DISCONTINUOUS GALERKIN TIME-STEPPING METHOD
IAIN SMEARS
Abstract. The discontinuous Galerkin time-stepping method has many ad-vantageous properties for solving parabolic equations. However, its practical use has been limited by the large and challenging nonsymmetric systems to be solved at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. We first construct a left preconditioner, based on inf-sup theory, that transforms the linear system to a symmetric positive definite problem that can be solved by the preconditioned conjugate gradient (PCG) algorithm. We then prove that the transformed sys-tem can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number κ bounded by 4 for any time-step size, any ap-proximation order and any positive self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.
1. Introduction
The discontinuous Galerkin (DG) time-stepping method is a single-step implicit scheme for the solution of dissipative evolution equations [10, 19], which generalizes the backward Euler method to higher-order approximations. This method features many advantages that make it an attractive choice for solving parabolic problems. • It permits arbitrarily high-order approximation to the solution, along with superconvergence at the time-step nodes [1, 3, 12], whilst remaining un-conditionally stable. Unlike linear multistep schemes, it is thus not con-strained by the Dahlquist barrier theorem and it does not require an aux-iliary scheme to compute the first few solution values.
• It also allows fully variable time-step sizes, approximation orders, and even spatial mesh refinement/coarsening between time-steps. It is well suited for adaptivity driven by a posteriori error analysis [2, 5, 6, 11, 17].
• It permits the temporal version of hp-refinement, where one varies both the time-step size as well as the approximation order on each time-step. This leads to exponential convergence rates for a broad class of solutions with singularities induced by the initial datum or by the source term [16].
2010 Mathematics Subject Classification. Primary 65F08; Secondary 65M22 65M60.
Key words and phrases. discontinuous Galerkin, time discretizations, parabolic PDE, precon-ditioning, conjugate gradient algorithm.
The author wishes to express his thanks to Paul Houston, Lorenz John, Christian Kreuzer, Endre S¨uli and Martin Vohral´ık for many helpful discussions. The author was supported by an EPSRC Doctoral Prize at the Mathematical Institute, University of Oxford, during the period in which this work was completed.
Despite the excellent theoretical qualities of the DG time-stepping method, it appears that its widespread use has been held back primarily by the significant challenge of solving the linear system encountered at each time-step. To understand the source of these difficulties, consider the DG time-stepping method applied to a semi-discrete evolution problem of the form
(1.1) M U0(t) + A U (t) = g in (0, T ),
where the solution U : [0, T ] → V, with V a finite dimensional space, where M and A are symmetric positive definite matrices. Such semi-discrete problems frequently arise from the application of standard spatial discretization methods (conforming or nonconforming finite elements, finite differences, etc) to uniformly parabolic self-adjoint partial differential equations (PDE).
In this context, at each time-step, the DG time-stepping method requires the solution of a linear system of the general block form
(1.2) b00M + c00τ A · · · b0pM + c0pτ A .. . . .. ... bp0M + cp0τ A · · · bppM + cppτ A u0 .. . up = f0 .. . fp ,
where τ denotes the time-step size, the polynomial degree p defines the approxi-mation order of the scheme, the solution coefficients uk ∈ V for k = 0, . . . , p, and,
after mapping the time-step interval to the reference interval (−1, 1), we have (1.3) bjk:= Z 1 −1 φ0kφjds + φk(−1)φj(−1), cjk := 1 2 Z 1 −1 φkφjds,
where {φk}pk=0is a chosen basis of Ppthe space of real-valued polynomials of degree
at most p. For the case p = 0, the DG method reduces to the backward Euler method and the system is therefore symmetric, up to the treatment of the source term. However, for p ≥ 1, the system matrix B of (1.2) is nonsymmetric, and its size is proportional to both the dimension of V and p. Even for moderate sizes of A and M , standard direct solution algorithms can be prohibitively expensive.
Although the theory of the DG time-stepping method has been extensively stud-ied, comparatively few works have sought to address the challenges of its practical solution. Sch¨otzau and Schwab propose in [16] an algorithm based on the Schur decomposition theorem: there exists a complex unitary transformation such that (cjk) is diagonalized and such that (bjk) is transformed to upper triangular form.
The solution is then obtained by solving p + 1 successive complex non-Hermitian backward Euler-type linear systems. In order to avoid complex arithmetic, Richter et al. propose in [15] a linear iterative fixed point scheme based on an approximation of the block LU factorization of B. Assuming exact spatial solvers, they analyse the contraction rates of their method for p ≤ 3.
Both of the above methods lead to a sequence of spatial problems. In practice, for large M and A, each of these is solved iteratively, for instance by preconditioned Krylov subspace methods: see e.g. the experiments in [20]. An alternative approach is to apply directly a preconditioned Krylov subspace method to (1.2). We are aware of only a couple of works in this direction, for either DG time-stepping methods or closely related implicit Runge–Kutta (RK) schemes.
In [13], Mardal et al. propose a block-diagonal preconditioner for RK schemes, with blocks of the form M + akkτ A, to be used with GMRES. They show that the
show that it is not robust with respect to p. In [22], Basting and Weller develop a preconditioner specifically for the DG time-stepping method for p = 1 (as well as for the related continuous Galerkin method of same size) based on an approximate Schur complement preconditioner for one of the unknown coefficients uj in (1.2).
1.1. Main contributions. In this work, we propose a robust and efficient pre-conditioning strategy for the DG time-stepping method. Our approach can be summarized as follows.
Applicability of PCG. In section 3.2, we construct and apply a left-preconditioner P>to the linear system Bu = f given by (1.2), resulting in a preconditioned system Lu = g with the key benefit that L := P>B is symmetric positive definite.
We immediately point out that P> 6= B>, i.e. we are not forming the
nor-mal equations of the system. Instead, the preconditioner P> simply represents a substitution for the test function appearing in the Galerkin formulation of the DG time-stepping method, motivated by inf-sup theory. The matrix L then gives the natural energy norm of the evolution equation: for example, in the context of second-order parabolic PDEs, L can be viewed as the Gram matrix of a discrete H1(H−1) ∩ L2(H1) inner product.
The transformed symmetric positive definite system can therefore be solved by the preconditioned conjugate gradient (PCG) algorithm [9, 21]. Our approach thus benefits from the higher computational efficiency of PCG as opposed to approaches based on applying preconditioned GMRES to the nonsymmetric matrix B.
Robustness and fast convergence. In section 3.3, we construct a spectrally equiv-alent and easy to compute preconditioner H for L, such that the condition number of the preconditioned system satisfies
(1.4) κ(H−1L) ≤ 4,
independently of all parameters τ , p, M and A. Therefore, the preconditioner is fully robust with respect to all problem and discretization parameters, and the fast convergence of the PCG algorithm is guaranteed. The analysis only assumes that the matrices M and A are symmetric positive definite and that solvers are available for systems involving A and M + µA, for positive µ.
