CC BY
64
MSC 12H20, 65L60, 93A30
DOI: 10.14529/ m m p160402
ASYMPTOTIC ESTIMATE OF A PETROV - GALERKIN METHOD FOR NONLINEAR OPERATOR-DIFFERENTIAL EQUATION
P. V. Vinogradova, Far Eastern State Transport University, Khabarovsk, Russian Federation, [email protected],
A.M. Samusenko, Far Eastern State Transport University, Khabarovsk, Russian Federation, [email protected],
I.S. Manzhula, Far Eastern State Transport University, Khabarovsk, Russian Federation, [email protected]
In the current paper, we study a Petrov - Galerkin method for a Cauchy problem for an operator-differential equation with a monotone operator in a separable Hilbert space. The existence and the uniqueness of a strong solution of the Cauchy problem are proved. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The minimal requirements to the operators of the equation were demanded, which guaranteed the convergence of the approximate solutions. There were no assumptions of the structure of the operators. Therefore, the method, specified in this paper, can be applied to a wide class of the parabolic equations as well as to the integraldifferential equations. The initial boundary value problem for nonlinear parabolic equations of the fourth order on space variables was considered as the application.
Keywords: Cauchy problem; operator-differential equation; Petrov - Galerkin method; orthogonal projection; convergence rate.
Introduction
Galerkin methods are widely used in various stages of mathematical modeling. These methods are used for study of the correctness of the mathematical model and the development of the approximate algorithms (see [1, 2], etc). One of these methods is the Petrov - Galerkin method. It is based on the use of two basises. The approximate solutions are constructed by the first basis, while the residual must be orthogonal to the second basis (see [3]). Petrov - Galerkin method was used in [4, 5] for the numerical solution of stationary models. This method has been investigated in [2, 6, 7] for parabolic mathematical models with a dominant convection.
When the two basises are linked through a linear operator, Petrov - Galerkin method is called the moments method. The analysis of the moments method applied to the linear ordinary differential equations was given in [8, 9]. A.V. Dzishkariani [10] used this method in the construction of projection-iterative method for linear elliptic boundary value problems. A.G. Zarubin [11] studied the same method for quasilinear stationary operator equations in Hilbert space. In [12, 13] the moments method was used to approximate determination of eigenvalues for various classes of operator equations. In given papers the convergence of approximate solutions was established and the convergence rate was obtained.
The study of nonlinear mathematical models is one of the difficult problems. The investigation of the differential equations in operator form is of special interest. The
Galerkin method for nonlinear differential-operator equations was studied in [14-16], where eigenvectors of the operator similar to the leading operator A of the equation were used as the basis functions. However, in specific mathematical models it is not always possible to find the explicit form of eigenvectors. In this paper, with the help of the moments method the approximate solution of the Cauchy problem for operator-differential equation is constructed in separable Hilbert space H. In contrast to [14-16] here the basis elements are not eigenvectors of a similar operator, which significantly expands the class of problems. In these papers a leading operator A and similar operator B satisfy the acute-angle inequality. In the present paper, this limitation is removed. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The results are illustrated by the initial-boundary value problem for a nonlinear parabolic equation.
1. Statement of the Problem and Auxiliary Assertions
Let Hi be separable Hilbert space densely and compactly embedded in a separable Hilbert space H with norm || • \\h = || • || and inner product (•, •). In the space H, we consider the Cauchy problem
u'(t) + Au(t) + K (u(t)) = h(t), u(0) = 0, (1)
where u(t) is the unknown function, h(t) is a given function. The functions u(t) and h(t) are determined on [0,T] (T < to). In the sequel, we assume that the operators A and K satisfy the following conditions:
(i) A is self-adjoint positive definite operator in H with the domain D(A) = Hi.
(ii) The operator K is subordinate to operator A with order a; i.e., D(K) D D(A) and for any element v from Hi, the inequality
\\K(v)||< v(IIAl/2v\l2)IIAvT, 0 < a< 1 (2)
holds, were p(£) is the nondecreasing continuous positive function on [0, to).
K
Let us define some spaces that will be necessary further.
Let L2(0,T; H) be a Hilbert space of all strongly measurable functions on [0,T] with finite norm
t ^ i
I|u||o,2 = ( i IHtWdt 2
(/ l\u(t)ll2dty. o
Consider the functions u(t) (0 < t < T) with values in Hi and with norm
T \ i
f \Au(t)\2dt I < to. Let the functions u(t) have derivatives u'(t) E L2(0,T; H) in o
the sense of distributions. Equip the set of these functions with the norm
t i
II u || 1,2 =(J(IIu'(t)II2 + \\Au(t)r)dty.
o
The completion of this set with respect to this norm is a Hilbert space W2i(H, H1).
