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Contractivity results for alternating direction schemes in Hilbert spaces
APPLIED NUMERICAL MATHEMATICS ELSEVIER
Applied
Numerical
Mathematics
15 (1994) 65-75
Contractivity results for alternating direction schemes in Hilbert spaces ’ J.C. Jorge a, F. Lisbona
b,*
a Departamento de Matema’tica e Informa’tica, Universidad Ptiblica de Nauarra, Campus Arrosadia s/n, 31006 Pamplona, Spain b Departamento de Matema’tica Aplicada, Facultad de Ciencias, Universidad de Zaragoza, Plaza San Francisco, 50009 Zaragoza, Spain
Abstract In this paper we obtain some contractivity results for operators R(- A,, . . ., - A,), where R(z,, . . , zn) are operators on a Hilbert space H. A rational approximations to exp(z, + . . . + z,J, and A, are maximal monotone general result is proved by using an extension to several variables of a result of Von Neumann for bounding f(A) (f a holomorphic function, A an operator on H). This theory is applied to the convergence analysis of Alternating Direction methods and, more generally, to Fractional Steps schemes. Key words: Linear
parabolic;
Alternating
Direction;
Fractional
Steps; Stability
1. Introduction
Let H be a complex Hilbert space, with scalar product (. , * > and associated norm 11. 11,and let A : D(A) c H -+ H be an unbounded linear maximal monotone operator on H. In this paper we will consider general initial boundary value parabolic problems formulated abstractly
u’(t) +Au(t i
4to)
=u
PO? TIT ho l D(A))*
)=O,
tE
0
It is well known (Hille-Yosida Theorem) satisfies the following contractivity property IlU(t
=GIh(s)II,
(1.1) that problem (1.1) has unique solution u(t) which
t,
* This work has been partially supported by the D.G.I.C.Y.T, * Corresponding author. E-mail: [email protected]. 016%9274/94/$07.00 0 1994 Elsevier SSDI 0168-9274(94)00009-6
Science
B.V. All rights reserved
Project
No. 89-0344.
66
J.C. Jorge, F. Lisbona /Applied
Numerical Mathematics
15 (1994) 65-75
Moreover, if A is coercive on H, with coercivity constant (Y> 0, i.e. (Au, u)2=crIIul12
(1.2)
VUED(A),
a stronger contractivity property holds, llu(t)ll
~e-“(‘-s)IIu(S)II,
toGsGt7
and the solution will tend to zero equally or faster than e-“’ when t tends to +m. The solution of (1.1) can be expressed as u(t)
= e -(f-wqt,),
where ePrA is the strongly continuous semigroup of contractions in In standard time discretization methods, approximations U” generated, the continuous transition eeAtA by some is a to the e*, Then, the defined by
H generated by -A. to u(t,) (t, mAt> are operator R( AtA) where A-acceptability properties.
u way to the convergence of discretization methods of and of the [4], Sanz-Serna [13], Sanz-Serna and Verwer Consistency means II
-AtA))&,)
of combining operator (see
II <
is
lR(t)l
< 1,
vt
is a necessary and sufficient condition to obtain (1.3) and strong A(O)-acceptability, i.e. (IR(t)l
I R(m) I < 1,
v’t
= -1,
i
is a sufficient condition to obtain (1.4) if A is also coercive. For a maximal dissipative operator, A-acceptability, ]R(z)]
~1,
Vz withRe(z)
67
.l.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
and strong A-acceptability (A-acceptability and I R(w) ( < 1) are sufficient to obtain (1.3) and (1.4) respectively. These stability questions have been widely studied by Crouzeix [4] for discretizations of problem (l.l), where the time discretization method used is a Runge-Kutta method. Palencia [ll], Brenner and Thomee [3], Crouzeix, Larsson, Piskarev and ThomCe [5] have studied more general stability questions for problems of type (1.11, where H is a Banach space and -A is a densely defined closed linear operator in H which generated a holomorphic semigroup of bounded linear operators in H. In order to consider Fractional Steps for discretizing the time variable in (1.1) we will assume also A = eAi
in D(A),
i=l
where Ai:D(A,)cH-+H,
i= l,...,
~1,are II maximal monotone
operators in H such that
D(A) = ;i D(Ai). i=l
For these methods, the continuous transition operator e-AtA is replaced by operators of type - AtA,), where R(z,, . . . , ZJ are rational approximations to e’l+ “’ +‘n holding R(-AtA,,..., suitable A-acceptability conditions. The interest of finding bounds of type )IR(-AtA,,...,
-AtA,)II
< 1
(1.5)
and ]IR(-AtA
,,...,
-AtA,)
II G epArp
(p > 0),
(1.6)
is obvious in order to analyze the convergence of Fractional Steps schemes combining consistency and contractivity. As a simple example let us examine the classical Alternating Direction method of Douglas [6] and Peaceman and Rachford [12] for discretizing the diffusion equation au a*u ----_=
a*u
at
um+l/*
_
U”
m+1’2 + A&J”), At
u
m+l
=
#lJm+w
-
At
+ AyyUm+l), = ;(AxFU m+1/2
where (A,,um)(x,
Y) =
(A,,u”)(x>
Y) =
uyx um(x,
-h,
y) - 2U”(x,
y -h)
h2 - 2U”(x, h*
Y) + Urn@ + h, Y) Y) + Urn@, Y + h)
7
F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
J.C. Jorge,
68
This method is based on a combination of a spatial discretization differences and a time discretization process of type
u m+l
-.4t(-a2/ax2),
=q
-q
process using central
-a2/ay2))um,
where R(%
1 + ;z2
= 22)
1 _
1, 2
.