For problems where M and A represent discretizations of some underlying spa-tial operators, the fact that the above bound is fully independent of M and A im-plies robustness with respect to the spatial discretization parameters. Our analysis thus covers a broad range of diffusive evolution problems and spatial discretiza-tion schemes, such as conforming/nonconforming finite elements, finite differences, spectral methods, etc. In practice the inverses of M + µA can be approximated by a few multigrid V -cycles, as shown by our numerical experiments.
Efficiency and applicability. In addition to the fast convergence of PCG, several further advantages contribute to the efficiency of our approach. All operations for L and H−1 can be composed entirely of operations on p + 1 blocks of the same size as M and A: thus we do not require the assembly of the full system matrices. Therefore, our method can be readily implemented on top of existing solvers for elliptic problems, without a significant increase in memory cost. Furthermore, the preconditioners are well-suited to parallelization over the blocks. For instance, H−1 is block-diagonal, so its action can be trivially parallelized over the blocks. There is also no need for complex arithmetic.
This paper is organized as follows: after setting the notation in section 2, we present the preconditioning strategy in section 3, where we establish (1.4). Sec-tion 4 considers the efficient implementaSec-tion of the method, and secSec-tion 5 presents the results of numerical experiments testing the robustness and efficiency of the preconditioners.
2. Preliminaries
2.1. Approximation space. Let V denote a finite dimensional real vector space, equipped with a given basis {ei}dim Vi=1 . Let V be equipped with two inner products
(·, ·)M and (·, ·)A, and let M and A be their matrix representations, given by Mij:=
(ei, ej)M and Aij := (ei, ej)A for all i, j = 1, . . . , dim V. The inner products (·, ·)M
and (·, ·)Ainduce the norms k·kM and k·kA on V.
Let Pp denote the space of real-valued polynomials of degree at most p, and let
Vp be the space of V-valued polynomials of a single real variable with degree at
most p. For example, if {φj} p
j=0 is a basis of Pp, then every v ∈ Vpis of the form
v : s 7→ v(s) =
p
X
j=0
vjφj(s),
with coefficients vj ∈ V for each j = 0, . . . , p. It follows that dim Vp, the dimension
of the space Vp, is equal to (p + 1) × dim V.
Remark 2.1. In this setting, it is natural to view functions in Vp as mappings from
time into V, and as a slight abuse of standard terminology, we will say that {φj} p j=0
forms a basis of Vp. Of course, this must be interpreted as the lengthier statement
that {eiφj} j=0,...,p
i=1,...,dim V forms a basis of Vp, where {ei}dim Vi=1 is a basis of V.
For a function v ∈ Vpand a given basis {φj} p
j=0, we will denote the vector of its
coefficients by v, i.e. v = (v0, . . . , vp) ∈ Vp+1, v = p X j=0 vjφj.
Naturally, the vector v ∈ Vp+1can be identified with a vector in R(p+1)×dim V once each coefficient vj is expressed in the basis {ei}dim Vi=1 of V.
If L is a linear operator between V and either itself or its dual V∗, then we can extend L to Vp by applying it coefficient-wise:
(2.1) Lv(s) := p X j=0 (Lvj) φj(s) ∀ v = p X j=0 vjφj∈ Vp.
The inner products (·, ·)M and (·, ·)Aalso extend to Vpin the natural way. Likewise,
if π : Pp→ Ppis a linear operator then we define πv =Ppj=0vj(πφj). For instance,
we define the time derivative v0 :=Pp
j=0vjφ 0 j.
Let (·, ·)L2 and k·kL2 denote respectively the L2-inner product and L2-norm over the interval (−1, 1). Let {Lk}k≥0 denote the set of Legendre polynomials, as
defined for instance in [8, Sec. 8.9]. The Legendre polynomials are orthogonal in the L2-inner product: for all k, j ≥ 0,
(2.2) (Lk, Lj)L2=
2 2k + 1δkj. Furthermore, Lk(1) = 1 and Lk(−1) = (−1)k for all k ≥ 0.
2.2. DG time-stepping. The DG time-stepping method leads to systems of the general form
(2.3) B(u, v) = F (v) ∀ v ∈ Vp,
where F is a bounded linear functional on Vpand the bilinear form B : Vp× Vp→ R
is defined by (2.4) B(u, v) := Z 1 −1 (u0, v)Mds + (u(−1), v(−1))M +τ 2 Z 1 −1 (u, v)Ads, where τ denotes the time-step size and p denotes the polynomial degree of the approximation. Given a basis {φj}
p
j=0for Vp, the problem (2.3) can be represented
by a linear system
(2.5) Bu = f ,
where B is a (p + 1) × (p + 1) block matrix, where u ∈ Vp+1 is the vector of coefficients of the expansion of u, and where f = (f0, · · · , fp) with each fj ∈ V∗,
j = 0, . . . , p, being the restriction of F to span φj⊗ V. The system (2.5) is thus of
the general form given by (1.2).
In practice, the time-step size, the polynomial degree, or even M and A, may vary between time-steps, although on any given time-step the linear system has the same general form. Therefore, our results can be used in conjunction with hp-refinement and adaptive refinement strategies [16].
Remark 2.2. It is well-known that (2.6) B(v, v) = 1 2kv(1)k 2 M + 1 2kv(−1)k 2 M + τ 2 Z 1 −1 kvk2 Ads,
which shows coercivity of B. Unfortunately, this norm does not control the time derivative, and thus coercivity and boundedness in the same norm can only be obtained through an inverse inequality for the finite dimensional space Vp, at the
expense of introducing constants that are not robust with respect to τ and p. This suggests that norm preconditioners for B that are based on the right-hand side of (2.6) are unlikely to be robust with respect to the discretization parameters. 2.3. A result of Pearson and Wathen. In addition to the norms k·kM and k·kA,
we will also use the negative norm k·kA−1 defined by
(2.7) kvkA−1 := sup
w∈V\{0}
(v, w)M
kwkA
∀ v ∈ V. We have the identity
(2.8) kvk2
A−1 = v>M A−1M v ∀ v ∈ V,
where v is identified with its vector representation in the basis {ei}dim Vi=1 . Indeed,
the upper bound kvk2A−1 ≥ v>M A−1M v follows from the choice of w = A−1M v in (2.7). The lower bound kvk2A−1 ≤ v>M A−1M v follows from the Cauchy–Schwarz inequality
w>M v = w>A A−1M v ≤ w>A w12 v>M A−1M v12,
where we have simplified (A−1M v)>A(A−1M v) = v>M A−1M v. The identity (2.8) shows that the norm k·kA−1 is in fact induced by an inner product (·, ·)A−1 repre-sented by the matrix M A−1M .
The following result due to J. W. Pearson and A. J. Wathen [14] will play a key part in the construction of our preconditioners.