Introduce the subspace W2l (H, Hl) = {u(t) e Wl(H, Hl),u(0) = 0}.
o
The solution of problem (1) is defined as a function u(t) eW\ (H,Hl) that satisfies t.
Condition (i) implies that the operator A has an inverse A-i : H ^ Hi. Futher, the following Bihary result (see [17]) will be necessary for us.
Lemma 1. Let g(s) be a positive funetion for s > 0 and let k,m > 0. Then from, the inequality
t
u(t) < k + m J v(s)g(u(s))ds, a < t < b
a
it follows that
t
u(t) < G-i^G(k) + mj v(s)ds^ ,
a
u
where G(u) = f d), u > u0 > 0.
uo
Theorem 1. Suppose that h(t) e L2(0,T; H), the operator K is compact in L2(0,T; H).
o
Then (1) a unique solution in W2l (H,Hi). Proof. It is known (see [18]) that the problem
v'(t) + Av(t) = g(t), v(0) = 0
o
with g(t) e L2(0,T; H) has a unique solution v(t) eW\ (H, Hl); moreover,
T T
i(\\v'(t)\\2 + \\Av(t)\\2)dt < M i \\g(t)\\2dt, (3)
where the positive constant M is independent of t. From (3) it follows that the linear operator
d )-i
(d + ^ : L2(0,T; H) (H, )
exists and it is bounded. Make a replacement (d + A u(t) = v(t) in (1). Then we obtain a functional equation
(6
v(t) + K\[jt + A) v(t))= h(t)
with the compact operator Ky (d + A j in L2(0,T; H). We shall use the Leray
\dt
Schauder principle. Consider the family of equations
u'(t) + Au(t) + XK(u(t)) = Xh(t), u(0) = 0, 0 < A < 1. (4)
Let u(A, t) be the solution of (4). By Mi denote various positive constants independent of A and t.
Take the inner product of equation (4) by Au(A,t) and integrate the resulting relation over the interval [0, t], t < T. By applying inequality (2) and e-inequality, we come to the estimate
1
1
-WA^M^m2 + iiAu(A,on2de <w-\\hmi2+
2e
£
- \|A«(A,£)||2^ + WAu(\,0Wl+av(WAl/2 u(\,0W2)dt.
2 _
o o
Using Holder inequality, we get
2WA1/2u(A,t)W2 + i WAu(A,£)W2d£ <
1
2£
0 , 2 + 2 WAu(A,C)W2dC+
l + a
/ t
l-a 2
WAu(A,e )W2dC
V — (WAl/2u(A,£ )W2d
Further, the Young inequality:
1 A A 8 — 1__^ a
ab < -e°a° +--— e bt-1, a,b,e> 0, 8> 1
8 8
is applied to the right-hand side of the preceding relation. Let 8 = a+i. Choosing
e > 0,
HAl/2u(AM2 < Mi ¡1+ v— (WA1/2u(A,^)W2)d£ I .
By using Lemma 1, we obtain
sup WAl/2u(A,t)W < M2.
0< t<T
(5)
From (2^d (3) we have
(
Wu(A,t)Wi ,2 < M3
T l/2 0^1 J WAu(A,t)W2av2(WA1/2u(A,t)W2)dtj
t
t
t
t
t
t
t
0,2 + \J \\Au(X,t)\\2adt
V V0 ' )
By using the last inequality and (5), we get
i f t \ i/2\
\\u(\,t)\\l,2 < M4 Since 0 < a < 1, then
\\u(\,t)\\l!2 < M5.
(1)
o
u(t) EW2, (H, Hi). Since the operator K(•) is monotone, the solution is unique. The proof of theorem is complete.
□
2. Petrov - Galerkin Method
Let {e^}^ be a complete orthogonal system of elements in H. The sequence E Hi
(s = 1, 2,...) is determined by the equation Aps = es.
Let Pn be the orthogonal projection in H onto the linear span Hn of the elements ei,e2,... ,en.
If F : H ^ H is a linear bounded operator, we have an operator norm
\\F\\H^H = sup \\Fx\\, x E H.
||z||=i
Let g(n) = \\A-i(I — Pn)\\H^H. The operator A-1 is compact operator in H, therefore, the function g(n) exists, moreover than g(n) ^ 0 as n ^ x>. In many cases the function g(n)
as n ^ x>.
