ETNA
Electronic Transactions on Numerical Analysis.
Volume 26, pp. 228-242, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF
HIGH-ORDER FINITE ELEMENT METHODS
SANG DONG KIM
Abstract. Bounds on eigenvalues which are independent of both degrees of high-order elements and mesh sizes
are shown for the system preconditioned by bilinear elements for high-order finite element discretizations applied to
a model uniformly elliptic operator.
Key words. multigrid, high-order finite element methods, piecewise linear preconditioning
AMS subject classifications. 65F10, 65M30
1. Introduction. High-order finite element methods for discretizing
second-order uni a
formly elliptic partial differential equation lead to a linear equation
which requires efficient iterative methods, such as Schwarz-based methods (see [5, 14, 17]), preconditioning methods related to multilevel methods, multigrid methods (see [6, 7, 8]), etc. This
is because such linear systems have large condition numbers which depend on the order of
the elements used and the mesh spacing. In particular, an algebraic multigrid (AMG) method
is useful in the case of irregular grids. However it was reported that a direct application of
AMG to
is not so efficient; see [8, 16]. The convergence factor degrades rapidly
as the order of the elements is increased. For the case of Stokes and elasticity equations, the
complexity from the high-order finite element discretizations for AMG is even worse than
that of a simple elliptic partial differential equation.
In [6] and [8], a preconditioner constructed by using the Legendre-Gauss-Lobatto quadrature points in each cell as mesh points for a bilinear discretization. The finite element preconditioning can be approximately inverted by one AMG V-cycle. This approach has several
advantages, including the possibility to avoid assembly of the high-order stiffness matrix. Numerical results show that this preconditioning is very effective, especially when accelerated
by a conjugate gradient method. It also has the advantage of a straightforward matrix-free
implementation for the fine grid high-order element matrix. In this paper, it is proven that
the preconditioning yields a preconditioned system with condition number bounded independently of the mesh space and the polynomial order of the spectral elements.
In order to show that such a bilinear preconditioning is effective, we will consider a
uniformly elliptic boundary value problem, like
"!$#&%(')+*-,."!$#&%('/
(1.1)
with boundary conditions
(1.2)
:
<;
on
in
#A@B:
<;
=?>
0
12435#63'782435#639'
on
=
?"!?#%('
where =
is a strictly
= >DC = ,.Ewith
!$#&%.' a nonempty = > . Further, we assume that
positive function and
is a nonnegative smooth bounded function on 0 . The piecewise bilinear finite element preconditioner will be constructed by another uniformly elliptic
boundary operator F , like
FHG
9
G
*-I
G
in
0
43J#39'7K43J#39'
L Received April 3, 2006. Accepted for publication January 1, 2007. Recommended by T. Manteuffel. This work
was supported by KRF R02-2004-000-10109-0 (KOSEF C00006).
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea (skim@knu.ac.kr).
228
ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
229
with the same boundary
(1.2). This operator F yields a matrix OMF N to reduce the condition
number of a matrix
induced by high-order elements applied to (1.1)
For convenience, we assume throughout this paper that the Dirichlet part of the boundary
= is
(1.3)
P543JQR7TSU43J#636V C SU43J#3WVX7YP543JQ5Z
= >
'\[X] MF N
The main object of this article is to prove that the eigenvalues of O
are independent of the degrees of high-order elements and the mesh sizes. As a result the condition
numbers of the preconditioned systems are fixed and small, so that the complexity is no
longer a problem when the AMG algorithm is applied. This allow one to employ multigrid
algorithms for solving problems like (1.1) with high-order element discretizations, which
guarantees convergence of the strategy of preconditioning the high-order matrix with a bilinear or trilinear matrix based on Legendre-Gauss-Lobatto quadrature nodes well suited to
a solution by multigird methods. For a single spectral element, this kind of preconditioning
was analyzed for Legendre spectral collocation methods in [3, 12, 13, 15], for example.
The goal of this paper can be achieved by extending the results of [12] to high-order
] _
ba
elements and by applying ^ ,
estimates in [18] of a local interpolation operator ` c to
]
N a
a global interpolation operator ` c . Further, we note that such an ^
semi-norm estimate
of the local interpolation operator
defined
on
a
space
of
piecewise
linear
functions can be
]
extended to the space of ^
by modifying the relevant results in [1]. We also note that the
discussions here can be extended to singular value results for general elliptic operators which
are not positive definite. For this, one should refer to [12].
This paper is organized as follows. In the next Section, we recall some known results
on piecewise polynomial bases, interpolation operators, etc. In Section 3, we extend the
results in [12] and [18] which lead to one and two dimensional preconditioning results for the
constant coefficients case in Section 4. The variable coefficients case is dealt with in Section 5
using the tensor representation that appeared in [4]. In Section 6, we provide some numerical
results which support the theories developed. Finally, we mention some conclusions in the
last Section.
D!
