CC BY
13
MSC 65N30, 65Z05
DOI: 10.14529/mmp 180109
NEW APPROXIMATE METHOD FOR SOLVING THE STOKES PROBLEM IN A DOMAIN WITH CORNER SINGULARITY
V.A. Rukavishnikov1, A.V. Rukavishnikov2
Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Khabarovsk, Russian Federation
institute of Applied Mathematics of Far-Eastern Branch, Russian Academy of Sciences,
Khabarovsk, Russian Federation
E-mail: [email protected], [email protected]
In this paper we introduce the notion of an ^-generalized solution to the Stokes problem with singularity in a two-dimensional non-convex polygonal domain with one reentrant corner on its boundary in special weight sets. We construct a new approximate solution of the problem produced by weighted finite element method. An iterative process for solving the resulting system of linear algebraic equations with a block preconditioning of its matrix is proposed on the basis of the incomplete Uzawa algorithm and the generalized minimal residual method. Results of numerical experiments have shown that the convergence rate of the approximate ^-generalized solution to an exact one is independent of the size of the reentrant corner on the boundary of the domain and equals to the first degree of the grid size h in the norm of the weight space (Q) for the velocity field components in contrast to the approximate solution produced by classical finite element or finite difference schemes convergence to a generalized one no faster than at an O(ha) rate in the norm of the space W21(Q) for the velocity field components, where a < ^d a depends on the size of the reentrant corner.
Keywords: corner singularity; weighted finite element method; preconditioning.
Introduction
The weak solution of Maxwell equations considered in a 2D polygonal domain with reentrant corner on the boundary does not belong to the Sobolev space W^Q). Such a problem is called a boundary value problem with strong singularity. For the Lame system, for a example, in a domain with a reentrant corner it is possible to define a weak solution in the space W\ (Q), but it does not belong to the space W22(Q). Such problem is called a problem with weak singularity.
According to the principle of coordinated estimates, the approximate solution to these problems by the classical finite difference and finite element methods converge to the exact one with a rate substantially smaller than one. In [1,2] it was proposed to define the solution of elliptic boundary value problems and Maxwell equations with strong singularity as an ^-generalized one. Such a new conception of solution allows to construct weighted finite element methods with first-order convergence rate estimate of the approximate solution to the ^-generalized one in the norms of the weighted Sobolev spaces.
In this paper we present our method for solving the Stokes problem. It is well known that the efficient numerical solution of problems in fluid mechanics is of significant engineering interest. There are basically three reasons why the finite element discretization of such problem turns out to be difficult.
Firstly, in the presence of reentrant corner u, u E (n, 2 n), on the boundary of the domain the solution of the problem is singular even though the input data are sufficiently
smooth. The two-dimensional flow of a viscous fluid near the corner was first examined in [3]. It is well known that the generalized solution of the Stokes problem: the velocity components and pressure in a two-dimensional domain Q with a boundary containing a reentrant angle does not belong to W22(Q) and W2^(Q) respectively (see e.g. [4]). Therefore, the approximate solution produced by standard finite element or finite difference schemes converges to a generalized solution no faster that at an O(ha) rate in the norm of the space W21(Q), where a < 1 depends on the size of the reentrant corner u for the velocity components (see [5]). In this case the so-called pollution effect can be observed in standard Sobolev and even in weighted Sobolev norms [6]. More recent results on the regularity theory and finite element approximations on domains with reentrant corners can be found in [7] and the references therein. By using special methods for extracting the singular part of the solution near corner points and applying grids refined towards the singularity point, it is possible to construct first-order accurate finite element schemes (see e.g. [8]).
Secondly, the design of LBB-stable method for a velocity and pressure spaces pairs [9].
Thirdly, the spaces enforce mass conservation strongly. Satisfying this criterion leads to more physically relevant solutions, decouples the pressure error from the velocity error, and removes possible instabilities that can arise from poor discrete mass conservation [10]. The specific element pair to achieve pointwise mass conservation of the discrete solution is the Scott-Vogelius element pair [11].
In the present paper we construct the weighted finite element method (see [1,2,12-15]) based on the conception of an ^-generalized solution [16-19] of the Stokes problem with a singularity due to a reentrant corner of u on the boundary of the domain and Scott-Vogelius element pair. Numerical experiments of the model problems have shown that the approximate ^-generalized solution produced by weighted finite element method converges to the exact one (velocity) with the rate O(h) in W\v(Q,S) norm for all considered sizes of the reentrant corner u in contrast to the approximate solution produced by classical finite element or finite difference schemes convergence to a generalized one no faster than at an O(ha) (a = a(u) < 1) rate. The simplicity of the solution determination is an additional benefit of the method for the numerical experiments.
The structure of the paper is as follows. In Section 1 define the ^-generalized solution of the Stokes problem with corner singularity. In Section 2 describe the proposed weighted finite element method. In Section 3 construct an iterative process with a block preconditioning matrix. In Section 4 present the results of numerical experiments. Finally, some concluding remarks are given in Section 5.
1. ^-Generalized Solution
Let R2 denote the two-dimensional Euclidean space, x = (x\,x2) be its arbitrary element, ||x|| = (xf + x2)1/2 and dx = dx\ dx2. Let Q C R2 be a bounded non-convex polygonal domain with a boundary r containing a reentrant angle with its vertex placed at the origin, and let Q be a closure of Q, i.e. Q = Q U r. Denote by Qs = {x G Q : ||x|| < 8 < 1,8 > 0} the part of a 8-neighbourhood of the point (0, 0) that lies in Q. Define a
• if • / \ ii / \ i ||x||,x G Q's, weight function p(x), such that p(x) = < ,
6 ' ' ^ 8 , x G Q\ Qs.