1 + $q
1- ;t2*
1
Since the operators a2/ax2 and a2/ay2 commute, condition (1.5) can be immediately deduced from the contractivity results for the Crank-Nicolson scheme. More difficult situations are presented if the rational function cannot be factorized as a product of single-variable rational functions. To show this question we will consider the stabilizing corrections scheme as a second example. This scheme, proposed by Douglas and Rachford [7] for the resolution of the three-dimensional diffusion equation, au -at
a2u a224 - ---= ax: ax,2
a2u 0,
a.+
has the following formulation (%&)
=
m+1/3
‘Oh, _
‘h
4Y
=Alh~r+1’3
At (
‘h
m+2/3
-
I
m+l
+A3hU;,
u7+1/3
= ( A2h~rf2/3
_ At
uh
+A2h~;:
- A,,u,“),
Um+2/3 h
=
At
( A3h~r+1
-A,,U,“),
where Aih=A,?. , ,
A time discretization u m+* =R( -at(
process of type -a2/axf),
-a2/ax,2),
-ht(
-at(-a2/a#Y,
where 1 + ZrZ2 + 2223 + R(Z,,
Z2,
Z3)
2123
-Z1Z2Z3
= (1
-
Zdl
-
Z2W
-
Z3)
can be associated to this method. In this case, the rational function cannot be factorized as a product of single-variable rational functions, and the study of the contractivity involves an extension of the stability theory for approximations of ePAtA, using rational functions of several variables.
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
69
In this paper we obtain some stability results of type (1._5-(1.6) in the situations where the operators Ai commute. These results permit the study of the contractivity and, consequently, the convergence of the classical Fractional Steps schemes. This theory is also the key for the construction and analysis of new schemes of this kind with better approximation properties, see [9]. Finally, in the last section, these results are used to prove the stability and the convergence of the time discretization process for some classical Alternating Direction schemes.
2. A-stability Let us consider a rational approximation R( - AtA,, . . . , - AtA,) of the continuous semigroup of contractions e ~ *lCA1+ ‘.. +An), where _CX? = {A,}i”_1 is a commutative system of maximal monotone operators on a complex Hilbert space H, i.e. Re(A,u, R(I+A,) &4;
U) > 0
Vu ED,
=H,
i E (l,...,
n},
(2.la)
iE {1,...,12},
=A,A,
Vi, jE(l,...,
(2.lb) n}.
(2.lc)
The purpose of this section is the determination of sufficient conditions for the approximations R(-AtA,,..., -AtA,> to be contractive operators in H (i.e. (1.5) is verified). In order to do this, we first introduce the concept of A-acceptability for a complex rational function of y1 variables. Definition IR(z,
2.1. Let R(z,, ,...,
. . . , z,J be a rational
zn)l ~1,
V(z, ,...,
zn)
complex
function.
suchthatRe(z,)
We say R is A-acceptable
if
l~i~n.
It is well known that the stability result (1.3) is deduced directly from a result of Von Neumann [19] by bounding f< T) = R( - A tA), where T = (I - A tA)( I + A tA)- ’ is a contraction in H and f< z) is holomorphic in a neighbourhood of D = {z E C I I z I < 1). In the case it > 1, obtaining (1.5) requires the use of a generalization to n variables, based in unitary dilations techniques, of the theorem of Von Neumann. A unitary dilation of a commutative system Y = {Ti}Lyz, of operators in H is a corresponding commutative system on a Hilbert space Z, containing H as a subspace, and Z= {u,)i”=i of unitary operators satisfying T;“I . . - Tnmnu= PJJ;I~ - * * Unmnu Vu E H, 172~20,
ViE{l,...,n}
( PH = orthogonal
projection).
Theorem 2.2 (Sz-Nagy [16]). If a commutative system F dilation, then we have, for every polynomial p,(h,, . . . , A,), IIP(T,,...,T,)II
G
sup (Z,,...,Z,)ED”
where ID = {z E G I I z I G l}.
I P(Z~,...,qJI,
of contractions in H has an unitary (2.2)
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
70
This result is obviously extended from polynomials to the set of functions which are analytical on a neightbourhood of D”. In particular, (2.2) is obtained for every rational function bounded in D”. In the same way as in the one-variable case, a previous stage for the application of Theorem 2.2 to obtain (1.5) is the construction of the commutative systems, of contractions in H, 7 = {7):):=1 where T=(Z-AtAJ(Z+AtAi)-‘,
l
(2.3)
Theorem 2.3. For every rational A-acceptable function R(z,, . . . , z,J and every commutative system LY’ of maximal monotone operators in H, such that the systems of contractions 7 constructed in (2.3) admit unitary dilations, (1.5) holds. Proof. Let us consider f(Zi,...,
z,)-R
the rational 1 -zi - 1 +zi i
Since R is A-acceptable, sup If@ 1,“‘, (Z,,...,ZJED” Now, taking into account f(TI, . ..,T,)
‘.“’
-~
f is bounded $)I
in ED” and
G 1.
the equality
= R( -AtA,,.
and (2.21, (1.5) is obtained.
function
. ., -AtA,) q
Corollary 2.4. For every rational A-acceptable function R(z,, ZJ and every commutative pair (A,, AJ of maximal monotone operators in H, (1.5) is satisfied. Proof. It is straightforward, tions in H admits a unitary
taking into account that every pair {T,, T2} of commuting 0 dilation (see [l]).
contrac-
For some years it had been hoped that the result of Ando [l] and (2.2) would be valid for commutative systems of more than two operators also but, in both cases, counterexamples exist even for n = 3 (see Sz-Nagy 1161 and Varopoulos [18]). Therefore, in the case n > 2 it is essential in our analysis to use some criteria available that permit us to know if the systems Y defined in (2.3) admit unitary dilations. Fortunately, a necessary and sufficient condition for a commutative system .Y= IT,, . . . , T,} to admit a unitary dilation was given by Brehmer [2]. This condition is that the inequality ]]u112-
c l
II~,ul12+
c
I17&ul12+
..* +(-l)rllq,
lgj
holds true for every u E H and for every subset {irl,, . . . , c,} c 7. In some cases, (2.4) is immediately verified (see [16]):
..* 7j,ul12)/0
(2.4)
11
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics I5 (1994) 65-75
Case 1. q are isometries (1
i.e. 717; = 7;.Ti and Tq* = ?;*T, (1 < i,j < n). This is the most important case in our situation because it covers the study of stability for parabolic equations with constant coefficients (see Example 5.1). Case 3. Cy=, ]I7;: I]2 < 1.
