ISSN 1064–5624, Doklady Mathematics, 2009, Vol. 79, No. 3, pp. 351–355. © Pleiades Publishing, Ltd., 2009. Original Russian Text © M.E. Bogovskii, 2009, published in Doklady Akademii Nauk, 2009, Vol. 426, No. 3, pp. 295–299.
MATHEMATICS
Certain Aspects of Applying Newton’s Method to the Navier–Stokes Equations M. E. Bogovskii Presented by Academician Yu.G. Evtushenko December 26, 2008 Received January 23, 2009
DOI: 10.1134/S1064562409030144
It is well known that the numerical solution of initial–boundary value problems for the nonlinear Navier– Stokes equations based on Newton’s method faces two practical problems: inverting the Fréchet derivative of the corresponding nonlinear mapping and choosing an initial approximation from a sufficiently small neighborhood of the desired solution. The goal of this paper is to develop new effective approaches to coping with these problems. In a bounded domain Ω ⊂ 3 with boundary ∂Ω ∈ 2 C , we consider the initial–boundary value problem for the nonlinear nonstationary Navier–Stokes equations
in Sobolev spaces the frequently encountered subspaces of solenoidal and potential vector fields, regarding them as linear function spaces with the correspond°1
ing induced norms. Let J 2 (Ω) denote the closure in ° 1
W 2 (Ω; 3) of its subspace °∞
v v
t=0
°1
dition [7], the space J 2 (Ω) admits a simpler constructive description
∂Ω
= 0,
°1
diva = 0,
t ∈ ( 0, T ),
a
(1)
x ∈ Ω, ∂Ω
HT 2, 1
{ v , ∇p } ∈ W 2, x, t ( Q T ; ) × L 2 ( Q T ; ), 3
3
that satisfies Eqs. (1) almost everywhere in the cylinder QT and satisfies the initial and boundary conditions in (1) in the sense of equalities of the corresponding 2, 1 traces, where W 2, x, t (QT; 3) is the anisotropic Sobolev space of vector fields v: QT → 3 with square integraα vt, D x v
∈ L2(QT; ble Sobolev distributional derivatives 3 ) for all multi-indices α such that |α| ≤ 2. When dealing with problems in viscous incompressible fluid dynamics, it is convenient to single out
Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198 Russia; e-mail:
[email protected]
3
(see [2, 5] for more detail). The subspace
= 0,
which describe the dynamics of a viscous incompressible fluid with a constant kinematic viscosity ν > 0 for 0 < t < T. The strong solution to problem (1) is defined as an ordered pair 2, 1
° 1
J 2 ( Ω ) = { u ∈ W 2 ( Ω; ): divu = 0 }
( x, t ) ∈ Q T = Ω × ( 0, T ),
= a ( x ),
3
For a bounded domain Ω ⊂ 3 satisfying the cone con-
v t + ( v , ∇ )v – ν∆v + ∇p = f ( x, t ), divv = 0,
°∞
J ( Ω ) = { u ∈ C ( Ω; ): divu = 0 }.
= { v ( x, t ) ∈ W 2, x, t ( Q T ; ): divv = 0, v 3
∂Ω
= 0}
is treated as a Hilbert space with the norm of the aniso2, 1 tropic Sobolev space W 2, x, t (QT; 3). In a similar manner, GT denotes the closed subspace of all potential vector fields ∇xψ ∈ L2(QT, 3) and is treated as a Hilbert space with the norm of the Lebesgue space L2(QT, 3). 2, 1
1
Recall that W 2, x, t (QT) → C([0, T]; W 2 (Ω)), where the sign → denotes topological embedding. Therefore, °1 HT → C([0, T]; J 2 (Ω)). The nonlinear mapping : °1
HT × GT → L2(QT, 3) × J 2 (Ω) corresponding to problem (1) is defined as
{ v , ∇p } def = { v t + ( v , ∇ )v – ν∆v + ∇p, v
t = 0 } (2)
for all {v, ∇p} ∈ HT × GT, where the curly brackets {·, ·} denote ordered pairs that are elements of the Cartesian product X × Y of the corresponding linear spaces X and Y. Thus, the strong solution to the initial–boundary value problem (1) is an ordered pair {v, ∇q} ∈ HT × GT satisfying the equation
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BOGOVSKII °
{ v , ∇p } = { f , a } ∈ L 2 ( Q T ; ) × J 12 ( Ω ) 3
(3)
with given f and a. When nonlinear problem (3) is solved by Newton’s method, the Fréchet derivative of nonlinear mapping (2) has to be inverted at each iteration step. Moreover, For each element {v, ∇q} ∈ HT × GT, the Fréchet derivative v of mapping (2) depends explicitly only on the component v ∈ HT of {v, ∇q} and is a linear continuous °1
operator v: HT × GT → L2(QT; 3) × J 2 (Ω) defined as
v { u, ∇q } = { u t + ( v , ∇ )u + ( u, ∇ )v def
– ν∆u + ∇q, u
t = 0}
(4)
for all {u, ∇q} ∈ HT × GT. This means that each iteration of Newton’s method as applied to nonlinear problem (3) involves the solution of a linear initial–boundary value problem of the form u t + ( u, ∇ )v + ( v , ∇ )u – ν∆u + ∇q = g, u
{ u, ∇q } ∈ H T × G T ,
= b,
t=0
(5)
°1
where g ∈ L2(QT, 3), b ∈ J 2 (Ω), and v ∈ HT is given and fixed. Specifically, in Newton’s method, the recur∞ rence sequence of approximate solutions {vm, ∇pm } m = 0 for m ≥ 0 is given by the equalities {v