The function g(n) is the main characteristics of the convergence rate of approximate solutions found by the moments method. The moments method for the approximate solution of the problem (1) leads to the following equation:
Pnu'n(t) + Aun(t) + PnK(un(t)) = Pnh(t), un(0) = 0, (6)
n
n
where Un(t) = ^n=1 a>s(t)<ps-
From now on, by С we denote various positive constants independent of n and t.
Lemma 2. Suppose that h(t) E L2(0,T; H). Then the inequalities
sup \\A1/2uJ < С, (7)
0<t<T
\\Aun\\o,2 < с (8)
hold.
Aun H
from 0 to £ < T. By using the e-inequality, we come to the relation
J (Rnu'n,Aun) dt + j\Aun\\2dt < (9)
00
Вестник ЮУрГУ. Серия «Математическое моделирование 21
и программирование» (Вестник ЮУрГУ ММП). 2016. Т. 9, № 4. С. 17-29
5 5
^\\h\\0,2 + e J \\Aunrdt + 2-J \\K(un)\\2dt. 0 0 Now we transform the first term on the left-hand side of (9):
J (Pnu'n,Aun) dt = J (u'n,Aun) dt. 00
Since u'(t) E H1 and condition (i) holds,
J (Pnu'n,Aun) dt = J (Al/2u'n,Al/2un) dt = 2 J dt ||A1/2un||2 dt = 0 0 0
= 2 HAl/2un(t)|2 - 2 HA1/2un(0)H2 = 2 ||A1/2un(Ol.
By applying (2) to the last term on the right-hand side of (9), using the Young inequality, we obtain
5
HAl/2un(i)H + J \\Aun\\2dt < C I \\h\0,2 + J V2/{1-a)(\\Al/2un\\2)dt I . (10) 00
By using Lemma 1, we obtain
\\A1/2un(e)\2 < G-1(G(C\\h(t)\\2022) + C J d^j,
where G(z) = f (p2/(idta)(t), z > z0 > 0. The existence of the function G-1(z) follows from
zo
the positiveness and continuity of the function ). Thus, estimate (7) is proved. Next, from (10) and (7) we obtain (8). The proof of lemma is complete.
□
We consider the problem
Pnv'n(t) + Avn(t) + PnK'(un(t))vn(t) = Pnh'(t), Vn(0) = 0. (11)
Lemma 3. Suppose that h(t) E L2(0,T; H), h'(t) E L2(0,T; H), h(0) = 0 and for any v E H1 the inequality
\\K'(z(t))v\\ < ^1(\\Al/2z\\)\\Avr\\v\\1-Y, 0 < 7 < 1 (12)
holds, where K' is the Freehet derivative, ) is a continuous positive function on [0; to). Then vn(t) = u'n(t).
t.
Pnu'n(t) + Au'n(t) + PnK' (un(t))u'n(t) = Pnh' (t), un(0) = 0, Pnu'n(0) = 0. (13)
5
Take the inner product of the equality Pnu'n(0) = 0 by Au'n(0)
0=(Pnu'n(0),Au'n(0)) = \\Al/2u'n(0)\\2.
Hence,
\K (0)\\ =0.
From (11) and (13) it follows that
Pn(v'n(t) - u'n(t)) + A(vn(t) - u'n(t)) + PnK'(un(t))(vn(t) - u'n(t)) = 0. (14)
Take the inner product of (14) by A(vn - u'n) and integrate the resulting relation over the interval [0,£ 1, £ < T. Then
t t 2\\A1/2(vn(0 - <m\2 + J \\A(vn - u'nWdt <J \\K'(un)(vn - u'n)\\\\A(vn - u'n)\\dt.
0 0
From this, together with (12) and (7), we have
t
2\\A1/2(Vn(0 - u'n(i))\\2 + J \\A(Vn - u'n)\\2dt <
0
t
< C J \\A(Vn - u'n)\\l+Y\\Vn - u'n\\l-Ydt. 0
By applying the Young inequality, we obtain
t t \\A1/2(vn(t) - vL(£))\\2 + J \\A(vn - u'n)\\2dt < C J \\vn - u'n\ \2dt.
00
A
Ik - u'n\\2dt < C ll^/V - un)\\2dt.
Thus,
\\Al/2(vn(£) - unm\2 < C J \\Al/2(vn - un)\\2dt.