%
2. Preliminary. With the
direction notation d
or , we assume that ef and ghi f are
P Qlk a
1Sr435#63WV
natural numbers. Let d&j jnmpo be the knots in the interval q
such that
43
) oHs ] s 6 s k a [X] s k a 35Z
d
d
d
d
a
a
Pt Q c
Pu Q c
Let j jnmXo and j jnmpo be the Legendre-Gauss-Lobatto (=:LGL) points in q
43
)vt oBs t ] s 6 s t ba [p] s t a w
3
c
c
arranged by
and its corresponding LGL weights respectively. Here ef denotes the number of subinter
xSr435#636V
N
vals of q
and gi f denotes the number of LGL points on a yzf subinterval by a
{
S
# V
N
translation of a q . a By the translation from q to a y f subinterval q i f
d i [p] d i we denote
| }P9~ i& Q ik ]   c
f j m jnpm o
as the €zf
(2.1)
~ i&f  i [p] ~ i&f  ] 6 ~ i&f  )l~ iƒf  b
a
s
s
s c a X[ ] s d i
d
o
c
enumerate them as
where
N LGL points in each subinterval qi f for y
3
~ ƒi f  …„ I if t j * I i [p] * i 'W#
d
d
j
„ if d i d i [p]
}3J#\I.#66p#
e‚f
and
ETNA
Kent State University
etna@mcs.kent.edu
230
S. D. KIM
(2.3)
~ i f [p]  # † if [p]  a <~ ƒi f  o Š
a <† &if  o # y DI(#66p# e f *D3JZ
c
c
a
P9† &i  Q c
and the corresponding LGL weights f j nj mpo are given by
† iƒf  „ I if u j #
35#ƒI.#6‡# f Z
(2.2)
y
e
j
~ a‰ˆ ]  k a
d , note that in (2.1) and (2.2)
With k f
o
'
dSU4defined
3J#3WV on q whose degrees are less than
consisting of piecewise polynomials
N a
basis for ‹ c is given by a piecewise Laof degree less than or equal to gŽi f in q5i f . a The
a

P? i [X]2‘ a ˆ 'nQ ik ]  c [p] ] C P o Q C P k a a Q ik a ]
P i a Q ik a ] [X]
grange polynomial basis
and
j
c
c
c m
m
m jnm
|
Let ‹Œj be the space of all polynomials j
N a
or equal to € and let ‹ c be the subspace of 
with respect to
which satisfy
‡’p"~9“J'”
•6’
 o E~6’('”
• o
 k a a E~6’('”
• k
c
 “z# where – y —39' g
 ’# – ;(#3J#66ž# g i f #
a a  ’ # – 1 e f ›3' g
c
i f * € #‡˜™
1"š?›3' g i f *œ,z#
i f #66X# e f g i f #
and
 i
a E~6’('”
D• i a  ’ # y 35#ƒI.#6‡# e f —3J#4 y —39' g i)
f Ÿ – Ÿ y *<3' g i f
c
c
35#ƒI.#6p# i 13J(# ,D
x;(#3J#66ž# i
š\# ¡3J#66
•
where €
ghf
gf and y
e}f and denotes the
Kronecker delta function. Even though this Lagrangian basis is actually rather easy to work
P¢“. 'nQ
with providing preconditioning results, one may prefer another basis
d ’v. “BThe
¤¢¥change
“¦"~6’('
of basis may be accomplished algebraically by use of the matrix £ given by £
.
N a
Then if ^ § is any matrix representation of an operator ^
‹ cO¨ ‹ bN c a given in terms of the
P9¢’‡Q
representations via the
P‡’‡Q
basis then
[X] the matrix representation ^ © of ^ in the Lagrangian
basis
is given by ^ ©
£ ^h§ £ . Hence it may be enough for us to use the above
piecewise Lagrangian basis functions in this paper. For two dimensional high-order space, let
S N V _ ŠN ª
N ¬ #
‹
‹ )
c « ‹ c
whose basis functions are given by tensor products of one dimensional piecewise Lagrange
® E!‡'
a
polynomials. Let
with
a ­ c be the space of all piecewise Lagrange linear functions § j
P9t Q c
N a as the space of all piecewise Lagrange linear functions
j a jnmpo on q . Define ­ c
a

|
P ® iƒ \' Q ik ]  c
f j d m jnmpo with respect to . For two dimensional piecewise linear space, let
S N V _ ŠN ª
#
­
­ c « ­ N c¬
respect to
whose basis functions are given by tensor products of one dimensional piecewise Lagrange
linear functions.
a SU43J#3WV ‹ a q ' such that
Define two interpolation operators ` c 
c
¨
ž'6"t '”
<¥"t 'W#A¯ Sr435#636V
` ca
j
j

ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
'
q such that
N a G 'W"~ i&f  j '”
G "~ iƒf  j 'n# G ¯
c
?# SU43J#3WVX7

Define a discrete inner product ± Gz² on 
a [p]
a
c
k
# ³ ³ ¥"~ iƒf  ' "~ i&f  '2† iƒf  *T¥"~
± G.²
j G j j
i m ] jWmpo
and `
N a  Sr435#636V ‹ N a
¨
c
c
`
231
SU43J#3WV°Z

rS 435#636V
as
kf a  c a ' G " ~ kf a  c a '2† fk a  c a
and its corresponding norm is given by
´ ´ ±
# #
²5µ
for
h¯

SU43J#3WV°Z
Finally, the notation ¶Y·¹¸ for any two real quantities ¶ and ¸ means by that there are
two positive constants which do not depend on mesh sizes and degrees of polynomials such
;
Z
Using this notation, the LGL numerical quadrature rule for a
that s»º Ÿ½¼ ¾ s  s»¿
I i
polynomial of degree ghf (see [1], for example) can be compared as
a
] _
³ c _ "t '/Ã #
¯
Z
'.Â
À X[ ]pÁ d d·
j j for all Á ‹ c a
(2.4)
Á
jnpm o
‰Š#ƒÄB'
‡Æ Æ
Ç
{E ] #66p#pÈ9'2É
Ä{
G for any two vectors
The
stands for Å
and
] #6notation
6p# È ' É
G ]
G _ where the superscript Ê denotes the transpose of a vector. The standard spaces
^ and will be used.