, ∇p
m+1
m+1
} = {v , ∇ p } m
m
– v m ( ( { v , ∇ p } – { f , a } ) ), –1
m
m
(6)
where v m is the inverse of Fréchet derivative (4) at the –1
element {v, ∇p} = {vm, ∇pm}, which, for m = 0, is called an initial approximation. A suitable initial approximation for Newton’s method can be effectively constructed using the Varnhorn regularization [9] of problem (1). Specifically, a delay is introduced into the convective term, which corresponds to a problem of the following form with a small parameter ε > 0: v t + ( ε v , ∇ )v – ν∆v + ∇p = f , divv = 0, v v
t=0 ∂Ω
( x, t ) ∈ Q T = Ω × ( 0, T ),
= a ( x ), = 0,
diva = 0,
t ∈ ( 0, T ),
a
x ∈ Ω, ∂Ω
(7)
= 0,
where ε is the shift operator with respect to t defined as εv(x, t) = v(x, t – ε) for t ≥ ε and εv(x, t) = v(x, 0) for t < ε. Any strong solution to problem (1) is approximated as ε → +0 in the norm of HT × GT by the sequence of regularized solutions (7). As a result, the Varnhorn regularization can be used to construct an initial approximation for Newton’s method.
This work consists of two sections. The first describes a new effective method for solving problem (5), i.e., a method for inverting Fréchet derivative (4). A solution to linear problem (5) is constructed in the form of a fast converging series. Its terms form a recurrence sequence and are determined by the solving operator of the linear initial–boundary value Stokes problem without convective terms. The Cauchy inequality for Banach-valued analytic functions is used to estimate the convergence rate of the constructed series in the norm of strong solutions. Although we eventually need only real solutions to the Navier–Stokes equations with real data, the convergence rate of the constructed series is estimated using the properties of complex-valued solutions to the corresponding initial–boundary value problems with complex-valued data. Section 2 describes a new effective method for constructing an initial approximation for Newton’s method with the help of regularized problem (7), whose solutions converge in the norm of strong solutions as ε → +0 to a strong solution of problem (1) if any. Given an arbitrary ε > 0, a regularized solution is constructed using the nonlinear analogue of the scheme described in Section 1, namely, in the form of a fast converging series whose terms form a recurrence sequence and are determined by the same solving operator of the linear initial– boundary value Stokes problem. The convergence rate of the constructed series is also estimated by applying the properties of complex-valued solutions. Therefore, Varnhorn regularization has to be extended to complexvalued solutions of the Navier–Stokes equations with complex-valued data. The efficiency of Varnhorn regularization is explained by the fact that it inherits the analytical dependence of solutions to the Navier–Stokes equations on the data of the problem. As a result, even in the complex-valued case, for every fixed ε > 0, the regularized problem has a global strong solution without assuming the existence of a global strong solution to reduced problem (1). 1. INVERSION OF THE FRÉCHET DERIVATIVE The basic advantage of Newton’s method is its superconvergence. The basic difficulty in the practical use of Newton’s method is associated with the inversion of the Fréchet derivative. An attempt to modify Newton’s method was made in [4], where the Fréchet derivative was inverted at the same initial element for all iterations. As a result, the basic advantage of Newton’s method, namely, its superconvergence was lost. In this paper, a new effective method is justified for inverting the Fréchet derivative (4) of nonlinear mapping (2) corresponding to the initial–boundary value problem for Navier–Stokes equations (1). Without resorting to any modification, the method converts the Fréchet derivative at all iterations of Newton’s method with the help of the same solving operator of the linear initial– DOKLADY MATHEMATICS
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CERTAIN ASPECTS OF APPLYING NEWTON’S METHOD
boundary value Stokes problem without convective terms. Introducing a complex parameter z ∈ , we consider the auxiliary initial–boundary value problem u t + z ⋅ ( u, ∇ )v + z ⋅ ( v , ∇ )u – ν∆u + ∇q divu = 0, u
t=0
u
∂Ω
= z ⋅ g ( x, t ), ( x, t ) ∈ Q T = Ω × ( 0, T ),
= z ⋅ b ( x ), = 0,
t ∈ ( 0, T ),
b
∂Ω
T = { u ( x, t ) ∈ v
∂Ω
= 0.
