Content uploaded by Carlo Garoni

Author content

All content in this area was uploaded by Carlo Garoni on Jan 16, 2018

Content may be subject to copyright.

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