3. A(O) -stability Let A be a maximal monotone self-adjoint operator. In standard-time discretization methods “U m+l = R( - AtA)Um”, obtaining (1.5) requires less restrictions for R(z)--A(O)-acceptability instead of A-acceptability-than in case that A is not self-adjoint. In this section we show that if {A,)i”=, is a commutative system of maximal monotone self-adjoint operators we can also weaken the restrictions for R(zr, . . . , zn> to obtain (1.5). R(z,, . . . , 2,) is A(O)-acceptable if
3.1. We say that a rational function
Definition
V(h, )...) A,)ER” IR(A,,.. .,A,)]
(3.1)
Theorem 3.2. For every commutative system { Ai}:= 1 of self-adjoint maximal monotone operators in H and every A(O)-acceptable rational function, (1.5) is satisfied. Proof. It is not a difficult task and we will give only a brief indication.
We shall first consider the commutative i=l,...,
q=(I+AtAi)-‘,
system IT,},“_,of self-adjoint contractions
where
n,
and the rational function l-z,
z,)=R
f(z,,...,
- -
i
-~
7 * . .,
l-z, 2,
=1
I .
Then we take, for each*T, a spectral decomposition q=
r A dPi,,
/ --E
llT’“l12=/1
IA12(dPihU, u),
PiA (see Hutson and Pym [8]) such that VUEH.
--E
so
R(-AtA,,...,
-AtA,)
= f(T,,
. . .,T,)
= /’
. . * /’ --E
f(A,,..
.,A,) dP,,] .
--E
dP,A n
and
11 f(TP
. . . , T,)u II2 = /’
. . . /’ --E
<
--E
I f(A,,
SUP (A,,...,A,)E[--E,ll”
. . . , A,) I 2(dP,,,
. . . dP,,,,U,
(If( A 1,...,A,)12)(I~112,
u)
vu~ff.
0
72
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
4. Strong A-stability
and strong A(O)-stability
As was mentioned in the introduction, the study of the strong stability properties pursues the obtention of bounds of type (1.6) in order to be able to study evolutionary problems in a infinite-length interval of time. Definition 4.1. We say that R(z,, . . . , zn) is strongly A-acceptable if it is A-acceptable exists c < 1 and A4 such that
R(z ,,“‘, zn)
Iz,I+
.** +lz,I
and there
>M.
The proofs of Lemmas 4.2 and 4.5 are extensions to n variables of the proofs of Crouzeix [4] for the case n = 1. Lemma 4.2. Let R(z, . . . ,z,,) be a strongly A-acceptable rational function, let (Y > 0 and At, > 0.
Then, for each i E { 1,. , . , n}, there exists pi > 0 (pi = p&R, CX,At,)) such that I R(
forevery
-h,At,.
. . , -(hi
+ c-u)At, . . . , -A,At)
(Al,...,
A,) with Re(Ai) > 0, i =
I G e-plAt, . .,
(4.1)
n and for every At < At,.
Theorem 4.3. Let { Aj],FCl be a commutative system of maximal monotone operators in H such n} ( Aiu, u) 2 (Y IIu II2 Vu E II( and such that the commutative system thatforan iE{l,..., (q.),51 of contractions in H, constructed as follows
~=(I-At(Ai-ul))(l+At(Ai-rul))-l, ?=(I-AtA;)(l+AtA,)-’
(j#i),
admits unitary dilations. Then, for every strongly A-acceptable rational function R(z,, . . . , zn> and At, > 0, there exists pi > 0 (pui = pi(R, (Y, At,)> such that IIR(-AtA,,...,
-AtA,)II
Proof. The process is analogue to the proof of Theorem Definition
is x,)with xiM, . . , n.
4.4. We say R(z,,
(b) ;herF+exists ... (c) Lemma
Let R(z,,
R( -AtA, ,...,
. . z,) be
A,)
-Athi_l,
0
. . z,) V(xl ,...,
...,
for every (AI,...,
3.2 but using the result (4.1).
n} there exists /Jo> 0 (pi c ~i(R, (Y, At,)), such that
-At(Ai
+a),
with hi 2 0, i = 1,. ..,n,
-Athi+I
,...,
-AtA,)
G eePL,*‘,
and for every At
.,
and
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
73
Theorem 4.6. Let {Aj},Y1 be a commutative system of maximal monotone self-adjoint operators such that for an iE{l,...,n) (A+, u)>c~lIuI(~. Then, for every rational A(O)-acceptable function R(z,, . . . , z,) and for every At,, > 0, there exists pi > 0 (pi = pi(R, (Y, At,) such that IIR(-AtA,,...,-AtA,)
~e-~~*’
VAtGAt,.
The proof is analogous to the proof of Theorem considering the commutative systems of self-adjoint
3.2, but using the result of Lemma 4.5 and contractions in H = {?;},Y, defined by
T,=(Z+At(Ai-cuZ))P’, T,=(Z+AtAj)-’
(j#i).
5. Applications In this section we show the application of the stability results of Sections 2-4 for analyzing the convergence of Alternating Direction schemes. We will consider two simple and classical examples like the method of decomposition and decentralization (Temam [17], Lions and Mercier [lo]), applied to a convection-diffusion problem and the stabilizing correction scheme, applied to the three-dimensional diffusion equation, seen in Section 1. Example 5.1. Let us consider
the following
ut-(dlu,,--v,u,-klu)-(d2uyy-v2uy-k2u)=0 written
two-dimensional
convection-diffusion inRx
problem
[0, T],
as well as
u, +A,u
+A,u
= 0
A, = - (d,a,, - v&I, - k,), where i
A, = - (d2ayy - v,ay - k2),
where d,, vl, k,, d,, v2 and k, are real and d,,d, > 0. The method of decomposition and decentralization consists discretization and the following time discretization process u m+l = R( -AtA,,
of combining
a standard
spatial
-AtA,)U”,
where R(z,,
1 ~2) = 1-z.