): divu = 0, 3
= 0 },
°
3
The solution of problem (8) at z = 1 coincides with the strong solution of problem (5) by virtue of its obvious uniqueness. °1
Theorem 1. For every g ∈ L2(QT, 3), b ∈ 2 (Ω), and z ∈ , problem (8) has a unique strong solution {u, ∇q} ∈ T × T. This solution is an entire function z ∈ with values in the Hilbert space T × T. The proof of Theorem 1 is based on an auxiliary result (Theorem 2 below). By Theorem 1, the strong solution to problem (5) can be represented as the Taylor series ∞
∑ ---k!- { u ( x, t ), ∇q ( x, t ) }, (9) 1
k
k
k=1
whose terms form a recurrence sequence constructed with the help of the solving operator of the linear Stokes problem without convective terms: DOKLADY MATHEMATICS
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1
2009
(10)
{ u , ∇q } ∈ T × T . 1
= b,
1
The index k = 0 is associated with the trivial solution and k
u
k
k t=0
k–1
, ∇ )v – k ⋅ ( v , ∇ )u
{ u , ∇q } ∈ T × T , k
= 0,
k
k–1
k ≥ 2.
, (11)
Theorem 1 and the Cauchy inequality (see [6, Ch. VII]) imply that series (9) converges in the norm of T × T as a geometric progression with a ratio arbitrarily close to zero. More specifically, Theorem 1 implies the following estimate for the remainder of the Taylor series ∞
∑
N ( x, t ) =
k = N+1
1 k k ---- { u ( x, t ), ∇q ( x, t ) }. k!
(12)
Corollary 1. Let v ∈ T, g ∈ L2(QT, 3), and b ∈ °1
J 2 (Ω) be given. Then, for any R > 0, there is a constant CR > 0 depending only on R and the norms v T , g
3
L2 ( QT ; )
, b
such that
3
W 2 ( Ω; )
N
1
H T × GT
C ≤ ------NR ∀ N ≥ 1. R
(13)
To prove Theorem 1, it is sufficient to establish the following result. Theorem 2. Let v ∈ T be given. Then, for every °1
g ∈ L2(QT; 3), b ∈ 2 (Ω), and z ∈ , problem (8) has a unique strong solution {u, ∇q} ∈ T × T satisfying the inequality u
2 ( Ω ) = { u ∈ W 12 ( Ω; ): divu = 0 }.