3. Basic estimates. In this section, _ we will discuss some estimates of global interpola]
¯}Sr435#636V
¯}S i [X] # i V
N a
d
d and a
tion operator ` c in terms of ^
and
norms. For d
and Ë
f
h¯ ]
function
^ , let
3
'w
D¥ '
̌͞„ I if * I i [p] * i 'Î¥Z
(3.1)
Gi d
Ëf
d
d
d
<;¦#635#66p# i f
g we have
Then for €
a i 'WEt '”
i "t '
Ì Í „ if t * 3 i [p] * i ' Î < E~ i&f  '
N a ž'6"~ iƒf  'n#
(3.2)
` cG
j
G j
I j I d
d
` c
j
j
which yields
Also, we have
(3.3)
and
(3.4)
6' '
1 ž'6 n' Z
` ca G i d
` N ca Ë f
k
]
_ Â * I ] ' _ Â
´ ´ _] ³ „ if
'
i
IÀ X[ ]Ð G d Ð d
À Gzi Ñ d Ð d°Ò
i m ]pÏ
„ if [p]Ð
k
]
if ] ´ W' ' _ Â * I
a
aG i 'Ñd' _ Â dÒ #
` N a ´ _] ³ „ I
i
À X[ ] Ð ` c G d Ð d
À
`
c
c
Ð
i m ]žÏ
„ if [p] Ð
ETNA
Kent State University
etna@mcs.kent.edu
232
S. D. KIM
´ ´
]
]
where
´ ´ Sobolev ^ _ norm. From now on, we will use Ð Ð ] as the
] denotes the standard
Sobolev ^
seminorm and
as the usual
norm. In order to discuss the piecewise linear

ba 
finite elements preconditioner,
it
may
be
required
to analyze the relations between ` c § and §
_
]l
in the sense of ^
and
norm. For this purpose, we recall the following lemma (see [1],
[18, Lemma 7.2]) which can be also extended to higher dimensions; see in [18, Theorem 7.3].
We remark that the similar estimates with Chebyshev weight and Chebyshev-Gauss-Lobatto
points are found in [1] and [11].
¯ a
L EMMA 3.1. It follows that for all G
­ c
a ]#
b
]
Ð G Ð · Ð` c G Ð
(3.5)
and
´ ´ ´ ´Z
` ca G · G
¯
a
]l '
^ q
Note that the result ` c G ] Ÿ  G ] in (3.5) can be verified for any function G
Ð
Ð
Ð Ð
by modifying Theorem 1.7, Corollary 1.9 in Chapter II and Corollary 1.16, Theorem 1.19 in
Chapter III with usages
´ ofa Theorem
´
´ 1.15,
´ Lemma 1.18 and Proposition 1.17 in Chapter III
therein in [1] where ` c G ] Ÿ  G ] is found. For reader’s convenience, we include the
statement here.
¯ ] '
P ROPOSITION 3.2. For all G
^ q , there is a positive constant  such that
aG ] Ÿ  G ] Z
`Ð c Ð
Ð Ð
bN a
Now the extension of Lemma 3.1 to the global interpolation operator ` c can be done
easily by combining (3.3), (3.4)
Lemma 3.1. Here we state it as theorem.
¯ with
­ bN c a
T HEOREM 3.3. For all
N a ] ] Z
Ð` c Ð · Ð Ð
L EMMA 3.4. For
Á
¯
N a SU43J#3WV , it follows that
‹ c
´ ´ ´ ´ Z
Á · Á
'
Proof. First note that d is a polynomial of degree
S
# Á V
(2.4) on the interval d i [X] d i and using (2.3) yield
(3.6)
k
´ ´_ ³
Á
im
³k
·
im
k
³
im
·± Á
g if
on
a
S i [p] # i V
d
d .
Then applying
fc ' _ Â
À /c Ó Ð Á d Ð d
]
a f ca µ
³
E~ i&f  ' _ † i&f 
j Ð j
] jnmpo Ð Á
a c a [X]
k a [p]
³
"~ iƒf  ' _ † i&f  * E ~ kf a  a ' _ † fk a  a * ³
" ~ i&f  a ' _ † iƒf  a
j
j
c
c
c Ð
c
ÐÁ
Ð
i m ] ÐÁ
] jnmpo Ð Á _ Ð
# ´ ´ Z
Á ²
Á
In the last equivalence in (3.6), the observations (2.3) were used.
ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
233
Ž¯
N a , we have
­ c
´ ´ ´ N a ´ #
´ ´ ´ N a ´ Z
· ` c
· ` c
and
T HEOREM 3.5. For all
(3.7)
N a <¯ ‹ N a SU43J#3WV , Lemma 3.4 yields ´ ` N a ´ · ´ ` N a ´ Z Hence for
c
c
c
c
¯ N a
the proof of (3.7) it is enough to show that for
­ c
´ ´ ´ bN a ´ Z
(3.8)
· ` c
¯ ba
h¯ N 
­ c and
­ c f , we have
Using (3.1) and (3.2) for functions G i
a
_
k³ a i
k³ a i
k³ a ³ c
_
_
_
´ ´ ´ i´ #
´ a i ´ N a p'WE~ i ' † i Z
„
„
I G
I ` cG
`
(3.9)
j
j
im ]
im ]
i m ] jnmXo5Ô c
ÔÔ
ÔÔ
Ô
Proof. Since `
Since (see [12, Theorem 3.1])
´ i´ ´ ´ #
G · ` ca G i
it follows that, using (3.9),
´ ´_
Therefore, it is enough to show
a
k³ a ³ c
·
i m ] jnmpo
for all
¯ a#
Gi ­ b
c
_
b
` N a ž'6"~ i ' † i Z
j
j
c
ÔÔ
ÔÔ
Ô
Ô
a ca
_
³ ` N a p'W"~ i ' † i · ´ ` bN a ´ _ Z
j
j
c
i m ] jnmpozÔ c
Ô
ÔÔ
ÔÔ
³k
(3.10)
Actually, because of (2.3) we can rewrite the left term of (3.10) as
a
a ca
_
_
_
k a ³ c [X]
³ b
i
i
³
N a p'WE~ i ' † i * bN a ž'6"~ k a ' † k a
` N a ž'6"~ ' † `
` c
j
j
j
j i
j
j
i m ] jnmpozÔ c
] jnmpoÕÔ c
m
Ô
Ô
Ô
Ô
ÔÔ
ÔÔ
Ô_
Ô
ÔÔ
k a [X] ÔÔ
i a ' Ô † ia Ô
* ³
N a p'WE~ `
c
c
im ] Ô c
ÔÔ
a
ÔÔ
_
k [X] Ô
i a ' † ia
´ N a ´ _ba * ³
N a p'WE~ ` c
c i ] ` c
c
c
m a ÔÔ
ÔÔ
_ Ô
k
Ô
´ N a ´ _ba * ³ bN a ž'6"~ i ' † i Z
` c
o
o
c i _ ` c
m ÔÔ
ÔÔ
Ô
Ô
³k
Then one may see that (3.10) holds. These arguments complete the proof of (3.8) and consequently (3.7).
