An analytical investigation is performed of the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation ...
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
80 (1990) 295-304
ON THE SPECTRUM OF THE ITERATION OPERATOR ASSOCIATED TO THE FINITE ELEMENT PRECONDITIONING OF CHEBYSHEV COLLOCATION CALCULATIONS P. FRANCKEN Physique Atomique
Theorique,
Universite Libre de Bruxelles, B-1050 Bruxelles, Belgium
M.O. DEVILLE Unite de Mecanique Appliquke,
Universite Catholique de Louvain, B-1348 Louvain-la-Neuve,
Belgium
E.H. MUND Service de Metrologie Nucleaire, Universite Libre de Bruxelles, B-1050 Bruxelles, Belgium, and Unite de Thermodynamique, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
Received 26 June 1989 Revised manuscript received December
1989
An analytical investigation is performed of the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation calculations, on a one-dimensional Dirichlet model problem. Use is made of the techniques developed by Haldenwang et al. for the finite difference preconditioning of the same problem. In the latter case the eigenvalues may be obtained as the diagonal elements of an upper-triangular matrix. This is not possible with finite element preconditioning, where the corresponding operator splits into two parts, one of which only is upper-triangular. Discarding the non-triangular part of the operator (which vanishes asymptotically for large values of N, partition size), this procedure yields an approximate expression of the eigenvalues in good agreement with the properties given earlier by Deville and Mund.
1. Introduction Low order preconditioning algorithms are widely regarded as the most efficient way to implement pseudospectral (i.e. collocation) methods for the solution of boundary value problems. This is due to several reasons, the most prominent one being the substitution of a non-sparse, non-symmetric, poorly conditioned matrix operator (the collocation operator) by a sparse, symmetric and well conditioned one (the preconditioner). The technique runs as follows. Let L,u =f
(1.1)
represent the discretized form of the basic problem with L, the collocation operator. Introducing a preconditioner Lap(=L,) and a solution guess u(O),the algebraic system (1.1) is solved iteratively by the Richardson method: 0045-7825/90/$03.50
@ 1990 - Elsevier Science Publishers
B.V. (North-Holland)
P. Francken
296
et al., Spectrum of the iteration operator for FEM preconditioning
L aP u@+‘) = L,,zbk) - ak(Lcdk) -f)
,
k = 0, 1, .
(1.2)
. . .
The operator Lii . L, is of paramount importance: its spectral radius p(L,-b . L,) governs the rate of convergence of the iterative procedure, (Yebeing such that p(Z - a,L,-d . L,) < 1. Stated this way, the preconditioning scheme is the same for all collocation methods. Its actual implementation however may vary from case to case. For instance, Chebyshev collocation allows FFT techniques to be used for the evaluation of the residues (L#‘) - f), whereas Legendre collocation uses matrix multiplications. Finite difference (FD) preconditioning was applied first by Orszag [l] and Morchoisne [2]. A theoretical analysis carried out by Haldenwang et al. for Chebyshev collocation showed that it requires underrelaxation with an optimal value of (Ye equal to 4. Finite element (FE) preconditioning appeared almost simultaneously in works by Canuto and Quarteroni [4] and Deville and Mund [5,6]. The latter authors showed numerically that for a large class of problems as well as for a large class of collocation schemes, the relaxation parameter cykcould be set equal to 1, yielding spectral accuracy (i.e. round-off error levels) in less than 20 iterations [7]. Linear and higher degree elements were analyzed, showing the same behaviour. In other words, FE preconditioning in practice sets the spectrum of p(Lii . L,) inside the unit circle. This has been recognized also by Canuto and Pietra for the linear FE preconditioning of a Fourier model problem (cf. [8,9]). In this paper we investigate the spectrum of the iteration operator L[i * L, associated to a linear FE preconditioning of the Chebyshev collocation procedure on a simple one-dimensional Dirichlet model problem. Use is made of the techniques developed by Haldenwang et al., which, in the case of FD preconditioning, yielded the exact analytical expression of the Lil . L, eigenvalues on the diagonal of an upper-triangular matrix. Unfortunately, in the FE case, only approximate closed form expressions may be derived. This is due, as will be shown in the next paragraph, to the presence of the mass matrix which introduces complex eigenvalues. However, the approximate expression one may derive by ignoring the nontriangular part of the operator already explains all the properties mentioned previously. 2. Linear FE preconditioning of a model problem Let us now consider the one-dimensional
with homogeneous u(-1)
Dirichlet boundary
second order elliptic model problem
conditions
= u(1) = 0.
(2.2)
We introduce GLC-N, the set of Gauss-Lobatto-Chebyshev the polynomial TN(x): xi = -cOsei,
fli=i
!!; N
quadrature
i=O,...,N.
Given a set of basis functions {cpi(x) 1 j = 0, . . . , N}, an approximation
nodes associated with
(2.3)
C(X) of the solution of
297
P. Francken et al., Spectrum of the iteration operator for FEiU preconditioning
(2.1), (2.2) writes
5 ajqj(x) .
ii(x) =
(2.4)
j=O
In the sequel, use will be made of two different sets of basis functions, interpolation polynomials on the Chebyshev collocation grid (2.3)
the Lagrange
with Sjk the Kronecker symbol, and the set of Chebyshev polynomials { Tj(x) 1 j = 0, . . . , N}. The collocation equations (1.1) are easily written in terms of the Lagrange interpolation set, the unknown vector u being made of the nodal values u(xi) at the GLC grid. The matrix operator L, becomes i=l,...,
(L,),=--$(p,(xi), i=O,
(L,),=Soj, tLc)ij
=
‘Nj
j=O ,...,
i=N,
7
N-l,
j=O ,...,
N,
N,
j=O ,...,
W-9
N.