1 1
-
l-z,’
This rational function is strongly A-acceptable. Note that R is factorized as a product of single-variable strongly A-acceptable rational functions but we have preferred to use this example because of the simplicity of its formulation. In this case, defining appropriate HA,) and HA,), operators A, and A, are maximal monotone on the spaces H = L*(n) if k, > 0 and k, > 0, and are coercive if k, > 0 and k, > 0. Note also that the operators A,, A,, AT, A; commute.
14
J.C. Jorge, F. Lisbona /Applied
Numerical Mathematics
For proving the convergence of the time discretization be introduced in the usual way e, = u(t,)
15 (1994) 65-75
process the concept of local error can
- ii”,
where U“m
=R(AtA,,
At@&,,_,).
Assuming enough smoothness for solution u(t) of the continuous problem, and using Taylor expansions for the exact and semidiscrete solutions, it is not difficult to show the following consistency result: IIe, II~2~0)G CAt2. So, if Et’ = u(t,> - 17” is the global error, at the instant t, = m At, of the semidiscretization process and we decompose it in the form E,$f = (u(tm)
-C;“)
+ (ii”
- Urn) = e, +R(
-AM,,
-AtA,)E,f,
using the results of Sections 2 and 4 we will obtain, for A, and A, maximal monotone
II Et’ II LZ(CQ G II e, II LZ(O) + II EiL, and if the operators
A,
II LEO),
(5-l)
or A, are also coercive
II Et’ II LEO)G II e, II L>(O) + emA@ II E,%, II L,Z(CQ (P > 0).
(54
Finally, taking into account that IIe, II G CAt2, 0 < i
(order one of convergence)
(5.3)
is obtained by induction for mAt < T < 03 if (5.1) is satisfied or for every m 2 0 if (5.2) holds. Example 5.2. Let us take again the stabilizing correction scheme for discretizing the three-dimensional diffusion equation. As we have seen in the introduction, this method is associated with the following time discretization process.
u m+l =R( -at( = R( -AtA,,
-a2/a+ -AtA,,
-At(
-a2/a_Q
-at(
-a2/ax,2))um
-AtA,)U”,
where 1 + z1z2 + z2zj + zrz3 + zrzzzj R(z,,
~2, 23) =
(l-21)(1
-z2w
-4
*
This rational function is A(O)-acceptable. Since the operators A,, A, and A, are maximal monotone, self-adjoint and commute in H = L2(0), (1.5) holds. To analyze the convergence, proceeding in the same way as in Example 5.1, (5.3) is obtained for mAt =GT < ~0.
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
75
References contractions, Acta Sci. Math. (Szeged) 24 (1963) 88-90. S. Brehmer, Uber vertauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961) 106-111. of semigroups, SIAM J. Numer. Anuf. 16 (1979) [31 P. Brenner and V. ThomCe, On rational approximations 683-694. des equations differentielles operationelles lineaires par des methodes de 141 M. Crouzeix, Sur I’approximation Runge-Kutta, Thesis, Universite Paris VI (1975). of analytic [51 M. Crouzeix, S. Larsson, S. Piskarev and V. ThomCe, The stability of rational approximations semigroups, BIT 33 (1993) 74-84. 161 J. Douglas Jr, On the numerical integration of u,, + uyr = ut by implicit methods, J. SOiM 3 (1955) 42-65. 171 J. Douglas Jr and H.H. Rachford Jr, The numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Sot. 82 (1956) 421-439. Ml V. Hutson and J.S. Pym, Applications of Functional Anulysis and Operator Theory (Academic Press, New York, 1980) 234-242. para la integration de problemas parabolicos lineales: 191 J.C. Jorge, Los metodos de Pasos Fraccionarios formulation general, analisis de la convergencia y disefio de nuevos metodos, Thesis, Universidad de Zaragoza, Spain (1992). 1101 P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SOlMJ. Numer. Anal. 16 (1979) 964-979. in Banach spaces, SIAM J. Numer. Anal. 30 (1993) 1111 C. Palencia, A stability result for sectorial operators 1373-1384. and H.H. Rachford Jr, On the numerical solution of parabolic and elliptic differential 1121 D.W. Peaceman equations, J. SOlM 3 (1955) 28-42. to partial differential equations and stability concepts of methods 1131 J.M. Sanz-Serna, Convergent approximations for stiff systems of O.D.E.‘s, Actas VI C.E.D.Y.A., Jaca, Universidad de Zaragoza, Spain (1984) 488-493. J.G. Verwer and W.H. Hundsdorfer, Convergence and order reduction of Runge-Kutta 1141 J.M. Sanz-Serna, schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1986) 405-418. at the PDE/stiff ODE interface, Appl. Nilmer. 1151 J.M. Sanz-Serna and J.G. Verwer, Stability and convergence Math. 5 (1989) 117-132. [I61 B. Sz-Nagy, Unitary Dilutions of Hilbert Space Operators and Related Topics, CBMS Regional Conference Series in Mathematics 19 (AMS, Providence, RI, 1974) 25-54. de la methode des pas fractionaires, Ann. Mat. Puru Appl. 74 [171 R. Temam, Sur la stabilite et la convergence
111 T. Ando, On a pair of commutative
121
(1968). N. Varopulos, Sur une inegalite de Von Neumann, C.R. Acud. Sci. Ser. A-B 277 (1973) A19-A22. [191 J. Von Neumann, Eine Spektraltheorie fir allgemeine Operatoren eines unitaren Raumes, Math.