ETNA
Kent State University
etna@mcs.kent.edu
234
S. D. KIM
39Ö
IvÖ
and
we discuss the one dimensional
4. Case of constant coefficients:
w
w
. First,
ef and g i
gi f for the one dimensional
case. For convenience, let e
243J#39' case only.
Consider two uniformly positive definite elliptic operators defined in q
such that
(4.1)
”×
H ] Ñ ' Ñ * _p #
¶
¶
in
H ] ' * _ #
¸ GzÑ Ñ ¸ G
in
and
(4.2)
#
FHG
#Ø¥2439'”
< Ñ 2 3'”
D;
q
#
q
G
2439'”
GzÑ
39'”
;
_#¸_
nonnegative constants, leading to two bil] are
'W#A
¥2439'
D 23'”
<;(Q
^ q
,
Ñ
]
]
Ú ] " # '”
* Â
" # '”
* Â Z
G
À p[ ] ¶ ] Ñ G Ñ ¶ _ G d and ¸ ] G
À [p] ¸ ] Ñ G Ñ ¸ _ G d
where ¶ ] ¸ ] are positive constants and ¶
7
P¯
Ù , where Ù
linear forms on Ù
For the high-order and piecewise linear approximations to (4.1) and (4.2), let
 Û P ¯
‹ N c
G
­ N c Û P¯
Pl ’ Q ’È
whose suitable basis functions
m
# 439'”
23'
<;(Q5#
‹ N c G
GÑ
­ N c #Ø¥ 243'
< Ñ 39'”
D;.Q
È
] and P ® “ Q “ m ] , respectively, can be given, with
ÂÕ
N  Û '
N  Û 'nZ
(4.3)
dim ‹ c
dim ­ c
|
Then the stiffness matrix M
with high-order elements based on of (4.1) is given by
#\˜('
Ú ] ܏‡’ž#\‡“5'n# #\˜Ý
}3J#\I.#66p#ƒÂ¦#
M –
–
|
and the stiffness matrix MF N associated with piecewise linear elements based on corresponding to (4.2) is given by
#ƒ˜¦'
] ® ’ # ® “ 'W#
MF N –
¸
#ƒ˜Õ
3J#\I.#66ž#&Z
P ’ Q ’È ]
P ® ’ Q ’È ]
Denote e Þ
and e Þ N by mass matrices with respect to
and
m
m , respectively.
#ƒ˜Õ
35#ƒI(#66ž#&#
That is, –
#ƒ˜('
܏‡’ž#\‡“5'n#
#\˜('
® ’ž# ® “J'WZ
eÞ –
eÞ N –
–
Since all the stiffness and mass matrices are symmetric and positive definite, the preconditioned matrix below also has all positive real eigenvalues.
‚
1" ] #&‡_J#66p# È ' É
T HEOREM 4.1. For every
, we have
N Š#\4' #n4'n#
#n4' #\H'nZ
MF
· M
· eÞ
and e Þ N
[p] Pߞ’‡Q ’È ]
Hence, the preconditioned matrix MF N M
has all positive real eigenvalues
m independent of mesh sizes „ i and degrees g i of polynomials. That is, there is absolute positive
constants º and  such that
; s›º Ÿ ¦ß ’ Ÿ
Z
 sD¿
ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
)
' ¯ N  Û
d
­ c
'4
d
Å
N p'W 'R
d
Å
polynomial interpolation can be written as ` c
Proof. Let
be represented as
bilinear forms yield that
È
235
’ m ] ’ ® ’ d ' . Then its piecewise
È
’ m ] ’  ’ d ' . The definitions of
#\H'¥
Ú ] N c # N c ž ' ´ N c ´ _] à
# #n4'
] " #p' ´ ´ _] #
M
`
`
· `
MF
¸
·
and
Š#\4'
1 N c ? # N c p'
´ N c ´ _ #
eÞ
`
`
`
and
N Š#\4'
1" #p'
´ ´ _ Z
eÞ
Then using Theorem 3.3 and 3.5 completes the proofs.
Ú ] N ? # N '
N # N '
`
`
G
and ` c ` c G
c
c
calculated at LGL points. Define two matrices
and e á
as
#ƒ˜('
Ú  ܏ ’ #\ “ 'n#
#\˜('
 ’ ƒ#  “ #
]
–
eá –
±
²
For actual computations, the bilinear form
where
Note that e á
will be
Ú ]  " # '
] # * _ ?# Z
G
¶ ± Ñ GzÑâ²
¶ ± Gz²
is the diagonal matrix which consists of LGL weights, that is
eá
diag
E† ƒif  '
j
and employing (2.4) leads to
Š
#\4' #n4'n#
#n4' Š#\4'
M
·
· eá
and e Þ
and these matrices
and e á
are symmetric and positive
[p] definite.