Lemma 2.3. Let A and M be arbitrary symmetric positive definite matrices and let µ ≥ 0 be a nonnegative real number. Then we have
(2.9) 1
2 ≤
v> M A−1M + µ A v
v> M +√µ A A−1 M +√µ A v ≤ 1 ∀ v ∈ V \ {0}.
Proof. For the original proof of this result, see [14, Thm 4]. We provide here an alternative proof. First, the upper bound of (2.9) is immediate from
kvk2A−1+ µkvk2A≤ kvk2A−1+ 2 √
µkvk2M+ µkvk2A. Now, for any v ∈ V, we have kvk2
M ≤ kvkA−1kvkA by (2.7). It follows from the inequality ab ≤ 12a2+1
2b 2
for any a, b ∈ R that
v>(M +√µ A) A−1(M +√µ A) v = kvk2A−1+ 2 √ µkvk2M + µkvk2A ≤ 2kvk2 A−1+ 2µkvk2A. (2.10)
Note that (2.10) is precisely the lower bound of (2.9).
3. Preconditioners
3.1. Reconstruction operator. Define the operator I : Pp→ Pp+1 by
(3.1) Iv := v − v(−1)(−1)
p(L
p− Lp+1)
2 ∀ v ∈ Pp.
The properties of the Legendre polynomials imply that, for any v ∈ Pp,
(3.2) Iv(1) = v(1), Iv(−1) = 0, Z 1 −1 Iv w ds = Z 1 −1 v w ds ∀ w ∈ Pp−1.
Substituting w for w0, which belongs to P
p−1 whenever w ∈ Pp, in (3.2) and using
integration by parts shows that (3.3) Z 1 −1 (Iv)0w ds = Z 1 −1 v0w ds + v(−1) w(−1) ∀ w ∈ Pp.
Remark 3.1. The operator I is the reconstruction operator commonly used in the a posteriori error analysis of the DG time-stepping method [11]. This operator is often defined by requiring that Iv(sk) = v(sk) for all k = 1, . . . , p + 1, where
−1 < sk ≤ sp+1 = 1 are the Gauss–Radau quadrature points, with a further
condition, chosen here as Iv(−1) = 0. The Gauss–Radau points {sk} p+1
k=1 are the
roots of the polynomial (1 − s)Pp(1,0)= Lp− Lp+1, see [7] and [8, eq. 8.961.5]. It is
thus found that the definition (3.1) implies the interpolation property (3.4) Iv(−1) = 0, Iv(sk) = v(sk) ∀ k = 1, . . . , p + 1.
This establishes the equivalence of the two definitions. Furthermore, (3.4) easily shows that if (Iv)0 ≡ 0 on (−1, 1), then v ≡ 0, since v ∈ Pp then has at least
p + 1 roots. We note however that we will not need the Gauss–Radau points for the analysis or computations in this work.
In the following, the inner product on Pp defined by
(3.5) (v, w) 7→
Z 1
−1
(Iv)0(Iw)0ds
will play an important role. Remark 3.1 shows that this bilinear form does indeed define an inner product, although we will show in section 4.1 below (see in particular (4.11)) the stronger result that there is a constant C, independent of p, such that (3.6) kvkL2 ≤ Ck(Iv)0kL2 ∀ v ∈ Pp.
3.2. Left preconditioner. The identity (3.3) implies that (3.7) B(u, v) =
Z 1
−1
(Iu)0, vM+τ
2(u, v)Ads ∀ u, v ∈ Vp.
Note that since Iu ∈ Vp+1 for u ∈ Vp, it follows that (Iu)0 ∈ Vp. Thus, we may
define the operator P : Vp→ Vp by
(3.8) P v := A−1M (Iv)0+τ
2v.
The fact that P v ∈ Vpfor any v ∈ Vp implies that the solution u of (2.3) also solves
(3.9) L(u, v) = G(v) ∀ v ∈ Vp,
where
(3.10) L(u, v) := B(u, P v), G(v) := F (P v) ∀ u, v ∈ Vp.
We note that the operator P can be viewed as defining a left preconditioner for the linear system (2.5) that represents (2.3). Indeed, let P denote the matrix representation of the linear operator P in the basis {φj}
p
j=0. Then,
B(u, P v) = v>P>Bu, F (P v) = v>P>f , so that (3.9) is equivalent to solving the linear system
(3.11) Lu = g,
with L = P>B and g = P>f .
As the following theorem shows, L has an improved structure over B.
Theorem 3.2. For p ≥ 0, let the bilinear form L and linear functional G be defined by (3.10). The bilinear form L is given by
(3.12) L(u, v) = Z 1 −1 (Iu)0, (Iv)0)A−1ds + τ 2(u(1), v(1))M +τ 2(u(−1), v(−1))M + τ2 4 Z 1 −1 (u, v)Ads.
Therefore, L is symmetric and positive definite, and for any symmetric positive definite matrices A and M , any p ≥ 0 and any τ > 0, we have
(3.13) L(v, v) ≥ kvk2
D, |L(v, w)| ≤ 2kvkDkwkD ∀ v, w ∈ Vp,
where k·kD :=pD(·, ·) is the norm induced by
(3.14) D(u, v) := Z 1 −1 (Iu)0, (Iv)0) A−1ds + τ2 4 Z 1 −1 (u, v)Ads.
Proof. The identities (3.15) (Iu)
0, A−1M (Iv)0
M = h(Iu)
0, M A−1M (Iv)0i = ((Iu)0, (Iv)0) A−1, u, A−1M (Iv)0
A= hu, AA
−1M (Iv)0i = (u, (Iv)0) M, imply that (3.16) L(u, v) = Z 1 −1 (Iu)0, (Iv)0) A−1ds + τ2 4 Z 1 −1 (u, v)Ads +τ 2 Z 1 −1 (Iu)0, v M + u, (Iv) 0 Mds. By (3.3), we have Z 1 −1 (Iu)0, vM + u, (Iv)0Mds = Z 1 −1 d ds(u, v)Mds + (u(−1), v(−1))M = (u(1), v(1))M+ (u(−1), v(−1))M, which implies (3.12). The lower bound L(v, v) ≥ kvk2
D is immediate, whereas the
upper bound |L(v, w)| ≤ 2kvkDkwkD follows from the application of the Cauchy–
Schwarz inequality to (3.16).
It follows that the problem (3.9) has a unique solution, momentarily denoted ˜
u ∈ Vp. Moreover, it is well-known that (2.3) has a unique solution u ∈ Vp. Since
the definition of L and G in (3.10) shows that u is a solution of (3.9), we deduce from uniqueness that ˜u = u. Therefore, the problems (2.3) and (3.9) are equivalent. Remark 3.3. We wish to thank C. Kreuzer for bringing to our attention the fact that Theorem 3.2 can be viewed as part of the inf-sup analysis of the DG time-stepping method: defining the energy norm k·kL:=pL(·, ·) and the norm k·kY :=
q R1 −1k·k 2 Ads, we have (3.17) kukL= sup v∈Vp\{0} B(u, v) kvkY , |B(u, v)| ≤ kukLkvkY ∀ u, v ∈ Vp,
where the first equality is attained by the choice v = P u since kP ukY = kukL.