[la
(1951)
258-281.
Nuchr.
4
Applied
Numerical
Mathematics
15 (1994) 65-75
Contractivity results for alternating direction schemes in Hilbert spaces ’ J.C. Jorge a, F. Lisbona
b,*
a Departamento de Matema’tica e Informa’tica, Universidad Ptiblica de Nauarra, Campus Arrosadia s/n, 31006 Pamplona, Spain b Departamento de Matema’tica Aplicada, Facultad de Ciencias, Universidad de Zaragoza, Plaza San Francisco, 50009 Zaragoza, Spain
Abstract In this paper we obtain some contractivity results for operators R(- A,, . . ., - A,), where R(z,, . . , zn) are operators on a Hilbert space H. A rational approximations to exp(z, + . . . + z,J, and A, are maximal monotone general result is proved by using an extension to several variables of a result of Von Neumann for bounding f(A) (f a holomorphic function, A an operator on H). This theory is applied to the convergence analysis of Alternating Direction methods and, more generally, to Fractional Steps schemes. Key words: Linear
parabolic;
Alternating
Direction;
Fractional
Steps; Stability
1. Introduction
Let H be a complex Hilbert space, with scalar product (. , * > and associated norm 11. 11,and let A : D(A) c H -+ H be an unbounded linear maximal monotone operator on H. In this paper we will consider general initial boundary value parabolic problems formulated abstractly
u’(t) +Au(t i
4to)
=u
PO? TIT ho l D(A))*
)=O,
tE
0
It is well known (Hille-Yosida Theorem) satisfies the following contractivity property IlU(t
=GIh(s)II,
(1.1) that problem (1.1) has unique solution u(t) which
t,
* This work has been partially supported by the D.G.I.C.Y.T, * Corresponding author. E-mail: [email protected]. 016%9274/94/$07.00 0 1994 Elsevier SSDI 0168-9274(94)00009-6
Science
B.V. All rights reserved
Project
No. 89-0344.
66
J.C. Jorge, F. Lisbona /Applied
Numerical Mathematics
15 (1994) 65-75
Moreover, if A is coercive on H, with coercivity constant (Y> 0, i.e. (Au, u)2=crIIul12
(1.2)
VUED(A),
a stronger contractivity property holds, llu(t)ll
~e-“(‘-s)IIu(S)II,
toGsGt7
and the solution will tend to zero equally or faster than e-“’ when t tends to +m. The solution of (1.1) can be expressed as u(t)
= e -(f-wqt,),
where ePrA is the strongly continuous semigroup of contractions in In standard time discretization methods, approximations U” generated, the continuous transition eeAtA by some is a to the e*, Then, the defined by
H generated by -A. to u(t,) (t, mAt> are operator R( AtA) where A-acceptability properties.
u way to the convergence of discretization methods of and of the [4], Sanz-Serna [13], Sanz-Serna and Verwer Consistency means II
-AtA))&,)
of combining operator (see
II <
is
lR(t)l
< 1,
vt
is a necessary and sufficient condition to obtain (1.3) and strong A(O)-acceptability, i.e. (IR(t)l
I R(m) I < 1,
v’t
= -1,
i
is a sufficient condition to obtain (1.4) if A is also coercive. For a maximal dissipative operator, A-acceptability, ]R(z)]
~1,
Vz withRe(z)
67
.l.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
and strong A-acceptability (A-acceptability and I R(w) ( < 1) are sufficient to obtain (1.3) and (1.4) respectively. These stability questions have been widely studied by Crouzeix [4] for discretizations of problem (l.l), where the time discretization method used is a Runge-Kutta method. Palencia [ll], Brenner and Thomee [3], Crouzeix, Larsson, Piskarev and ThomCe [5] have studied more general stability questions for problems of type (1.11, where H is a Banach space and -A is a densely defined closed linear operator in H which generated a holomorphic semigroup of bounded linear operators in H. In order to consider Fractional Steps for discretizing the time variable in (1.1) we will assume also A = eAi
in D(A),
i=l
where Ai:D(A,)cH-+H,
i= l,...,
~1,are II maximal monotone
operators in H such that
D(A) = ;i D(Ai). i=l
For these methods, the continuous transition operator e-AtA is replaced by operators of type - AtA,), where R(z,, . . . , ZJ are rational approximations to e’l+ “’ +‘n holding R(-AtA,,..., suitable A-acceptability conditions. The interest of finding bounds of type )IR(-AtA,,...,
-AtA,)II
< 1
(1.5)
and ]IR(-AtA
,,...,
-AtA,)
II G epArp
(p > 0),
(1.6)
is obvious in order to analyze the convergence of Fractional Steps schemes combining consistency and contractivity. As a simple example let us examine the classical Alternating Direction method of Douglas [6] and Peaceman and Rachford [12] for discretizing the diffusion equation au a*u ----_=
a*u
at
um+l/*
_
U”
m+1’2 + A&J”), At
u
m+l
=
#lJm+w
-
At
+ AyyUm+l), = ;(AxFU m+1/2
where (A,,um)(x,
Y) =
(A,,u”)(x>
Y) =
uyx um(x,
-h,
y) - 2U”(x,
y -h)
h2 - 2U”(x, h*
Y) + Urn@ + h, Y) Y) + Urn@, Y + h)
7
F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
J.C. Jorge,
68
This method is based on a combination of a spatial discretization differences and a time discretization process of type
u m+l
-.4t(-a2/ax2),
=q
-q
process using central
-a2/ay2))um,
where R(%
1 + ;z2
= 22)
1 _
1, 2
.