CÈ OROLLARY 4.2. The preconditioned matrix MF N
has all positive real eigenvalues
Pß ’ Q ’ ]
i
i
g
independent
of
mesh
sizes
and
degrees
of
polynomials. That is, there are
m
„
absolute positive constants º and  such that
; s›º Ÿ ߦ’ Ÿ
Z
 sD¿
(4.4)
Proof. Let
}
E ] &# _ #6X#pÈ'É
be any nonzero vector. Since
Š#\4' #n4' #\H'
#
M
N #n4'
#n4' N Š#\4'
MF
M
MF
using the min-max theorem, Theorem 4.1 and (4.4) we have the conclusion. This argument
completes the proof because all involved matrices are symmetric and positive definite.
ator
We now turn to the two dimensional case. For this, we consider the model elliptic oper
such that
where =
(4.5)
>
”×
}HS žãã*œ‡ä6ä6V.*-Iv?#åæ
D;
on
= >
#A@Ž)æ
<;
on
is the boundary described in (1.3), which leads to the bilinear form
Ú E?# ”' Ü)?#\ '$*›I(E?# 'W#
G
G
G
for
?# ¯ ] 'W#
G ^ > 0
=
#
ETNA
Kent State University
etna@mcs.kent.edu
236
S. D. KIM
where
Let
] 'bw
P¯ ] ' ×
D;
^ > 0
^ 0 Ð
Q5Z
= >
on
 Û #àS N  Û V _ ŠN  ª Û
Û Z
S N  Û V _ ŠN  ª Û
‹
‹ c « ‹ N c¬
­
­ c « ­ N c¬
P9çèbQ èÈ ]
m
#\˜Õ
35#ƒI.#6‡#ƒÂ¦#
Let us order the LGL points by horizontal lines and we list all LGL points
çè8
1"~6’ž#&~“J'W#
where
é
–
*-‡ܘ+—39'n#
–
as
Â
èŠ"!?#%('¯TS N  Û V _
is defined
in
(4.3).
Accordingly,
we
order
the
basis
vectors
and
ê
‹
_
ëRèŠ"!?#%('B¯S N  Û V
M ì ì
MF
in the
same
order.
Let
and
be
the
stiffness
matrices
induced
­
N
S N  Û V _
S N  Û V _
by (4.5) on the space ‹
and ­
, respectively. From now on, assume that
Æ Æ }3J#ؚ¥
35#ƒI
¶
¸
in the operators ] and F ] in (4.1) and (4.2). Then using the one dimensional stiffness
a
a
a , e Þ a , we have
matrices M c , MF N c and mass matrices e Þ
c Nc
Šª #
ì b¬ Šª * b¬
(4.6)
M
eÞ c « M c M c « eÞ c
ì ª *
ªZ
(4.7)
MF N e Þ N c¬ « MF N c MF N c¬ « e Þ N c
¤
1" ] #66X# È '2É
where
L EMMA 4.3. For every vector
(4.8)
and
(4.9)
, we have
Í e Þ ¬ M ª '&#n Î · Í e Þ ¬ MF N ª '&Š#\ Î
c
Nc «
c
c «
ª 'ƒ#\ Î
ª '&Š#\ Î Z
Í M ¬ e Þ Š
· Í MF N c¬ « e Þ N c
c
c «
Proof. First note that all the matrices here are symmetric and positive definite. Hence it is
enough to estimate (4.8) and (4.9) in terms of eigenvalues. Now because of Theorem 4.1 the
conclusions (4.8)
and
B]O
»(4.9)
E ] # can
6X#&verified
žÈ9'2É byÄ)following
]O
í ] # Lemma
#6p# 5.4
È'2É in [12]. The details are
_ #6be
G G_
G . Theorem 4.1 implies
as follows: Let
and
that
(4.10)
M a ] #n ] ' MF a ] #n ] 'W#$ Þ Ä ] #\Ä ] ' Þ a Ä ] #\Ä ] 'W#
·
e ca
· e Nc
Nc
c
Now consider eigenvalue problems
(4.11)
] î a ]#
M ca
MF N c
î
ß
and
a Ä ] ß e Þ a Ä ] Z
eÞ Nc
c
where
d
<!$#&%‡Z
ã # ä
From (4.10)
and
ã # we ä know that and are uniformly bounded in terms of mesh sizesB„ ’ ] i „ i Ä)
“]
i
i
degrees g
#ƒ˜Õ
3J#66pg#ƒÂ . Note that each in (4.11) has a complete set of eigenvectors and ,
–
. Therefore the vectors and eigenvalues
ï’v“4
¤ ’ ]
«
Ä “ ] #ØðՒv“H
Ä “ ]
«
’ ] #Añb’v“O
¤î’z߇“
ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
237
are complete set of eigenvectors and eigenvalues of the eigenvalue problem
ª 'ƒ}
DñR ¬
Šª 'ƒ‚
ñR ¬
¬ Š
'&Š#
¬
'&ŠZ
eÞ c « M c
e Þ N « MF N ª
MF N « e Þ N ª
and M c « e Þ
c
ñŒ’v“
ã # ä of eigenvaluesã # ä because of the uniform bounds of
Hence
î
ß one can see the uniform bounds
and in terms of mesh sizes „ i „ i and degrees g i g i .