This observation highlights the fact that the operator P is fundamental to the inf-sup analysis of the DG time-stepping method, although this is apparently not well-known. We also comment that an equivalent but possibly more natural choice of norms would be to rescale the norms with respect to τ and choose v = P u/τ , although the current choice of scaling simplifies the presentation of the precondi-tioners in this paper.
3.3. Spectrally equivalent norm preconditioner. As shown by Theorem 3.2, the bilinear form L is symmetric and positive definite. Therefore, the linear sys-tem (3.11) can be solved iteratively by the preconditioned conjugate gradient (PCG) algorithm [9, 21]. In this section, we construct a spectrally equivalent and easily applicable preconditioner for L, thereby leading to the robust and fast convergence of the PCG algorithm.
Recall that the bilinear form L is spectrally equivalent to the bilinear form D defined by (3.14). The first step in our construction of a preconditioner is to choose an advantageous basis for Vp that block-diagonalizes the matrix D representing D.
In a second step, we construct our preconditioner, denoted by H, under this basis, and we apply Lemma 2.3 to show robust spectral equivalence between H and L.
As shown in section 3.1, the symmetric bilinear form defined in (3.5) is posi-tive definite. Therefore, there exists a set of linearly independent eigenfunctions {qj}
p
j=0⊂ Ppand corresponding real positive eigenvalues {λj} p j=0⊂ R>0such that (3.18) λj Z 1 −1 (Iqj)0(Iv)0ds = Z 1 −1 qjv ds ∀ v ∈ Pp, j = 0, . . . , p.
We note that the eigenvalues and eigenfunctions generally depend on the value of the polynomial degree p. The eigenfunctions are chosen to be orthonormalized: (3.19)
Z 1
−1
(Iqj)0(Iqk)0ds = δjk ∀ 0 ≤ j, k ≤ p.
We will show how to compute these eigenfunctions and eigenvalues in section 4.1. The following result characterizes the distribution of the eigenvalues.
Theorem 3.4. Let p ≥ 0 be a nonnegative integer, and let λ0 ≥ · · · ≥ λp be
the eigenvalues of the generalized eigenvalue problem (3.18). Then, there exists a positive constant C, independent of j and p, such that
(3.20) λj ≤ C(j + 1)−2 ∀ j = 0, . . . , p, ∀ p ≥ 0.
Moreover, there exists a constant c, independent of p, such that (3.21) λ0≥ · · · ≥ λp≥ c (p + 1)−4.
The proof of Theorem 3.4 is given in Appendix A since it is based on Weyl’s Theorem [4, Ch. 5] and on some results presented in section 4.1. Figure 2 shows that the orders of the bounds of Theorem 3.4 are sharp. The eigenfunctions for p = 4 are shown in Figure 1.
Let D denote the matrix representation of the bilinear form D, defined in (3.14), under the basis {qj}pj=0 of Vp, where we recall the terminology from Remark 2.1.
The matrix D has a simple block-diagonal structure:
(3.22) D = diag M A−1M + τ 2λ j 4 A p j=0 , i.e. D is block-diagonal, with blocks M A−1M + λjτ
2
4A for j = 0, . . . , p.
The norm preconditioner H is defined in matrix form by (3.23) H := diag ( M +τpλj 2 A ! A−1 M +τpλj 2 A !)p j=0 .
The matrix H is symmetric positive definite, and its inverse is trivially given by H−1= diag M + τpλj 2 A !−1 A M +τpλj 2 A !−1 p j=0 .
We propose to use H as a preconditioner for the PCG algorithm applied to (3.11). Each PCG iteration requires the application of H−1, which comprises two applica-tions of a solver for a weighted backward Euler step and a multiplication by A for each of the p + 1 blocks.
We now give the central result of this work, which shows that L and H are spectrally equivalent with fully robust bounds.
Theorem 3.5. Let p ≥ 0, and let {qj}pj=0 ⊂ Pp and {λj}pj=0 ⊂ R>0 be the
eigenfunctions and eigenvalues of (3.18). Let L be the matrix representing L in the basis {qj}
p
j=0. Then, for any symmetric positive definite matrices A and M and
any τ > 0 , we have (3.24) 1 2 ≤ v>L v v>H v ≤ 2 ∀ v ∈ V p+1\ {0}.
Proof. Theorem 3.2, in particular (3.13), shows that
(3.25) 1 ≤ v>L v
v>D v ≤ 2 ∀ v ∈ V
p+1\ {0}.
Lemma 2.3, applied block-wise with µ = λjτ2/4, implies that
(3.26) 1 2 ≤ v>D v v>H v ≤ 1 ∀ v ∈ V p+1 \ {0}.
Therefore, (3.25) and (3.26) imply (3.24).
Theorem 3.5 immediately implies that the condition number κ of the precondi-tioned system, defined as the ratio of extremal eigenvalues of H−1L, satisfies
(3.27) κ ≤ 4.
Therefore, we may expect very fast convergence from the PCG algorithm for Lu = g using an initial guess u0 ∈ Vp and the preconditioner H. The iterates {uk}k≥0 of
the PCG algorithm satisfy the well-known bound [21] (3.28) ku − ukkL ku − u0kL ≤ 2 √κ − 1√ κ + 1 k ≤ 2 3k ∀ k ≥ 0,
where we recall the energy norm k·kL=pL(·, ·).
The fact that the condition number κ ≤ 4 shows that the left preconditioner P> and the norm preconditioner H are very effective at preconditioning the original system matrix B, despite B being nonsymmetric and poorly conditioned. This is a key aspect of the efficiency of the proposed preconditioners.
Furthermore, we highlight that (3.27) and (3.28) are valid for any time-step size τ , any polynomial degree p, and any symmetric positive definite matrices M and A. Thus the preconditioners are fully robust with respect to all discretization and problem parameters.
Remark 3.6. The fact that the energy norm k·kL appears in the bound (3.28) is
advantageous in practice. Since the endpoint value of the solution at s = 1 serves as initial datum for the next timestep, it is beneficial to be able to control the accuracy to which it is computed. In particular, (3.12) shows that the energy norm controls the value at s = 1 in the M -norm1, which is the natural norm for this quantity of interest. Therefore (3.28) is a guaranteed convergence rate for this quantity of interest in the physically relevant norm.