0
Further, using the Gronwall inequality, we obtain vn(t) = u'n(t). The proof of lemma is complete. ^
Lemma 4. Let the assumptions of Lemma 3 hold. Then
IK\\o,2 < C. (15)
e
e
Proof. Take the inner product of (11) by Avn and integrate the resulting relation over the interval [0,£ 1, £ < T. Then
? / t
2\\Al/2Vnm2 + J \\Avn\\2dt < C l\\h'\\0t2 + J \\K'(un)vn\\2dt 0 \ 0
From the last inequality, by using (12) and estimates (7) and (8), we obtain
sup \\A1/2Vn\\ < C,
0<t<T
\\ AVn \\ 0,2 < C.
Next, using Lemma 3, we obtain (15). The proof of lemma is complete.
□
Theorem 2. Let the assumptions of Lemma 3 hold. Then
sup \\un - u\\ < Cgl/2(n), (16)
0<t<T
\\A1/2(un - u)\\o,2 < Cg1/2(n). (17)
Proof. For the solutions of problems (1) and (6) we have
(u - Un)' + A(u - Un) + K(u) - K(un) = (Pn - I)(u'n - h + K(un)).
Let us multiply this equation by u - un in the sense of the inner product in H and then integrate from 0 to £ < T. By using the monotonicity of the operator K, we come to the relation
1 \\u(£) - un(£)\\2 + \\A1/2(u - un)\\2dt < | ((Pn - I)(u'n - h + K(un)),u - un) \dt.
Since u - un E H1} we have
1 \\u(0 - Un(0\\2 + /\\A1/2(u - un)\\2dt < J \\u'n - h + K(un)\\\\(Pn - I)(u - Un)\\dt. 00
Since
\\(Pn -1 )(u - un)\\ = \\(Pn - I)A-1/2A1/2(u - un)\\ = \\(Pn - I)A-1/2\\h^h\\A1/2(u - un)\\ = \\A-1/2(Pn - I)\\h^h\\A1/2(u - un)\\< g1/2(n)\\A1/2(u - un)\\,
24 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2016, vol. 9, no. 4, pp. 17-29
ç
then
ç ç
2И£) - un(0\\2 + J \\A1/2(u - Un)\\2dt < g1/2(n)J \\u'n - h + K(un)\\\\A1/2(u - un)\\dt. 0 0
Next, applying e-inequality and choosing sufficiently small e > 0, we come to the inequality
T
\K£) - un(C)\2 + J \\Al/2(u - un)\\2dt < C (Щ^ + \\u'n\\2,2 + \\KMfe) g(n). 0
From (2^d (15) we obtain (16) and (17). The proof of the theorem is complete.
□
3. Application to Initial-Boundary Value Problems
In this section, the projection method is applied to a nonlinear parabolic equation with discontinuous boundary conditions.
Let П = [0,1] x [0,1], Q = П x [0,T]. In Q, consider the following initial-boundary value problem:
du(xi,x2,t) +A2u(XuX2,t)+ u(xi,x2,t)\u(xi,x2,t)\p = h(x\,x2,t), (xi,x2,t) G Q, (18)
u(xi,x2, 0) = 0, (xi,x2) G П, (19)
du(0,x2,t) du(1,x2,t) д2u(xl, 0,t) 82u(xl, 1,t) ^ = 0, dv = dv = dv2 = dv2 =0 (20) where p > 1, v is external normal to дП.
Let H = Ь2(П), Hi =W24 (П), where W4 (П) = {v(xi,x2) G Wi(Q),v(xi,x2)\dn =
0, ^^ = ^^ = ^^^ = = 0}, W24(H) is the Sobolev space (see [19]).
On Hi define the operators A = A2 and K = (I•)\I • \p.
It is known that the system of the functions ekj(xi,x2) = sinknxi sin jnx2, (k =
1, 2,... ,j = 1, 2,...) is a complete orthogonal system in L2(Q). Let Pn be the orthogonal projection in L2(Q) onto the linear span Hn of the fonctions {ekj(xi,x2)}'kj=i.
о
The functions ф^(xi,x2) GW4 (П) are the solution of the equation
A2pkj (xi,x2) = ekj (xi,x2). From the form of functions фkj(xi,x2), we obtain
g(n)= \\(A2)_i(I - Вп)\\Ып^Ып) < C. The approximate solution for (18) - (20) is defined as
n
un(xi,x2,t) = akj (t)fkj (xi, x2), k,j=i
where the unknown functions akj (t) are the exact solution of the Cauchy problem Pnu'n + A2un + PnUn\Un\P = Pnh, Un(xi,X2, 0) = 0. Let us check inequality (2).