}
E ] #66X#& È 'É
P ROPOSITION 4.4. For every
, it follows that
ì Š#\4' ì #\H'nZ
MF N
· M
ì '\[X] bì ä bounded. The bounds are indeHence the eigenvalues of ã MF N ä M
are all positive
ã # and
#
pendent of the mesh sizes „ i „ i and the degrees g i g i of polynomials.
Proof. It follows from Lemma 4.3 with (4.6) and (4.7).
bì For actual computations of (4.5), that is, for computations of M
, we use LGL quadrature formula. For this, consider
Ú E?# ”' ) ƒ# *
G
±
.G ²
_

Û
# ¯8S N V
which can be written as, for G
‹
É
Ú E?# '”
¤Ä ¬ ª *
G
eá c «
c
»
E ] #66X# È '2É
Ä
]
G
where the vectors
and
# #
± .G ²
ª 'ƒ
¬
c « eá c
#66p# È '2É
G
are vector representations
È
³
E!$#%('”
è èŠ"!?#%('nZ
G
è m ]G ê
of
È
³
¥"!?#%('”
ž è èŠ"!?#%('
and
è m ] ê
ì
Now we will use the matrix MF N in (4.7) as the preconditioner for
ì ‚
where
ª * ¬
Šª Z
ì w ¬ Š
eá c «
c
c « eá c
ì '\[p] ì ä
ä
MF N
ã
Then weã can
show
that
the
eigenvalues
of
are bounded well in terms of
#
#
mesh sizes „ i „ i and degrees g i g i .
‚
1" ] #6p# È 6'2É
L EMMA 4.5. For every vector
, we have
ª 'ƒ#n4'
& ¬ Šª '&Š#\4' ¬
eá c «
· e Þ N c « MF N c
c
(4.12)
and
ª 'ƒ#n4' & ¬
ª '&Š#\4'WZ
¬
· MF N c « e Þ N c
c « eá c
Proof. Recall that ò « F is symmetric and positive definite if the matrices ò and F
symmetric, positive definite. Note that
& eá c¬ «
¬
e Þ N c « MF
ª '&#n4' & ¬
c
eá c «
& ¬
N cª 'ƒ#\H'
eÞ c «
M
ª &' Š#\4' b
eÞ c¬ « M
ª '&Š#\4' ¬
e Þ N c « MF
c
c
ª '&Š#\4'
c
#
ªN '&#n4'
c
are
ETNA
Kent State University
etna@mcs.kent.edu
238
and
S. D. KIM
& b
¬ e á ª '&Š#\4' c
c «
ªN ƒ' #n4'
¬
MF N c « e Þ c
M
¬ e á ª &' Š#\4' M
c
c «
¬ e Þ ª &' Š#\4' & MF
c
c «
¬ e Þ ª 'ƒ#\H'
c
c «
Z
¬ e Þ N ª &' Š#\4'
c
Nc «
Therefore, due to min-max theorem and Lemma 4.3, it is enough to show that
Šª 'ƒ#\H'
¬ ª ƒ' #n4'
¬
eá c «
c
c « eá c
#
Šª 'ƒ#\H'
¬ ª ƒ' #n4' and ¬
eÞ c « M c
M c « eÞ c
ã # ä
ã# ä
are bounded independently of degrees g i g i and mesh sizes „ i „ i . Due to (4.4), this can
be done by following the similar arguments of Lemma 4.3. These arguments complete the
proof.
‚
E ] #66p#& È 6'É
T HEOREM 4.6. For every vector
, it follows that
(4.13)
ì
ì Š#\4' ì #n4'nZ
·
MF N
'nä [p] ì ã ä and bounded.
are all positive
i and the degrees g i g i .
Hence the eigenvalues of F ã § N
#
pendent of the mesh sizes „ i „
Proof. It comes from (4.12) and (4.7) and Lemma 4.5
IvÖ
5. Case of variable coefficients:
and (1.2) as
The bounds are inde-
. Consider the bilinear form corresponding to (1.1)
Úó " # ”' "!?#%(') #ƒ ' *DÜ,."!?#%('/ # '
?# ¯ ] 'nZ
G
G
G for G ^ > 0
Ö a
Let us denote
c by the differentiation matrix at LGL points (see [4] for example) and
diagonal matrices é and ô by
žõ:w
T ’v“ 'n# Ü,õw
, ’v“ 'W#÷öY
*-‡ܘ+—39'
ô
–
é
diag
diag
?"!?#%('
,.E!$#&%.'
IvÖ
occurred from the expansions of the
of
"!?#%('
’v“  ’ E!‡'& “ and
"%('
,."!?#in%('øterms
, ’v“  ’ "Lagrangian
!ž' “ E%(' basis
Å
Å
functions such that
and
, respec_
tively.