1The factor of τ appearing alongside kv(1)k2
M in the energy norm can be scaled out as it
−1 −0.5 0 0.5 1 −0.5 0 0.5 1 q0 q1 q2 q3 q4 Figure 1. Eigenfunctions {qj} p j=0 defined by (3.18) and (3.19),
computed for p = 4 by the method of section 4.1 and ordered by decreasing eigenvalue λj≥ λj+1. Note that the j-th eigenfunction
qj need not be of degree at most j. Furthermore, the set of
eigen-functions generally depends on p, although they are independent of A, M and τ . 2 4 6 10−2 10−1 1 j λj Eigenvalues for p = 5 1 10 1 10−1 10−2 10−3 10−4 (j +1) −2 (j+ 1)− 4 j + 1 λj Eigenvalues for p = 10 1 10 100 10−8 10−4 100 (j +1) −2 (j+ 1)− 4 j + 1 λj Eigenvalues for p = 100 1 10 100 1000 1 10−4 10−8 10−12 (j +1) −2 (j + 1)− 4 j + 1 λj Eigenvalues for p = 1000 Figure 2. Eigenvalues {λj} p
j=0 computed by the method of
sec-tion 4.1 and arranged in decreasing order, for p = 5, 10, 100 and 100. All plots are on a logarithmic scale, except for p = 5 which is given on a semilogarithmic scale. It appears that the bounds of Theorem 3.4 are sharp.
4. Implementation
4.1. Computation of eigenfunctions and eigenvalues. The algorithm requires the computation of the eigenfunctions {qj}
p
j=0 and eigenvalues {λj} p
j=0 defined in
(3.18), which amounts to solving a symmetric positive definite eigenvalue problem of dimension p + 1. This poses little difficulty, as in practice p is usually small, especially in comparison to dim V. Since this step only depends on p and is fully independent of A, M and τ , it can be pre-computed to very high accuracy.
We now show how to assemble this eigenvalue problem in a form where it can be solved numerically. Since the cases p = 0 and p = 1 can be easily computed by hand, we present a general method for p ≥ 2. First, it can be shown from [8, Sec. 8.914] that the Legendre polynomials {Lk}pk=0satisfy
(4.1) Lk=
L0k+1− L0 k−1
2k + 1 ∀ k ≥ 0.
Lemma 4.1. Let p ≥ 2 be a nonnegative integer. Define the polynomials
(4.2) ψ0:= 1 √ 2(L1+ L0) ψp:= Lp− Lp−1 √ 4p + 2 , ψk := Lk+1− Lk−1 √ 4k + 2 , k = 1, . . . , p − 1. Then, there holds
(4.3) (Iψk)0=
q
k +12Lk ∀ k = 0, . . . , p.
Therefore, we have, for each 0 ≤ k, j ≤ p, (4.4)
Z 1
−1
(Iψk)0(Iψj)0ds = δkj.
Proof. Consider the case k = 0: ψ0(−1) = 0, and L01 = L0. Therefore Iψ0 = ψ0,
and thus (Iψ0)0 = L0/
√
2, which is (4.3) for k = 0. Now consider the general case k = 1, . . . , p − 1. Since Lj(−1) = (−1)j for all j ≥ 0, we have ψk(−1) = 0 and thus
Iψk= ψk. Now, the identity (4.1) implies that
(4.5) (Iψk)0= 2k + 1 √ 4k + 2Lk= q k +12Lk,
thus verifying (4.3) for k = 1, . . . , p − 1. For the case k = p, we use the fact that Lp(−1) − Lp−1(−1) = 2(−1)p to compute
(4.6) I(Lp− Lp−1) = Lp− Lp−1−
2(−1)p(−1)p
2 (Lp− Lp+1) = Lp+1− Lp−1. Therefore, similarly to (4.5), the identity (4.3) for k = p follows from (4.1).
Define the matrix Z ∈ R(p+1)×(p+1) such that ψ
k = Ppj=0ZkjLj as in (4.2).
Define the diagonal matrix D := diag{2/(2j +1)}pj=0, which represents the L2-inner
product in the Legendre polynomial basis. Define the matrix T ∈ R(p+1)×(p+1)by
(4.7) Tkj:=
Z 1
−1
Note that T = ZDZ>. It follows from (4.4) that the eigenvalue problem (3.18) can be expressed as: find V ∈ R(p+1)×(p+1)and {λ
j}pj=0such that
(4.8) T V = V diag{λj} p
j=0, V V>= V>V = Id.
The eigenvalue problem (4.8) can thus be solved numerically by standard eigenvalue solvers. The eigenfunctions {qj}pj=0 can then be recovered as
(4.9) qj(s) = p X k=0 Vkjψk(s) = p X k=0 QkjLk(s), where Q := Z>V .
Observe that T is pentadiagonal, i.e. Tkj= 0 if |k − j| > 2. Moreover, it is easy
to show that there exists a constant C independent of p such that (4.10) kψkk2L2 = Tkk≤ C(k + 1)−2 ∀ k = 0, . . . , p.
The Cauchy–Schwarz inequality implies that |Tjk| ≤pTkkTjj, and thus all entries
of T are uniformly bounded. Since T has at most 5 nonzero entries per row, and all entries are uniformly bounded, the Gershgorin discs of T are bounded independently of p. The Gershgorin Disc Theorem therefore implies that there is a constant C, independent of p, such that λ0, the maximal eigenvalue of T , satisfies
(4.11) λ0= sup u∈Rp+1\{0} u>T u kuk2 2 ≤ C. This is equivalent to the bound (3.6).
4.2. Implementation of preconditioners. In order to compute the action of the left-preconditioner P>, we use the matrix representation of the mapping v 7→ (Iv)0 in the basis given by {qj}
p
j=0. Thus we need to find the matrix K such that
(Iqk)0 = P p
j=0Kkjqj for all k = 0, . . . , p. We show below that this is easily
computed. Inverting (4.1) by induction yields
(4.12) L0k = k−1 X j=0 1 − (−1)k−j j + 12 Lj ∀ k ≥ 0. Therefore, we have (4.13) (ILk)0= p X j=0 Kkj∗Lj ∀ k = 0, . . . , p,
where, after some calculation, it is found that the matrix K∗ ∈ R(p+1)×(p+1) can
be defined in terms of ˜K ∈ R(p+1)×(p+1), x ∈ Rp+1, and y ∈ Rp+1 by K∗:= ˜K − x y>, K˜kj:= ( 1 − (−1)k−j j +12 if j ≤ k, 0 otherwise, (4.14) xj := (−1)j, yk:= (−1)1−k k +12 . (4.15)
Lemma 4.2. Let p ≥ 2 be an integer, and let {qj}pj=0be the eigenfunctions defined
by (4.9). Then, for each k = 0, . . . , p, there holds (4.16) (Iqk)0 =
p
X
j=0
where we recall that D = diag{2/(2j + 1)}pj=0.
Proof. The orthogonality of the matrix V in (4.8) implies that ψk =P p
m=0Vkmqm.