1 + $q
1- ;t2*
1
Since the operators a2/ax2 and a2/ay2 commute, condition (1.5) can be immediately deduced from the contractivity results for the Crank-Nicolson scheme. More difficult situations are presented if the rational function cannot be factorized as a product of single-variable rational functions. To show this question we will consider the stabilizing corrections scheme as a second example. This scheme, proposed by Douglas and Rachford [7] for the resolution of the three-dimensional diffusion equation, au -at
a2u a224 - ---= ax: ax,2
a2u 0,
a.+
has the following formulation (%&)
=
m+1/3
‘Oh, _
‘h
4Y
=Alh~r+1’3
At (
‘h
m+2/3
-
I
m+l
+A3hU;,
u7+1/3
= ( A2h~rf2/3
_ At
uh
+A2h~;:
- A,,u,“),
Um+2/3 h
=
At
( A3h~r+1
-A,,U,“),
where Aih=A,?. , ,
A time discretization u m+* =R( -at(
process of type -a2/axf),
-a2/ax,2),
-ht(
-at(-a2/a#Y,
where 1 + ZrZ2 + 2223 + R(Z,,
Z2,
Z3)
2123
-Z1Z2Z3
= (1
-
Zdl
-
Z2W
-
Z3)
can be associated to this method. In this case, the rational function cannot be factorized as a product of single-variable rational functions, and the study of the contractivity involves an extension of the stability theory for approximations of ePAtA, using rational functions of several variables.
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
69
In this paper we obtain some stability results of type (1._5-(1.6) in the situations where the operators Ai commute. These results permit the study of the contractivity and, consequently, the convergence of the classical Fractional Steps schemes. This theory is also the key for the construction and analysis of new schemes of this kind with better approximation properties, see [9]. Finally, in the last section, these results are used to prove the stability and the convergence of the time discretization process for some classical Alternating Direction schemes.
2. A-stability Let us consider a rational approximation R( - AtA,, . . . , - AtA,) of the continuous semigroup of contractions e ~ *lCA1+ ‘.. +An), where _CX? = {A,}i”_1 is a commutative system of maximal monotone operators on a complex Hilbert space H, i.e. Re(A,u, R(I+A,) &4;
U) > 0
Vu ED,
=H,
i E (l,...,
n},
(2.la)
iE {1,...,12},
=A,A,
Vi, jE(l,...,
(2.lb) n}.
(2.lc)
The purpose of this section is the determination of sufficient conditions for the approximations R(-AtA,,..., -AtA,> to be contractive operators in H (i.e. (1.5) is verified). In order to do this, we first introduce the concept of A-acceptability for a complex rational function of y1 variables. Definition IR(z,
2.1. Let R(z,, ,...,
. . . , z,J be a rational
zn)l ~1,
V(z, ,...,
zn)
complex
function.
suchthatRe(z,)
We say R is A-acceptable
if
l~i~n.
It is well known that the stability result (1.3) is deduced directly from a result of Von Neumann [19] by bounding f< T) = R( - A tA), where T = (I - A tA)( I + A tA)- ’ is a contraction in H and f< z) is holomorphic in a neighbourhood of D = {z E C I I z I < 1). In the case it > 1, obtaining (1.5) requires the use of a generalization to n variables, based in unitary dilations techniques, of the theorem of Von Neumann. A unitary dilation of a commutative system Y = {Ti}Lyz, of operators in H is a corresponding commutative system on a Hilbert space Z, containing H as a subspace, and Z= {u,)i”=i of unitary operators satisfying T;“I . . - Tnmnu= PJJ;I~ - * * Unmnu Vu E H, 172~20,
ViE{l,...,n}
( PH = orthogonal
projection).
Theorem 2.2 (Sz-Nagy [16]). If a commutative system F dilation, then we have, for every polynomial p,(h,, . . . , A,), IIP(T,,...,T,)II
G
sup (Z,,...,Z,)ED”
where ID = {z E G I I z I G l}.
I P(Z~,...,qJI,
of contractions in H has an unitary (2.2)
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
70
This result is obviously extended from polynomials to the set of functions which are analytical on a neightbourhood of D”. In particular, (2.2) is obtained for every rational function bounded in D”. In the same way as in the one-variable case, a previous stage for the application of Theorem 2.2 to obtain (1.5) is the construction of the commutative systems, of contractions in H, 7 = {7):):=1 where T=(Z-AtAJ(Z+AtAi)-‘,
l
(2.3)
Theorem 2.3. For every rational A-acceptable function R(z,, . . . , z,J and every commutative system LY’ of maximal monotone operators in H, such that the systems of contractions 7 constructed in (2.3) admit unitary dilations, (1.5) holds. Proof. Let us consider f(Zi,...,
z,)-R
the rational 1 -zi - 1 +zi i
Since R is A-acceptable, sup If@ 1,“‘, (Z,,...,ZJED” Now, taking into account f(TI, . ..,T,)
‘.“’
-~
f is bounded $)I
in ED” and
G 1.
the equality
= R( -AtA,,.
and (2.21, (1.5) is obtained.
function
. ., -AtA,) q
Corollary 2.4. For every rational A-acceptable function R(z,, ZJ and every commutative pair (A,, AJ of maximal monotone operators in H, (1.5) is satisfied. Proof. It is straightforward, tions in H admits a unitary
taking into account that every pair {T,, T2} of commuting 0 dilation (see [l]).
contrac-
For some years it had been hoped that the result of Ando [l] and (2.2) would be valid for commutative systems of more than two operators also but, in both cases, counterexamples exist even for n = 3 (see Sz-Nagy 1161 and Varopoulos [18]). Therefore, in the case n > 2 it is essential in our analysis to use some criteria available that permit us to know if the systems Y defined in (2.3) admit unitary dilations. Fortunately, a necessary and sufficient condition for a commutative system .Y= IT,, . . . , T,} to admit a unitary dilation was given by Brehmer [2]. This condition is that the inequality ]]u112-
c l
II~,ul12+
c
I17&ul12+
..* +(-l)rllq,
lgj
holds true for every u E H and for every subset {irl,, . . . , c,} c 7. In some cases, (2.4) is immediately verified (see [16]):
..* 7j,ul12)/0
(2.4)
11
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics I5 (1994) 65-75
Case 1. q are isometries (1
i.e. 717; = 7;.Ti and Tq* = ?;*T, (1 < i,j < n). This is the most important case in our situation because it covers the study of stability for parabolic equations with constant coefficients (see Example 5.1). Case 3. Cy=, ]I7;: I]2 < 1.