{ u ( x, t ), ∇q ( x, t ) } =
u
(8)
and is treated as a Hilbert space with the norm of the 2, 1 anisotropic Sobolev space W 2, x, t (QT; 3) of vector fields u: Ω → 3. In a similar manner, T denotes the closed subspace of all potential vector fields ∇xψ ∈ L2(QT; 3), which is treated as a Hilbert space with the norm of the Lebesgue space L2(QT; 3). The strong solution to problem (8) is defined as an ordered pair {u, ∇q} ∈ T × T that satisfies Eq. (8) almost everywhere in QT and satisfies initial condition (8) in the sense of the equality of the corresponding trace on the basis of QT. Define °1
1
1
k
The transition to complex-valued solutions requires the introduction of the corresponding linear spaces over the field of complex numbers. Note that this transition is caused by the use of the Cauchy inequality for estimating the convergence rate of Taylor series in powers of z. The subspace HT over the field is denoted by 2, 1 W 2, x, t ( Q T ;
u t – ν∆u + ∇q = g,
u t – ν∆u + ∇q = – k ⋅ ( u
x ∈ Ω,
divb = 0,
353
≤ C( g
T
+ ∇q 3
L2 ( QT ; )
3
L2 ( QT ; )
+ b
3
W 2 ( Ω; ) 1
),
(14)
where C > 0 is a constant depending only on Ω, T, |z|, and v T . The existence of a strong solution in Theorem 2 is easily established using the Galerkin method and a priori estimates for the stationary Stokes problem. The uniqueness of a strong solution in Theorem 2 is obvious. Theorem 2 implies that the strong solution to problem (8) is -differentiable with respect to z in the norm of T × T. Then, by the uniqueness guaranteed by Theorem 2, the derivative of the strong solution to problem (8) with respect to the complex conjugate parameter z is the unique trivial solution of the corresponding homogeneous problem, which means that the strong solution of problem (8) is holomorphic with respect to z in T × T (see [6, Ch. VII]). The roots of the method proposed for inverting the Fréchet derivative are easy to trace in the theory of lin-
354
BOGOVSKII
ear operators (see, e.g., [3]). Therefore, this method can be viewed as new only in the context of applying to the Navier–Stokes equations. Note, however, that this method can be applied not only to the Navier–Stokes equations but also to a wide class of quasilinear evolution problems with analytical nonlinearities. 2. INITIAL APPROXIMATION FOR NEWTON’S METHOD An initial approximation for Newton’s method (6) from a sufficiently small neighborhood of the desired solution is easy to construct if the norms of f and a or 1 one of the numbers T and --- are small. However, Newν ton’s method is not necessary in this case, since, due to the smallness conditions, we can effectively use contraction mappings. The need for the fast converging Newton method arises primarily when contraction mappings are insufficient. In addition to the inversion of the Fréchet derivative, there is another difficulty in the use of Newton’s method (6) as applied to problem (3). Namely, we have to choose an initial approximation {v0, ∇p0} that belongs to a sufficiently small neighborhood of the desired solution. The goal of this section is to show that such a choice can be made if the desired strong solution exists. The approach proposed for constructing an initial approximation for Newton’s method is underlain by the following convergence theorem for Varnhorn regularizations. °1
Theorem 3. Let f ∈ L2(QT, 3) and a ∈ 2 (Ω) be given for which problem (1) has a strong solution [v, ∇p} ∈ T × T. Then, as ε → +0, the strong solutions to problem (7) converge in the norm of T × T to a strong solution of problem (1) at a convergence rate no worse than o(ε1/4). There are several ways of proving Theorem 3, but we do not describe them. Note only the importance of the assumption that problem (1) has a strong solution {v, ∇p} ∈ T × T, since the existence of even a weak solution to problem (1) is not guaranteed when f and a are complex-valued. Note also that the convergence rate estimate no worse than o(ε1/4) hardly has any practical value for choosing a suitable regularization parameter ε, since the constant in this estimate depends on the norm v T of the desired solution. Introducing a complex parameter z ∈ , we consider the auxiliary initial–boundary value problem v t + z ⋅ ( ε v , ∇ )v – ν∆v + ∇q = z ⋅ f , v
t=0
= z ⋅ a,
{ v , ∇p } ∈ T × T ,
(15)
whose solution at z = 1 coincides with a strong solution of problem (7) by virtue of its obvious uniqueness.
°1
Theorem 4. Let f ∈ L2(QT; 3), ‡ ∈ 2 (Ω), z ∈ be given. Then, for every ε > 0, problem (15) has a unique strong solution that is an entire function z ∈ with values in the Hilbert space T × T. Note that the uniqueness of the Varnhorn regularization follows from the unique global solvability of problem (15) with any ε > 0 in the class of strong complexvalued solutions as guaranteed by Theorem 4. Due to this global solvability, the solution of regularized problem (15) can be represented as a series (fast converging in the norm of the strong solution) whose terms form a recurrence sequence and are constructed using the same solving operator of the linear initial–boundary value Stokes problem without convective terms. By Theorem 4, for each ε > 0, the strong solution {v, ∇p} ∈ T × T of problem (7) can be represented as a Taylor series { v ( x, t, z ), ∇p ( x, t, z ) } ∞
∑ ---k!- { v ( x, t ), ∇ p ( x, t ) },
=
1
k
k
(16)
k=1
which converges in the norm of T × T on any compact set in . Moreover, the terms of series (16) form a recurrence sequence that is constructed using the solving operator of the linear Stokes problem v t – ν∆v + ∇ p = f , 1
v
1 t=0
1
1
(17)
{ v , ∇ p } ∈ T × T , 1
= a,
1
where, by definition, the indices k = 0 and 2 correspond to trivial solutions: k–2
k vt
v
– ν∆v + ∇ p = – k! k
k t=0
k
( ε v , ∇ )v -------------------------------------------, m! ( k – 1 – m )! (18) m=1
∑
m
k–1–m
{ v , ∇ p } ∈ T × T , k
= 0,
k
k ≥ 3.