Since the basis {ψk} p
k=0 is orthonormal in the inner product of (3.5), there holds
Lj= p X r=0 Z 1 −1 (ILj)0(Iψr)0ds ψr= p X r=0 p X m=0 Kjm∗ (r +12)1/2 Z 1 −1 LmLrds ψr = p X r=0 Kjr∗ r + 12−1/2 ψr= p X r=0 (K∗D1/2)jrψr= p X m=0 (K∗D1/2V )jmqm,
where we have used (4.3) and (4.13) in the first line. Therefore, we compute (Iqk)0= p X j=0 Vjk(Iψj)0= p X j=0 Vjk(j +12)1/2Lj= p X j=0 p X m=0 Vjk(D−1/2K∗D1/2V )jmqm,
which completes the proof.
We now show how the matrix K is used in applying the preconditioner P>.
Recall the original linear system (2.5), expressed in the basis {qj} p j=0,
Bu = f ,
where, f = (f0, . . . , fp) is the vector of restrictions of F to the subspace span qj⊗ V,
i.e. for any vector v ∈ V and any j = 0, . . . , p, we have F (vqj) = fj>v.
The left preconditioner P> transforms the original linear system into (3.9).
For v =Pp k=0vkqk, there holds P v = p X j=0 A−1M wj+ τ 2vj qj, wj := p X k=0 Kkjvk, F (P v) = p X j=0 fj>A−1M wj+ τ 2vj = p X k=0 M A−1 p X j=0 Kkjfj + τ 2fk > vk.
Therefore, g := P>f , g = (g0, . . . , gp), can be computed componentwise by
(4.17) gk= M A−1 p X j=0 Kkjfj + τ 2fk, k = 0, . . . , p.
The action of P> requires that p + 1 independent elliptic systems be solved, which
can be performed in parallel. If we ignore communication costs, then the cost of computing g is independent of the polynomial degree p on a parallel machine with sufficiently many computing nodes.
After application of the preconditioner P>, the linear system has the form Lu = g,
where L = P>B, which can be solved by the PCG algorithm with preconditioner H, as suggested in section 3. The PCG algorithm requires the action of the matrices L and H−1 at each iteration. There are two ways to implement the action of L, the first being the application of B followed by P>. The downside of this approach
machine where each node holds in memory only a few coefficients of u: the matrices B and P> are generally block-dense, so all vector components must be exchanged between all computational nodes for both steps.
The second approach is to use (3.12), which shows that L can be expressed as a diagonal matrix plus a “rank-two” term: in the basis {qj}
p j=0, we have (4.18) L = diag M A−1M +τ 2λ j 4 A p j=0 +τ 2q1q > 1 ⊗ M + τ 2q−1q > −1⊗ M,
where q±1 := (q0(±1), . . . , qp(±1)) are the vectors of endpoint values of the qj.
Therefore, we may compute Lu, with u = (u0, . . . , up), as follows:
wj:= M uj, j = 0, . . . , p, z±1:= p X j=0 qj(±1)wj, (4.19a) (Lu)j= M A−1wj+ τ2λ j 4 Auj+ τ 2qj(1)z1+ τ 2qj(−1)z−1. (4.19b)
This approach requires the same number of matrix-vector products as the first approach outlined above, but the communication costs are greatly reduced. Indeed, it is seen that the above procedure requires the parallel computation of the wj,
which are then gathered and reduced by one or two nodes tasked with computing z±1. The results are then broadcast back to all nodes, after which all subsequent
computations can be performed in parallel.
Finally, the application of H−1 is trivially parallel, as it requires only that each
node solves two linear systems involving the matrix M + τpλj/2A, and one
ap-plication of A. Theorem 3.4 shows that the eigenvalues remain uniformly bounded from above, and that the smaller eigenvalues approach zero as p increases. There-fore, for large p and j, the matrix M + τpλj/2A becomes increasingly similar to
M , implying that these systems will become increasingly easy to solve in many applications.
5. Numerical experiments
5.1. Condition numbers. In this experiment, we study the dependence of the condition numbers κ of the preconditioned system H−1L on the problem and dis-cretization parameters. In order to compute accurately the condition numbers, we consider first a one-dimensional problem. Let M and A be the mass and stiff-ness matrices obtained by applying piecewise-linear continuous finite elements on a uniform subdivision of the domain Ω = (0, 1) into elements of size h = 2−k, k = 5, . . . , 10. We note that M and A thus depend on h, although this is is left implicit in our notation. For this experiment, we use direct solvers to implement the action of A−1 and (M + τpλj/2A)−1.
Table 1 shows the condition numbers κ as a function of τ for a wide range of parameters. For this experiment, we set p = 2 and h = 2−5. It is found that for either very large or very small τ , κ → 1. This is easily explained by the fact that
(5.1) lim τ →∞ v>L v v>H v = limτ →0 v>L v v>H v = 1.
Therefore the condition number κ → 1 as τ → 0 or τ → ∞. Thus the maximal condition number is found for intermediate values of the time-step size τ , although in all cases it satisfies the theoretical bound κ ≤ 4 as shown in (3.27).
τ 10−6 10−5 10−4 10−3 10−2 10−1 1 10 κ 1.011 1.103 1.749 2.031 2.028 2.019 1.693 1.089 Table 1. Dependence of the condition number κ of H−1L on a wide range of values for the time-step size τ . Observe that for either very large or very small τ , the condition number is asymptotically 1, in agreement with (5.1). This explains the observation that the maximum condition number is observed for intermediate values.
κ h = 2−5 h = 2−6 h = 2−7 h = 2−8 h = 2−9 h = 2−10 p = 1 1.318 1.319 1.319 1.319 1.319 1.319 p = 2 2.019 2.019 2.019 2.019 2.019 2.019 p = 3 2.243 2.243 2.243 2.243 2.243 2.243 p = 4 2.353 2.353 2.353 2.353 2.353 2.353 p = 5 2.416 2.417 2.417 2.417 2.417 2.417 p = 6 2.493 2.493 2.493 2.493 2.493 2.493
Table 2. Dependence of the condition number κ for p = 1, . . . , 6 and h = 2−5, . . . , 2−10, with τ = 0.1. For fixed p, the condition number is insensitive to the mesh size h, i.e. variation of M and A. The condition number slowly increases with p towards its as-ymptotic value of approximately 2.686, see Table 3.
p 8 16 32 64 128 256
κ 2.558 2.643 2.674 2.684 2.686 2.686
Table 3. Asymptotic behaviour of the condition number with re-spect to the polynomial degree p = 2m, m = 3, . . . , 8, with fixed h = 2−5 and τ = 0.1. The asymptotic value of κ appears to be κ ≈ 2.686. This suggests that the theoretical bound κ ≤ 4 of (3.27) may not be quantitatively sharp in all cases, although it is nevertheless very close.
Table 2 shows that the condition number has little to no dependence on mesh refinement, i.e. variation of M and A. As p increases, the condition number ap-proaches an asymptotic value, as shown by Table 3.