3. A(O) -stability Let A be a maximal monotone self-adjoint operator. In standard-time discretization methods “U m+l = R( - AtA)Um”, obtaining (1.5) requires less restrictions for R(z)--A(O)-acceptability instead of A-acceptability-than in case that A is not self-adjoint. In this section we show that if {A,)i”=, is a commutative system of maximal monotone self-adjoint operators we can also weaken the restrictions for R(zr, . . . , zn> to obtain (1.5). R(z,, . . . , 2,) is A(O)-acceptable if
3.1. We say that a rational function
Definition
V(h, )...) A,)ER” IR(A,,.. .,A,)]
(3.1)
Theorem 3.2. For every commutative system { Ai}:= 1 of self-adjoint maximal monotone operators in H and every A(O)-acceptable rational function, (1.5) is satisfied. Proof. It is not a difficult task and we will give only a brief indication.
We shall first consider the commutative i=l,...,
q=(I+AtAi)-‘,
system IT,},“_,of self-adjoint contractions
where
n,
and the rational function l-z,
z,)=R
f(z,,...,
- -
i
-~
7 * . .,
l-z, 2,
=1
I .
Then we take, for each*T, a spectral decomposition q=
r A dPi,,
/ --E
llT’“l12=/1
IA12(dPihU, u),
PiA (see Hutson and Pym [8]) such that VUEH.
--E
so
R(-AtA,,...,
-AtA,)
= f(T,,
. . .,T,)
= /’
. . * /’ --E
f(A,,..
.,A,) dP,,] .
--E
dP,A n
and
11 f(TP
. . . , T,)u II2 = /’
. . . /’ --E
<
--E
I f(A,,
SUP (A,,...,A,)E[--E,ll”
. . . , A,) I 2(dP,,,
. . . dP,,,,U,
(If( A 1,...,A,)12)(I~112,
u)
vu~ff.
0
72
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
4. Strong A-stability
and strong A(O)-stability
As was mentioned in the introduction, the study of the strong stability properties pursues the obtention of bounds of type (1.6) in order to be able to study evolutionary problems in a infinite-length interval of time. Definition 4.1. We say that R(z,, . . . , zn) is strongly A-acceptable if it is A-acceptable exists c < 1 and A4 such that
R(z ,,“‘, zn)
Iz,I+
.** +lz,I
and there
>M.
The proofs of Lemmas 4.2 and 4.5 are extensions to n variables of the proofs of Crouzeix [4] for the case n = 1. Lemma 4.2. Let R(z, . . . ,z,,) be a strongly A-acceptable rational function, let (Y > 0 and At, > 0.
Then, for each i E { 1,. , . , n}, there exists pi > 0 (pi = p&R, CX,At,)) such that I R(
forevery
-h,At,.
. . , -(hi
+ c-u)At, . . . , -A,At)
(Al,...,
A,) with Re(Ai) > 0, i =
I G e-plAt, . .,
(4.1)
n and for every At < At,.
Theorem 4.3. Let { Aj],FCl be a commutative system of maximal monotone operators in H such n} ( Aiu, u) 2 (Y IIu II2 Vu E II( and such that the commutative system thatforan iE{l,..., (q.),51 of contractions in H, constructed as follows
~=(I-At(Ai-ul))(l+At(Ai-rul))-l, ?=(I-AtA;)(l+AtA,)-’
(j#i),
admits unitary dilations. Then, for every strongly A-acceptable rational function R(z,, . . . , zn> and At, > 0, there exists pi > 0 (pui = pi(R, (Y, At,)> such that IIR(-AtA,,...,
-AtA,)II
Proof. The process is analogue to the proof of Theorem Definition
is x,)with xi
4.4. We say R(z,,
(b) ;herF+exists ... (c) Lemma
Let R(z,,
R( -AtA, ,...,
. . z,) be
A,)
-Athi_l,
0
. . z,) V(xl ,...,
...,
for every (AI,...,
3.2 but using the result (4.1).
n} there exists /Jo> 0 (pi c ~i(R, (Y, At,)), such that
-At(Ai
+a),
with hi 2 0, i = 1,. ..,n,
-Athi+I
,...,
-AtA,)
G eePL,*‘,
and for every At
.,
and
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
73
Theorem 4.6. Let {Aj},Y1 be a commutative system of maximal monotone self-adjoint operators such that for an iE{l,...,n) (A+, u)>c~lIuI(~. Then, for every rational A(O)-acceptable function R(z,, . . . , z,) and for every At,, > 0, there exists pi > 0 (pi = pi(R, (Y, At,) such that IIR(-AtA,,...,-AtA,)
~e-~~*’
VAtGAt,.
The proof is analogous to the proof of Theorem considering the commutative systems of self-adjoint
3.2, but using the result of Lemma 4.5 and contractions in H = {?;},Y, defined by
T,=(Z+At(Ai-cuZ))P’, T,=(Z+AtAj)-’
(j#i).
5. Applications In this section we show the application of the stability results of Sections 2-4 for analyzing the convergence of Alternating Direction schemes. We will consider two simple and classical examples like the method of decomposition and decentralization (Temam [17], Lions and Mercier [lo]), applied to a convection-diffusion problem and the stabilizing correction scheme, applied to the three-dimensional diffusion equation, seen in Section 1. Example 5.1. Let us consider
the following
ut-(dlu,,--v,u,-klu)-(d2uyy-v2uy-k2u)=0 written
two-dimensional
convection-diffusion inRx
problem
[0, T],
as well as
u, +A,u
+A,u
= 0
A, = - (d,a,, - v&I, - k,), where i
A, = - (d2ayy - v,ay - k2),
where d,, vl, k,, d,, v2 and k, are real and d,,d, > 0. The method of decomposition and decentralization consists discretization and the following time discretization process u m+l = R( -AtA,,
of combining
a standard
spatial
-AtA,)U”,
where R(z,,
1 ~2) = 1-z.