Theorem 4 and the Cauchy inequality imply that series (16) converges in the norm of T × T as a geometric progression with a ratio arbitrarily close to zero. More specifically, Theorem 4 implies the following estimate for the remainder of the Taylor series ∞
∑
N ( x, t ) =
k = N+1
1 k k ---- { v ( x, t ), ∇ p ( x, t ) }. k!
(19)
Corollary 2. Let v ∈ T, f ∈ L2(QT; 3), and ‡ ∈ °1
2 (Ω) be given. Then, for any R > 0, there is a constant CR > 0 depending only on R and the norms v g
3
L2 ( QT ; )
, and b
N
3
W 2 ( Ω; ) 1
H T × GT
such that
C ≤ ------NR ∀ N ≥ 1. R
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To prove Theorem 4, it is sufficient to establish the following result. °1
Theorem 5. Let f ∈ L2(QT; 3), ‡ ∈ 2 (Ω), z ∈ be given. Then, for every ε > 0, problem (15) has a unique strong solution {u, ∇q} ∈ T × T. The uniqueness of a strong solution in Theorem 5 is obvious. The existence of a strong solution in Theorem 5 is easily derived from the a priori boundedness of strong solutions to problem (15) that correspond to each fixed regularization parameter ε > 0. Theorem 5 implies that the strong solution to problem (15) is -differentiable with respect to z in the norm of T × T. Then, in view of the uniqueness guaranteed by Theorem 5, the derivative of the strong solution to problem (15) with respect to the complex conjugate parameter z is the unique trivial solution of the corresponding homogeneous problem, which means that the strong solution to problem (15) is holomorphic with respect to z in T × T. The choice of an initial approximation is based on the approximation of a regularized solution to problem (7) by partial sums of series (16) at z = 1. The accuracy of the approximation is guaranteed by the estimate of the remainder of the series based on the Cauchy inequality and Theorem 4, which also imply that such an approximation is possible for any given regularization parameter ε > 0. Reducing (if necessary) the value of ε > 0, we can ensure that the constructed approximation of the regularized solution belongs to a sufficiently small neighborhood of the desired solution. The approximation of the regularized solution thus constructed is a suitable initial approximation {v0, ∇p0} for Newton’s method (6). In
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this way, an initial approximation for Newton’s method can be chosen on any prescribed time interval (0, T) without dividing (0, T) into small subintervals, since the division procedure leads to error accumulation for sufficiently large T. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 08-01-00661) and by an innovation educational project of the Peoples’ Friendship University of Russia. REFERENCES 1. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (Wiley, New York, 1978; Nauka, Moscow, 1996). 2. M. E. Bogovskii, Dokl. Akad. Nauk SSSR 248, 1037– 1040 (1979). 3. N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory (Interscience, New York, 1958; Inostrannaya Literatura, Moscow, 1962). 4. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Fizmatlit, Moscow, 1977; Pergamon, Oxford, 1982). 5. V. N. Maslennikova and M. E. Bogovskii, Sib. Mat. Zh. 24 (5), 149–171 (1983). 6. L. Schwartz, Analyse Mathématique: cours professé à l’Ecole Polytechnique (Hermann, Paris, 1967; Mir, Moscow, 1972), Vol. 2. 7. R. A. Adams and J. F. Fournier, Sobolev Spaces (Academic, New York, 2003). 8. H. Sohr, The Navier–Stokes Equations: An Elementary Functional Analytic Approach (Birkhäuser, Basel, 2001). 9. W. Varnhorn, The Stokes Equations (Akademie, Berlin, 1994).