The proposed preconditioners are thus fully robust with respect to the parame-ters, in agreement with the theoretical bound κ ≤ 4 from (3.27). Furthermore, this experiment suggests that the efficiency of the preconditioners can exceed theoretical expectations, as shown by condition numbers κ ≤ 2.686 throughout these tests. 5.2. Iteration counts and multigrid preconditioning. The condition number bound (3.27) holds for the preconditioner H, which assumes that exact solvers are used to apply the inverses of the matrices M + τpλj/2A. However, in
Mesh size DoF Direct 1 V-cycle 2 V-cycles 3 V-cycles h = 2−6 11 907 7 8 7 7 h = 2−7 48 387 7 8 7 7 h = 2−8 195 075 7 8 7 7 h = 2−9 783 363 7 8 7 7 h = 2−10 3 139 587 7 8 7 7
Table 4. Number of PCG iterations required for convergence of the PCG algorithm in the experiment of section 5.2, for various mesh sizes and corresponding number of degrees of freedom. Fast convergence of PCG is observed in all cases, with the iteration counts being mesh-independent. The method using 1 V-cycle ap-pears as the most efficient in this experiment.
M + τpλj/2A, such as multigrid, which leads to an approximation of the ideal
preconditioner H−1.
In this experiment, we study the effect of this approximation, in particular when the inverse of M + τpλj/2A is approximated by a small number of multigrid
V-cycles. Let M and A be defined as the mass and stiffness matrices of the piecewise-linear finite element space defined on a uniform triangulation of size h = 2−k, k = 6, . . . , 10 in the two-dimensional domain Ω = (0, 1)2. The finest mesh thus
leads to more than one million degrees of freedom (DoF) for V.
We consider the system Lu = g, with a chosen exact solution u. We approximate solvers for M + τpλj/2A by applying n V-cycles, n ∈ {1, 2, 3}, with symmetric
Gauss–Seidel smoothers. Thus each application of the preconditioner requires 2n V-cycles per block.
To obtain a fair comparison of all the preconditioners, a relative tolerance of 10−6 of the true error in the energy norm was used to determine convergence of PCG, and zero initial guesses were used for all computations.
Table 4 shows the PCG iteration counts for both ideal and approximate pre-conditioners; in this experiment, we set p = 2 and τ = 0.1, and thus there are more than 3 million degrees of freedom for Vp on the finest mesh. Fast and
ro-bust convergence of PCG is observed in all cases, showing the effectiveness of these preconditioners. For this experiment, the method using 1 V-cycle appears as the most efficient, as the reduction in cost per iteration outweighs the additional PCG iteration required.
In Table 5, we study the dependence of the iteration counts on the polynomial degree. It is found that the use multigrid preconditioners retains the robustness of the preconditioner with respect to the approximation order p. This is indeed the expected result, since Theorem 3.4 shows that as p is increased, for large j the matrices M + τpλj/2A are spectrally closer to M and thus the efficiency of the
multigrid V-cycle increases. Moreover, this experiment shows that it is possible to compute with very high approximation orders, as required by hp-refinement strategies in time [16].
We have also computed the results of this experiment using smaller time-step sizes τ , and we observed a decrease in the iteration counts, as predicted by (5.1).
h = 2−9 h = 2−10 Order Iterations DoF Iterations DoF
p = 4 8 1 305 605 8 5 232 645 p = 6 9 1 827,847 9 7 325 703 p = 8 9 2 350,089 9 9 418 761 p = 10 9 2 872 331 9 11 511 819 p = 12 9 3 394 573 10 13 604 877 p = 14 9 3 916 815 9 15 697 935
Table 5. Dependence of the PCG iteration counts on the poly-nomial degrees p = 4, . . . , 14 in the experiment of section 5.2, with approximate solvers using 1 multigrid V-cycle. The number of PCG iterations remains uniformly bounded, showing full robust-ness with respect to the polynomial degree, even up to 15 million DoF. The additional iteration for the case p = 12, h = 2−10 is explained by the fact that the error on the 9-th iteration was very close to, but above, the convergence criterion.
The conclusion drawn from these results is that in practice, standard approxi-mate solvers can be used while retaining the key properties of the ideal precondi-tioner H, namely the fast convergence of the PCG algorithm and the robustness with respect to the discretization parameters.
5.3. Parabolic problems with inexact solvers. The action of L as given in (4.19) requires the action of A−1. In many practical applications, it is desirable to approximate this step by using an inexact solver, such as a fixed number of multigrid V-cycles, leading to an approximation ˆL ≈ L. This raises the question of whether this can be performed without affecting firstly the accuracy of the solution, and secondly the performance of the preconditioning strategy. This experiment provides evidence that inexact solvers can indeed be used without compromising these important objectives.
Consider the piecewise-linear simplicial conforming finite element approximation of the heat equation in the unit square Ω = (0, 1)2, final time T = 0.1, imposed with
initial datum u0(x, y) = x(1 − x) sin(πy) and with homogeneous Dirichlet lateral
boundary conditions. This initial datum is chosen as it leads to a solution with decreased temporal regularity, see [16]. To compare near-exact and inexact solvers, we consider two approaches:
(D) direct solvers are used in the application of P>, H−1 and L,
(MG) 5 multigrid V-cycles are used to approximate the application of A−1 in L and P>, and 1 multigrid V-cycle is used to approximate (M +τpλj/2A)−1
in H−1, as in section 5.2.
We point out that the method (MG) does not use any direct solvers throughout the entire computation, and that each PCG iteration costs 7 V-cycles per block.
Table 6 shows the final time errors ku(T ) − uτ(T )kL2(Ω), for varying τ , where u denotes the exact solution of the PDE, and where uτ denotes the discrete solution.
Here, we used h = 2−8and p = 1. For comparison, we also show the errors attained for p = 0, i.e. the backward Euler (BE) method. The inexact solvers for (MG) retain the accuracy of the method, as the difference in solutions between (D) and (MG) is
τ /T Error (BE) Error (D) Error (MG) (D)–(MG) 1 2.546 × 10−2 3.078 × 10−3 3.078 × 10−3 1.303 × 10−8 1/2 1.475 × 10−2 3.934 × 10−4 3.934 × 10−4 2.129 × 10−8 1/4 8.008 × 10−3 5.444 × 10−5 5.441 × 10−5 3.219 × 10−8 1/8 4.178 × 10−3 8.718 × 10−6 8.641 × 10−6 8.126 × 10−8 Table 6. Final time errors ku(T ) − uτ(T )kL2(Ω) obtained by the DG time-stepping method for the problem of section 5.3, for p = 0 (the backward Euler method) and for p = 1 with direct solvers (D) and inexact multigrid (MG) solvers. The last column gives kuτ(T ) − ˆuτ(T )kL2(Ω), where uτ, respectively ˆuτ, denotes the
so-lution obtained by (D), respectively (MG).