Linear FE on the GLC grid yields an algebraic system:
Ku=Mf,
L,,=iwk,
where K and M, respectively
Kij = Soj ,
i=O,
Kij = SNj,
i=N,
(2.7) stiffness and mass matrix, are given by
j=O ,..., j=O ,...,
N,
(2.8)
N,
and Mij=&.
%+aij hivllhi+l
IP-1 6
7
,N-1,
j=O ,...,
N, (2.9)
Mij=o,
i=OorN,
with hi=xi+l
-xi.
(2.10)
P. Francken et al., Spectrum
298
of the iteration operator for FEM preconditioning
It turns out that the stiffness matrix K and the second-order have similar structures. One has
operator
L,, on the same grid
P -Q=LDF’
(2.11)
P being a diagonal matrix whose non-zero elements Pii=$(hi_l+hi),
i=l,...,
are equal to
N-l, (2.12)
Pii=l,
i=OorN.
Putting together those various expressions yields the iteration operator of linear FE preconditioning L=L~~.L,::K-l*M*L,=Lo~.~.L
(2.13)
C’
with matrix J equal to P-’ * M. Let us now turn to the set of Chebyshev polynomials. of { Tj(x) 1j = 0, . . . , N} will be written ii(X>= ,$I0ujrj Introducing
the expansion
2 uj=
j
j=O
Any approximation
(2.14)
*
(2.14) into the boundary
$(u+ +u_)=O,
of U(X) in terms
i
uj=
conditions
$(u+ -u_)=O,
(2.2) yields (2.15)
j=o
j odd
even
where we have set U, = u(? 1). For the sake of convenience, the vector of unknowns (uo, , . Us)’ in (2.14) will be denoted by u’. The unknowns in the two sets of basis functions are related by the linear transformation (2.16) with (2.17)
A similar transformation exists between the ~ght-hand sides in both representations, which differs from (2.17) at the boundaries only. Withf’ = (fO, . . . , fN-2, $(u+ + u_), $(u+ - u_))‘, one has f’=
?.f,
(2.18)
P. Francken et al., Spectrum of the iteration operator for FEM preconditioning
299
with
T=
(2.19) 0
0
4 -
-1
2
0
.... .....
0
4
0
. . . . . . .. .
0
14
and 0
T-’
1 0
-1 0
.
=
. 0 0
1
(2.20)
,
0 1
where, for convenience, S represents the order N - 1 submatrix of S related to the inner nodes of the partition. It is now a simple matter of algebraic manipulations to derive the iteration operator L’ of linear FE preconditioning in the basis of Chebyshev polynomials. One gets L’ = (L;,)-‘J’L;
,
(2.21)
with LLF = TL,,S-’
,
L; = TL,S-’
,
J’ = TJT-’ .
The explicit forms of Lf , LLF and L&l can be found in [lo] and [3, eqs. (A.18)-(A.20)]. the inner nodes of the partition one has (iC)ij=-$
j(j’-i’)
(2.23)
otherwise ,
and
(L”;& = -ci_2
For
ifjai+2andi+jeven,
I (L”6)ij = 0
(2.22)
sin’ 7rl2N cos 7rl2N 4 sin(i - 1)7r/2N sin ivl2N
’
j=i-2,
300
P. Francken et al., Spectrum of the iteration operator for FEM preconditioning
(Lx&’ =
sin2?r/2N cos* 7~/2N 2 sin(i + l)lr/ZN sin(i - 1)73/2N ’
‘= i ’ (2.24)
(L”&’
= -
(L”;,);’
=0,
sin* 7~/2N cos 7~/2N 4 sin(i + 1)7r/2N sin i7r/2N ’
j=i+2,
otherwise .
The main problem consists in the evaluation of the matrix J’. First, one should remark that, because of the structure of the mass matrix M, J has rows identically equal to 0. Consequently, J’ given by (2.22) reduces to -1
J J’z
0
0 where J”‘, restriction
. . . . . . . . . . . .
. ..f........
0
(2.25)
,
01
of J to the inner nodes, is equal to (2.26)
s being the (N - 1) x (N + 1) matrix containin> the non-zero rows of J. Using the properties of the Chebyshev polynomials, one can split J’ into two terms (2.27) Wi .t:h
(2.28)
and 1 0
...... .,....
-1 0 (2.29)
0
0
1
-1
calculations (cf. Both matrices J”:, and i& may be evaluated after lengthy but str~ghtfo~ard whereas ?f,, is not. One obtains for J”&: ill]). It turns out that J& is also upper-triangular, (J”LrJij= 5 (2 f cos 2
- i tan & t
sin $)
,
i = j ,
P. Francken et al., Spectrum
(J”:,), = - 5 $ tan &
of the iteration operator for
sin ‘$ ,
FEM preconditioning
301
(2.30)
i + 2 S j S N - 2 , i+jeven,
I
(J”:,), = 0 ,
otherwise . term J”ior, the basic method of Haldenwang
If we discard the correction and the product
et al. remains valid
LLp= (L;JIJ:& ‘X
X
. . .
)(
. . .
x
\
1 0 0
0
. . .
. . * 0 0 IF 0
1d \ \
0
0 0
0
. =. . , .
. . . . .
x x
0
*‘-
-a-
0
0 * *
0 . . .
: 0 0
0
0 0 1 0 0 1
1 0 ...... ......
0!I
x x
60 is such that the eigenvalues
A, =
of L& are read along the main diagonal.
sin* 7rl2N cos 7r/2N sin(n - l)T/2N sin ml2N -
(n-2)7F , tan -Resin 2N I N
n(n - 1) 3
One gets
2 + cos (n k2)=
2
