 1.By (3.40)–(3.42),
C[q,p,k]
N,n=TN(eiθ)⊗B[q,p,k]
n+IN⊗A[q,p,k]
n
=−TN(eiθ)⊗J[q]⊗Mn,[p,k]+IN⊗(H[q]+C[q])⊗Mn,[p,k]
+∆t
2IN⊗M[q]⊗Kn,[p,k].
Consider the decomposition of 2N nd−2C[q,p,k]
N,ngiven by
2Nnd−2C[q,p,k]
N,n=X[q,p,k]
N,n+Y[q,p,k]
N,n,
where
X[q,p,k]
N,n= 2N nd−2∆t
2IN⊗M[q]⊗Kn,[p,k]= Tnd−2IN⊗M[q]⊗Kn,[p,k],
Y[q,p,k]
N,n=−2N nd−2TN(eiθ)⊗J[q]⊗Mn,[p,k]+ 2N nd−2IN⊗(H[q]+C[q])⊗Mn,[p,k].
By Lemma 4.1 and the equation n=αn, we have
nd−2Kn,[p,k]=1
P(α)
d
X
s=1 s−1
O
r=1
nrMnr,[pr,kr]!⊗α2
sn−1
sKns,[ps,ks]
(5.3)
⊗ d
O
r=s+1
nrMnr,[pr,kr]!,
nd−2Mn,[p,k]=n−2
P(α)
d
O
r=1
nrMnr,[pr,kr].(5.4)
By property (2.5), Lemma 4.3, and the equation kTN(eiθ)k= 1, we have
kX[q,p,k]
N,nk ≤ C, kY[q,p,k]
N,nk ≤ CN/n2.
Since N/n2→0 by assumption, we have kY[q,p,k]
N,nk → 0 as n→ ∞. Hence, by (2.3),
kY[q,p,k]
N,nk1≤ kY[q,p,k]
N,nknN =o(nN ).
Since X[q,p,k]
N,nis symmetric, by Theorem 2.4 the thesis is proved if we show that
(5.5) {X[q,p,k]
N,n}n∼λf[α]
[q,p,k].
By (2.7), there exists a permutation matrix Πn,N , depending only on nand N, such
that
X[q,p,k]
N,n= Tnd−2IN⊗M[q]⊗Kn,[p,k]=TN(TM[q])⊗nd−2Kn,[p,k]
= Πn,N nd−2Kn,[p,k]⊗TN(TM[q])ΠT
n,N .
20 P. BENEDUSI, C. GARONI, R. KRAUSE, X. LI, S. SERRA-CAPIZZANO
Proving (5.5) is then equivalent to proving that
(5.6) {nd−2Kn,[p,k]⊗TN(TM[q])}n∼λf[α]
[q,p,k]=f[α]
[p,k]⊗TM[q].
By Lemma 2.6 and Theorem 2.7, the spectral distribution (5.6) is established as soon
as we have proved that
(5.7) {nd−2Kn,[p,k]}n∼λf[α]
[p,k].
The next step is devoted to the proof of (5.7).
Step 2.For p, n ∈Nand 0 ≤k≤p−1, let νbe defined as in (3.22) and let
Pn,[p,k]∈C(n(p−k)+k−1)×(n−ν)(p−k)be the matrix having I(n−ν)(p−k)as the submatrix
corresponding to the row and column indices i, j =k+ 1, . . . , k + (n−ν)(p−k) and
zeros elsewhere. Let
Pn,[p,k]=Pn1,[p1,k1]⊗ · ·· ⊗ Pnd,[pd,kd].
Noting that PT
n,[p,k]Pn,[p,k]=I(n−ν)(p−k), by (2.4) we have
PT
n,[p,k]Pn,[p,k]=I(n1−ν1)(p1−k1)⊗ · ·· ⊗ I(nd−νd)(pd−kd)=IP((n−ν)(p−k)).
By (5.3), Lemma 4.2, and (2.4),
PT
n,[p,k](nd−2Kn,[p,k])Pn,[p,k]
=1
P(α)
d
X
s=1
α2
s s−1
O
r=1
Tnr−νr(h[pr,kr])!⊗Tns−νs(f