Iterations τ = T τ = T /2 τ = T /4 τ = T /8
(D) 4 3.5 3 3
(MG) 6 5 5 5
Table 7. Average number of PCG iterations per time-step for the problem of section 5.3, with p = 1 and with either direct (D) or inexact multigrid (MG) solvers.
several orders of magnitude smaller than the difference to the exact solution. For p = 1, the expected third order super-convergence rate is observed.
Table 7 shows the average number of PCG iterations per time-step required by both approaches in order to obtain a residual tolerance of 10−6, as well as the final time error between the approximate solution uτ for (D) and ˆuτ for (MG). This
shows that the number of PCG remains robust with respect to the approximation of L entailed by (MG). This experiment shows that the use of inexact solvers can retain both the accuracy of the DG time-stepping method as well as the efficiency of the proposed preconditioners.
Conclusion
We have developed efficient and robust preconditioners enabling the fast solu-tion of the DG time-stepping method by the precondisolu-tioned conjugate gradient algorithm. The analysis and numerical experiments show that the ideal and ap-proximate preconditioners are robust with respect to all discretization parameters and lead to low condition numbers for the preconditioned system. Thus the high-order solution of large problems by the DG time-stepping method is tractable under the proposed approach.
Appendix A. Analysis of eigenvalues
In this section, we present the proof of Theorem 3.4. Without loss of generality, it is sufficient to consider only p ≥ 2 throughout this section. We will make use of Weyl’s Theorem from eigenvalue perturbation theory [4].
Theorem A.1 (Weyl). For a positive integer n, let T and ˆT denote symmetric n×n matrices. Let α1 ≥ · · · ≥ αn denote the eigenvalues of T and let ˆα1 ≥ · · · ≥ ˆαn
denote the eigenvalues of ˆT . Then, for each j = 1, . . . , n, we have (A.1) |αj− ˆαj| ≤ kT − ˆT k2,
where k·k2 denotes the matrix 2-norm.
We are now ready to prove Theorem 3.4.
Proof of Theorem 3.4. Recall that the matrix T is defined by (4.7). The case j = 0 follows immediately from (4.11), so we consider now j ≥ 1. Without loss of generality, we may assume that p ≥ 2. For 0 ≤ q ≤ p − 1, we define the orthogonal projector πq: Pp→ Pq by (A.2) πq: p X j=0 vjψj 7→ q X j=0 vjψj,
where the polynomials ψj are defined by (4.2), and v = (v0, . . . , vp) ∈ Rp+1.
Define the matrix ˆTq ∈ R(p+1)×(p+1) by ( ˆTq)kj := (πqψk, πqψj)L2. Note that ( ˆTq)kjis zero if either k or j is greater than q, and equals Tkjotherwise. Thus, the
matrix ˆTq has the general form
(A.3) Tˆq = ˜ T 0 0 0 ,
where ˜T denotes the (q + 1) × (q + 1) principal submatrix of T . The main step of the proof is to use (4.10) repeatedly to find an upper bound for
kT − ˆTqk2= sup u∈Rp+1\{0} sup v∈Rp+1\{0} v>(T − ˆTq)u kuk2kvk2 . For arbitrary u and v ∈ Rp+1\ {0}, define the polynomials u := Pp
j=0ujψj and v =Pp j=0vjψj. Then, we have v>(T − ˆTq)u = (u, v)L2− (πqu, πqv)L2 = (u − πqu, v − πqv)L2+ (πqu, v − πqv)L2+ (u − πqu, πqv)L2. (A.4)
It follows from the definition of πq in (A.2) that
ku − πquk2L2 = p X k=q+1 p X j=q+1 uk(ψk, ψj)L2uj.
Since (ψk, ψj)L2 = Tkj = 0 if |k − j| > 2, the strengthened Cauchy–Schwarz
inequality yields (A.5) p X k=q+1 p X j=q+1 uk(ψk, ψj)L2uj≤ 5 p X k=q+1 kψkk2L2|uj| 2 ≤ C (q + 1)2kuk 2 2.
Note that this implies the following a priori estimate for the orthogonal projector: (A.6) ku − πqukL2≤ C(q + 1)−1k(Iu)0kL2 ∀ u ∈ Pp.
Thus (A.5) and the Cauchy–Schwarz inequality imply that (A.7) |(u − πqu, v − πqv)L2| ≤ C(q + 1)−2kuk2kvk2.
For k ≤ q and j ≥ q + 1, we have (ψk, ψj) = 0 if k < q − 1 and j > q + 2, and thus (πqu, v − πqv)L2 = q X k=0 p X j=q+1 uk(ψk, ψj)vj = q X k=max(q−1,0) min(p,q+2) X j=q+1 uk(ψk, ψj)vj.
The Cauchy–Schwarz inequality thus implies that
|(πqu, v − πqv)L2| ≤ 2 q X k=max(q−1,0) |uk|2kψ kk2L2 1 2 min(p,q+2) X j=q+1 |vj|2kψ jk2L2 1 2 ≤ C(q + 1)−2kuk2kvk2. (A.8)
An identical argument shows that
(A.9) |(u − πqu, πqv)L2| ≤ C(q + 1)−2kuk2kvk2.
Combining (A.7), (A.8) and (A.9) implies that there exists a constant C, indepen-dent of p and q, such that
(A.10) kT − ˆTqk2≤ C(q + 1)−2.
Therefore, Weyl’s Theorem implies that the eigenvalues λjof T and ˆλj of ˆTqsatisfy
(A.11) |λj− ˆλj| ≤ C(q + 1)−2 ∀ j = 0, . . . , p,
where the constant C is independent of j, q and p. However, it is clear from (A.3) that ˆλj= 0 for j ≥ q + 1, and thus there exists a constant C independent of p and
q such that
(A.12) λj= |λj− ˆλj| ≤ C(q + 1)−2 ∀ j ≥ q + 1.
The left-hand side of this inequality is independent of q while the right-hand side is valid of all q ≤ j − 1. Therefore, the choice q + 1 = j thus yields (3.20) for j = 1, . . . , p.
The lower bound (3.21) follows from inverse inequalities. Indeed, λp satisfies
λp= min v∈Pp\{0} kvk2 L2 k(Iv)0k2 L2 .
To obtain (3.21), it is therefore enough to show the inverse inequality (A.13) k(Iv)0kL2 ≤ C(p + 1)2kvkL2 ∀ v ∈ Pp.
For any v ∈ Pp, standard inverse inequalities [18] imply that
k(Iv)0kL2≤ kv0kL2+ kvkL∞ 1 2k(Lp− Lp+1) 0k L2 . (p + 1)2+ (p + 1) k(Lp− Lp+1)0kL2 kvkL2. (A.14)
It follows from (4.12) that
(A.15) k(Lp− Lp+1)0k2L2 = p X j=0 8(j + 12)2 2j + 1 = 2 (p + 1) 2 .
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