By using the embedding of the space W2 (0) into C2(Q), we obtain
\\K(v)\\L2{n) = \\v\v\p\\L2{n) < CM^. (21)
Since
\\v\\c(Q) < C\\v\\wm, \\v\\c(Q) < C\\Av\\l2(h) = C\\Al/2v\\L2in).
(22)
\\v\v\P\\L2(Q) < C\\A/2vC2Iq).
Thus, (2) holds if a = 0.
Let us check inequality (12). Write down the Frechet derivative of the operator K :
K'(z(x\, x2,t))v(xi, X2) = (1 + p)\z(x\, x2,t)\pv(x\, x2).
Next,
\\K'(z)v\\L2(Q) < (1 + p)\z\Pc{Q.)\\v\\l2(Q) < C\\Az\L(Q)\\v\\l2(Q).
Hence, (12) holds if 7 = 0.
Let h(xl,x2,t) E L2(Q) and h(xl,x2,0) = 0. Then from Theorem 2 it follows that
sup \\Un(xi,x2,t) - u(xi,x2,t)\\L2(Q) < Cn-3/2, (22)
0<t<T
\\A(un(xi,x2,t) - u(xi,x2,t))\\L2(Q) < Cn-3/2.
The numerical realization of the Petrov - Galerkin method for (18) - (20) is made using the Matlab software package. The implementation of this algorithm in the form
(22)
function f (xi,x2,t) and the value of p. For testing of the algorithm, we put p = 1, T = 1. We choose h(xi, x2, t) so that the exact solution of (18) - (20) is u(xi,x2,t) = t2(xi — xi)s(x'2 — x2)3. The program finds an approximate solution and compares it with the known exact solution. Table 1 presents the results of numerical experiments containing sup \\un(xi,x2,t) — u(xi;x2, t)\\L2(Q) for different values of n. The results of numerical
0<t<T
experiments are consistent with the theoretical estimate.
Table
n =10 n =15 n = 20 n = 25 n = 30 n = 35
0, 253^10-6 0,153-10-7 0,10740-7 0, 793-10-8 0,51M0-8 0,411•10-8
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Received 4 June, 2016
УДК 517.9+517.6 DOI: 10.14529/mmpl60402
АСИМПТОТИЧЕСКАЯ ОЦЕНКА МЕТОДА ПЕТРОВА - ГАЛЕРКИНА ДЛЯ НЕЛИНЕЙНОГО ДИФФЕРЕНЦИАЛЬНО-ОПЕРАТОРНОГО УРАВНЕНИЯ
П.В. Виноградова, A.M. Самусенко, И.С. Маноюула
В работе исследуется метод Петрова - Галеркина ДЛЯ ЗвДе1ЧИ Коши для дифференциально-операторного уравнения с монотонным оператором в сепарабель-ном гильбертовом пространстве. Доказано существование и единственность сильного решения исследуемой задачи. Получены новые асимптотические оценки скорости сходимости построенных приближенных решений к точному решению в равномерной топологии. На операторы уравнения накладываются минимальные требования, необходимые для сходимости построенных приближенных решений. Отсутствуют какие-либо предположения о структуре операторов. Таким образом, метод исследуемый в данной работе, может быть применен к широкому классу параболических уравнений, а также, интегро-дифференциальных уравнений. В качестве приложения, исследуемый в работе метод, применяется к модельному параболическому уравнению четвертого порядка по пространственным переменным.
Ключевые слова: задача Коши; дифференциально-операторное уравнение; метод Петрова - Галеркина; оператор ортогонального проектирования; скорость сходимости.
Литература
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Полина Витальевна Виноградова, доктор физико-математических наук, доцент, кафедра «Высшая математика:», Дальневосточный государственный университет путей сообщения (г. Хабаровск, Российская Федерация), [email protected].
Александр Маркович Самусенко, К^НДИДсХТ физико-математических наук, доцент, кафедра «Высшая математика», Дальневосточный государственный университет путей сообщения (г. Хабаровск, Российская Федерация), samusenkoalexander@gmail .com.
Илья Сергеевич Манжула, магистрант, кафедра «Высшая математика», Дальневосточный государственный университет путей сообщения (г. Хабаровск, Российская Федерация), [email protected].
Поступила в редакцию 4 июня 2016 г.