1 1
-
l-z,’
This rational function is strongly A-acceptable. Note that R is factorized as a product of single-variable strongly A-acceptable rational functions but we have preferred to use this example because of the simplicity of its formulation. In this case, defining appropriate HA,) and HA,), operators A, and A, are maximal monotone on the spaces H = L*(n) if k, > 0 and k, > 0, and are coercive if k, > 0 and k, > 0. Note also that the operators A,, A,, AT, A; commute.
14
J.C. Jorge, F. Lisbona /Applied
Numerical Mathematics
For proving the convergence of the time discretization be introduced in the usual way e, = u(t,)
15 (1994) 65-75
process the concept of local error can
- ii”,
where U“m
=R(AtA,,
At@&,,_,).
Assuming enough smoothness for solution u(t) of the continuous problem, and using Taylor expansions for the exact and semidiscrete solutions, it is not difficult to show the following consistency result: IIe, II~2~0)G CAt2. So, if Et’ = u(t,> - 17” is the global error, at the instant t, = m At, of the semidiscretization process and we decompose it in the form E,$f = (u(tm)
-C;“)
+ (ii”
- Urn) = e, +R(
-AM,,
-AtA,)E,f,
using the results of Sections 2 and 4 we will obtain, for A, and A, maximal monotone
II Et’ II LZ(CQ G II e, II LZ(O) + II EiL, and if the operators
A,
II LEO),
(5-l)
or A, are also coercive
II Et’ II LEO)G II e, II L>(O) + emA@ II E,%, II L,Z(CQ (P > 0).
(54
Finally, taking into account that IIe, II G CAt2, 0 < i
(order one of convergence)
(5.3)
is obtained by induction for mAt < T < 03 if (5.1) is satisfied or for every m 2 0 if (5.2) holds. Example 5.2. Let us take again the stabilizing correction scheme for discretizing the three-dimensional diffusion equation. As we have seen in the introduction, this method is associated with the following time discretization process.
u m+l =R( -at( = R( -AtA,,
-a2/a+ -AtA,,
-At(
-a2/a_Q
-at(
-a2/ax,2))um
-AtA,)U”,
where 1 + z1z2 + z2zj + zrz3 + zrzzzj R(z,,
~2, 23) =
(l-21)(1
-z2w
-4
*
This rational function is A(O)-acceptable. Since the operators A,, A, and A, are maximal monotone, self-adjoint and commute in H = L2(0), (1.5) holds. To analyze the convergence, proceeding in the same way as in Example 5.1, (5.3) is obtained for mAt =GT < ~0.
J.C. Jorge, F. Lisbona /Applied Numerical Mathematics 15 (1994) 65-75
75
References contractions, Acta Sci. Math. (Szeged) 24 (1963) 88-90. S. Brehmer, Uber vertauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961) 106-111. of semigroups, SIAM J. Numer. Anuf. 16 (1979) [31 P. Brenner and V. ThomCe, On rational approximations 683-694. des equations differentielles operationelles lineaires par des methodes de 141 M. Crouzeix, Sur I’approximation Runge-Kutta, Thesis, Universite Paris VI (1975). of analytic [51 M. Crouzeix, S. Larsson, S. Piskarev and V. ThomCe, The stability of rational approximations semigroups, BIT 33 (1993) 74-84. 161 J. Douglas Jr, On the numerical integration of u,, + uyr = ut by implicit methods, J. SOiM 3 (1955) 42-65. 171 J. Douglas Jr and H.H. Rachford Jr, The numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Sot. 82 (1956) 421-439. Ml V. Hutson and J.S. Pym, Applications of Functional Anulysis and Operator Theory (Academic Press, New York, 1980) 234-242. para la integration de problemas parabolicos lineales: 191 J.C. Jorge, Los metodos de Pasos Fraccionarios formulation general, analisis de la convergencia y disefio de nuevos metodos, Thesis, Universidad de Zaragoza, Spain (1992). 1101 P.L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SOlMJ. Numer. Anal. 16 (1979) 964-979. in Banach spaces, SIAM J. Numer. Anal. 30 (1993) 1111 C. Palencia, A stability result for sectorial operators 1373-1384. and H.H. Rachford Jr, On the numerical solution of parabolic and elliptic differential 1121 D.W. Peaceman equations, J. SOlM 3 (1955) 28-42. to partial differential equations and stability concepts of methods 1131 J.M. Sanz-Serna, Convergent approximations for stiff systems of O.D.E.‘s, Actas VI C.E.D.Y.A., Jaca, Universidad de Zaragoza, Spain (1984) 488-493. J.G. Verwer and W.H. Hundsdorfer, Convergence and order reduction of Runge-Kutta 1141 J.M. Sanz-Serna, schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1986) 405-418. at the PDE/stiff ODE interface, Appl. Nilmer. 1151 J.M. Sanz-Serna and J.G. Verwer, Stability and convergence Math. 5 (1989) 117-132. [I61 B. Sz-Nagy, Unitary Dilutions of Hilbert Space Operators and Related Topics, CBMS Regional Conference Series in Mathematics 19 (AMS, Providence, RI, 1974) 25-54. de la methode des pas fractionaires, Ann. Mat. Puru Appl. 74 [171 R. Temam, Sur la stabilite et la convergence
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(1968). N. Varopulos, Sur une inegalite de Von Neumann, C.R. Acud. Sci. Ser. A-B 277 (1973) A19-A22. [191 J. Von Neumann, Eine Spektraltheorie fir allgemeine Operatoren eines unitaren Raumes, Math.
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Nuchr.
4