Then
we
will
use
the
tensor
representation
for
the
approximations
to
(5.1)
on
the
space

Û
S N V
‹
given by (see detailed discussions in [4])
É ª ' è Üù ¬ Ö ª '$*D Ö É ¬ ù ª ' è Ö ¬ ù ª '$* R
Eù ¬ Ö áú}û
c
c áú
c
c «
c «
c «
c áú
c «
ù a
Â
where
c is the identity matrix of order
ª 'n#
w
ýŠ ¬
ýT
Z
where
áú ü
eá c « eá c
é or ô
(5.1)
Note that the matrix úá
ýŠ°
# '
é ô and e á c ¬
ü
« eá
is diagonal whose elements are positive because the matrices
ª'
c are diagonal with positive elements. Further, if the ma-
trix é is the identity and the matrix
(4.12) with
ô
is
IJù
, then the matrix
a Ö a Ö a#
É a e á c
c c
c
d
<!?#%žZ
is the same as
ì in
ETNA
Kent State University
etna@mcs.kent.edu
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
}
" ] 6# 6X#
' è Ü ù ¬ Ö Šª ' Üù ¬
áú
c ·
c «
c «
É ¬
' è Ö ¬ ù ª ' Ö úá
c ·
c «
c «
I¦ ¬
ú}
á û · eá c
L EMMA 5.1. For any vector
Eù b
¬ Ö
c «
Ö É ¬ ù
(5.3)
c «
(5.2)
(5.4)
É ª
c
ª
c
where the matrix equivalence ò¤·<F
239
È 2' É , it follows that
É ª 'W ¬
Šª W' Üù ¬ Ö Šª '
Ö Š
c
c «
c eá c « eá c
ª 'W Ö ¬ ù ª '
ù ª 'W ¬
c eá c « eá c
c
c «
Šª 'W#
« eá c
should be understood as
#n4' Š#\4'WZ
ò
· F
Proof. Note that the variable coefficients
¬
e
tions and the matrices úá ü ¬þ and e á
á
c
«
for any vector it follows immediately that
" !?#%('
,."!?#%('
and
are positive bounded funcª'
c are diagonal with positive elements. Hence
ý # Z
Í áú ü Š#\ Î · Í e á b
¬ e á ª &' Š#\ Î # where T
é ô
c
c «
ª 'ƒ
Ä
1Eù ¬ Ö This argument complete (5.4). Let
c . Since
c «
è
É ª ' Üù ¬ Ö ª '&Š#\ Î Í è Ä #ƒÄ Î #
Í Üù ¬ Ö Š
áú
c
c «
c «
c áú
(5.5)
we have the conclusion (5.2) with help of (5.5). The similar arguments complete (5.3).
main goal of this section is to show that the eigenvalues of the preconditioned matrix
The
ì ' [X] MF N
are real and bounded as follows.
}
E
T HEOREM 5.2. For any vector
] #66X#& È 'É , we have
ì Š#\4' #\H'nZ
MF N
·
In the sense of eigenvalues, it follows that all eigenvalues of the matrix ã
positive
ã # ä and bounded. The bounds are independent of the mesh sizes „ i
i
g g i of piecewise polynomials.
Proof. Note that
ì ä '\[p] are real
#MF N i
„ and the degrees
Eù b¬ Ö ¬ e á ª 'WÜù ¬ Ö ª '”
e á b
¬ ¬
É ª W' e á b
c
c
c «
c «
c «
c «
c
c
and
É ¬ ù Š
ª 6' ¬
ª '6 Ö ¬ ù Šª '”
¬
Ö ¬Z
c eá c « eá c
c
c «
c « eá c
c «
Therefore, using Lemma 5.1 one may see that
b9#\H'
From Theorem 4.6,
·
ì #n4'nZ
ì #n4' MF ì #\H'nZ
·
N
These arguments complete the proof.
ETNA
Kent State University
etna@mcs.kent.edu
240
S. D. KIM
35
30
25
diamond : n=6, h=1/5
20
Condion numbers
15
plus : n=10, h=1/3
10
5
star : n=4, h=1/2
0
−5
−10
−15
−0.6
−0.4
−0.2
0
gamma
F IG . 6.1. Condition numbers of
0.2
ÿ 0.4
0.6
0.8
according to
6. Computational results. Because there are numerical results reported already in [6]
and [8], we discuss a few numerical experiments for one dimensional problems. Consider
(6.1)
with boundary conditions
(6.2)
æ
}Hr?E!‡'/ ' *-,."!‡'2?#Ø!Ž¯KÜ;(#63'
Ñ Ñ
¥E;'”
<¥23'
D;
F
. Let us take the preconditioner operator F
×
Š Ñ Ñ * ‡ #å!h¯8E;(#39'
as
ã In
with the same homogeneous Dirichlet boundary conditions
iã where will be chosen
later.
i for
g
g
ã
the
following
numeric
tests,
the
uniform
mesh
size
and
uniform
degree
„
„
3J#6p#
y
e are used.
?E!‡'4
3Œ*Ì! _
,.E!‡'H
ã
E XAMPLE 6.1. With a chosen
and
(6.1), we discuss the
. Weinwill
T
€ 7Ž;(Zó3
optimal preconditioning operator
(6.2)
by
considering
several
take
where € is an integer between
[p] and . The Fig. 6.1 shows that the condition
; number of the
preconditioned matrix MF N
becomes small relatively if we choose near among other
’s. These phenomena were pointed out in [9] if (6.1) is discretized by a finite difference
scheme, that is to say, may be chosen by considering the advection coefficient in (6.1).
D;
Hence one may choose
in this case.
¤;
E XAMPLE 6.2. Stimulated by the numerical results in Example 6.1, we will take
?E!‡' ,."!ž'×
{3
for the preconditioner operator
in (6.1) are taken. The condi F while
tion numbers of the matrix §
are shown in Fig. 6.2 for various mesh sizes and degrees
[p] of
polynomials. Then we enumerate condition numbers of the preconditioned matrix MF N
in
Fig. 6.2 also, which shows that the condition numbers can be fixed for mesh
sizes and degrees
_[
of polynomials. We
point
' out that the condition numbers grow as „ for a fixed degree of
polynomial and g for a fixed mesh size. These also can be proven in a standard finite
element theory and in spectral methods; see [1]. These phenomena are depicted in Fig. 6.3.
7. Conclusion. As we have shown that the condition numbers of the preconditioned
systems are well bounded, it is possible to use AMG algorithms without increasing complexity for solving elliptic boundary value problems with high-order discretizations based on
LGL quadrature nodes. As an immediate application, one may apply the techniques here
to a first-order system of least-squares for an elliptic problem whose systems are found in
[2], combining the results here and in [12] (for example, one may refer to [10]). In order to
avoid the non-nested property of irregular LGL nodes occurring in the multigrid method, one
ETNA
Kent State University
etna@mcs.kent.edu
241
PIECEWISE BILINEAR PRECONDITIONING OF HIGH-ORDER FINITE ELEMENT METHODS
1D conditionnumber of high−order elements
6
x 10
4
3.5
2.5
2.4
Number of subintervals
plus + : 2
2.3
star * : 4
3
condition numbers
condition number
circle o : 8
cross x : 16
2.5
diamond : 32
2
1.5
1
2.1
2
Number of subintervals
plus + : 2
star * : 4
circle o : 8
cross x : 16
diamond : 32
1.9
1.8
0.5
0
2.2
1.7
0
5
10
15
20
25
deg of polynomial
30
35
1.6
40
F IG . 6.2. Condition numbers of
change of ratio of condition numbers for a meshsize
7
0
5
(left) and
10
15
20
25
deg of polynomials
ÿ 30
35
40
(right).
change of ratio of condition numbers for a fixed degree
4.2
degree of polynomial
6
star * : 8
change of ratio of condition numbers
1/2
3
1−d ratio of condition number O(n )
diamond :
star: 1/4
5
plus: 1/8
cross: 1/16
4
circle : 1/32
3
2
1
plus + : 4
4.15
meshsize
circle o : 16
4.1
cross x : 24
diamond : 32
4.05
4
3.95
5
10
15
20
25
deg of polynomial (n)
30
35
40
3.9
0
0.05
F IG . 6.3. Increasing rates of condition numbers of
0.1
mesh size
0.15
0.2
0.25
.
may use Chebyshev-Gauss-Lobatto nodes for discretizations which has a nested property in
communications at each level. This case will be dealt in a forthcoming paper.
@ "!$# %'&# ()#z@
*JZ
The author deeply thanks Professor Manteuffel for his kind guidance in preparing the present topic and also thanks to anonymous referees for pointing out
some corrections.
REFERENCES
[1] C. B ERNARDI AND Y. M ADAY , Approximation Spectrales de Problémes aux Limites Elliptiques, Springer,
Paris, 1992.
[2] Z. C AI , T. M ANTEUFFEL , AND S. M C C ORMICK , First-order system least squares for second-order partial
differential equations: Part II, SIAM J. Numer. Anal., 34 (1997), pp. 425–454.
[3] M. O. D EVILLE AND E. H. M UND , Finite element preconditioning for pseudospectral solutions of elliptic
problems, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 311–342.
[4] M. O. D EVILLE , P. F. F ISCHER , AND E. H. M UND , High-order Methods for Incompressible Fluid Flow,
Cambridge Monographs on Applied and Computational Mathematics, Cambridge Univ. Press, Cambridge, UK, 2002.
[5] M. D RYJA , B. S MITH , AND O. W IDLUND , Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 1662–1694.
[6] P. F ISCHER , An overlapping Schwarz method for spectral element solution of the incompreesible NavierStokes equations, J. Comput. Phys., 133 (1997), pp. 84–101.
[7] P. F ISHER AND E. R ONQUIST , Spectral element methods for large-scale parallel Navier-Stokes calculations,
Comput. Method Appl. Math., 116 (1994), pp. 69–76.
[8] J. H EYS , T. M ANTEUFFEL , S. M C C ORMICK AND L. O LSON , Algebraic multigrid(AMG) for high-order
ETNA
Kent State University
etna@mcs.kent.edu
242
S. D. KIM
finite elements, J. Comput. Phys., 204 (2005), pp. 520–532.
[9] T. M ANTEUFFEL AND J. O TTO , Optimal equivalent preconditioners, SIAM J. Numer. Anal., 30 (1993),
pp. 790–812.
[10] S. D. K IM , H.-C. L EE , AND B. C. S HIN , Pseudospectral least-squares method for the second-order elliptic
boundary value problem, SIAM J. Numer. Anal., 41 (2003), pp. 1370–1387.
[11] S. D. K IM AND S. PARTER , Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations, SIAM J. Numer. Anal., 33 (1996), pp. 2375–2400.
[12] S. PARTER AND E. R OTHMAN , Preconditioning Legendre spectral collocation approximations to elliptic
problems, SIAM J. Numer. Anal., 32 (1995), pp. 333–385.
[13] S. PARTER , Preconditioning Legendre spectral collocation methods for elliptic problems I: Finte element
operators, SIAM J. Numer. Anal., 39 (2001), pp. 348–362.
[14] L. PAVARINO AND O. W IDLUND , A polylogorithmic bound for an iterative substructuring method for spectral element in three dimensions, SIAM J. Numer. Anal., 33 (1996), pp. 1303–1335.
[15] A. Q UARTERONI AND E. Z AMPIERI , Finite element preconditioning for Legendre spectral collocation approximations to elliptic equations and systems, SIAM J. Numer. Anal., 29 (1992), pp. 917–936.
[16] J. R UGE , AMG for high-order discretiztions of second-order elliptic problems, in: Abstract of the Eleventh
Copper Mountain Conference on Multigrid Methods (http://www.mgnet.org/mgnet/Conference /CopperMtn03/Talks/ruge.pdf), 2003.
[17] A. S T-C YR , M. G ANDER , AND S. T HOMAS , On optimized Schwarz preconditioning for high-order spectral
methods, Twelfth Copper Mountain Conference on Multigrid Methods, 2005.
[18] A. T OSELLI AND O. W IDLUND , Domain Decomposition Methods- Algorithms and Theory, Springer Series
in Computational Mathematics, Springer, Berlin, Heidelberg, 2005.