Let L2,p(Q) denote the weighted space of functions with bounded norm
1/2
Ы\ь2ф (П) = (/ P213 (x)v2(x)dx)
and Wlp (Q) denote the weighted space of functions with bounded norm
1/2
p (x)|D"Mx)|||L2(n)J
|m|<1
\\"\\Ч?(") = (£ P(x)l^mv(x)|\|2(Q^ , (1)
where Dmv(x) = , |m| = m1 + m2,mi > 0,i = 1, 2 - integer. For vector functions
v = (v1;v2) we define weighted spaces L2,p(Q) and W1, p(Q) with norms ||v||L2 3(n) =
"'2, в (
ч 1/2 ( ч 1/2
I 2 I ||,11_|12\л^^||жг11 - - /11/11.112 I ll/11_ll2 \
||vi|L2,3(n) + ||v2|L2,3(n)J and ||v||w2,K(n) = (JM^3(n) + ^Mwi3(n)) respectively. Let W21p(Q,A), for 3 > 0, denote a set of functions v(x) from the space W21 p(Q) satisfying the following conditions:
//A \ p+m
p2p(x)v2(x)dx > Ci > 0, |Dmv(x)| < C^p(x^) x G Q5, (2)
n\n'5
where m = 0,1 and C2 > 0 be a constant independent of m, with a norm (1). Let L2 ,p(Q, A) be a set of functions from the space L2, p (Q), which satisfy the conditions (2) (for m = 0)
with a norm of the space L2 ,p(Q). Also, L^ ,p(Q, A) = {q G L2 ,p(Q, A) : / pp qdx = 0}.
n
o o
The set W21,p (Q,A)(W21,p (Q, A) C W^p(Q, A)) is defined as a closure in norm (1) of the
Q,
(2). We say that ^(x) G W2 p(r, A), if there exists a function $(x) from W^p(Q,A) such that $(x)|r = (x) and ||^|wi/2(rs) = inf 3(n,sy
2,3 ^ ' ' <|r—^ '3
For vector functions v = (v1 ,v2) define sets L 2, p(Q,A) and W1, ,p(Q,A) such that vi G L2,p(Q,A) and vi G W21 p(Q, A) with a bounded norm of spaces L2 ,p(Q^d W1, ,p(Q)
o
respectively. Similarly for vector functions we define sets W^ >p (^,A) and We/2(r,A).
The Stokes problem is to find the velocity field u = (u1,u2) and pressure p which satisfy the system of differential equations and the boundary conditions
— uAu + Vp = f, div u = 0, in П, (3)
u = g, on Г. (4)
Here f = (f1, f2) and g = (g1, g2) are defined in П and on Г respectively, V is the kinematic viscosity which is related to the Reynolds number Re of the flow by V = ReDefine the bilinear and linear forms
a(u, v) = J v Vu • V(p2vv)dx, b(v,p) = — Jp div (p2vv)dx,
П П
c(u,q) = —J p2v q div u dx, l(v) = J p2v f • vdx.
ПП
Definition 1. The pair of functions (uv(x),pv(x)) G W^ v(Q,5) x L2 v(Q,5) is called an Rv-generalized solution of the Stokes problem (3), (4), if uv(x) satisfies the boundary condition (4) almost everywhere on r and integral identities
a(uv, v) + b(v,pv) = l(v), c(uv ,q) = 0
o
hold for any pair (v(x), q(x)) GW^ v (Q, 5) x L2> v (Q, S), where the right-hand side functions f G L2 ,p(Q, S), g G Wl /|(r,5), v > /3.
2. The Weighted Finite Element Scheme
The weighted finite element scheme for the Stokes problem (3), (4) is constructed relying on the definition of an Rv-generalized solution. For this purpose, we construct the triangulation Yh which is a barycenter refinement of a quasi-uniform triangulation Th of Q [20]. The domain Q is divided into a finite number of triangles Li, Li G Th (macroelement). Each Li is refined as stated above into three triangles Kj (finite element), Kij G Yh (barycenter refinement) with vertices Ri and midpoints Sk. Then,
1)Rvel = RVfl U RTel = {Rl U Sk}, where RQe1 and RjTel are sets of triangulation nodes for the velocity components in Q and on r respectively;
2)Rpres = {Ql} is the set of triangulation nodes for the pressure, where a node Ql coincides with a node Rk on the corresponding Kij.
Denote by Qh = (J Ks ^te union of all finite elements with sides of order h. Now
Ks&Th
we introduce Scott-Vogelius (SV) element pair [11] (case k = 2). In short, polynomials of degree two and one are used to approximate the velocity components and pressure, spaces Xh and Zh respectively:
Xh = {vh G C(Q) : vh\K G P2(K),VK G Yh}(Xh = Xh x Xh);
Zh = {zh G L2(Q) : zh\K G Pi(K), VK G Yh, f zhdx = 0}.
q
The SV element is very interesting from the mass conservation point of view since
Xh Zh
property, namely div Xh C Zh. Thus, using SV elements, weak mass conservation via
f div whzhdx = 0 Vzh G Zh implies strong (pointwise) mass conservation. We can choose q
the special test function zh = div wh to get || div wh|L2(Q) = 0. In [21] it was shown that the SV space pair (k = 2) is LBB-stable.
Now we introduce a special set of basis functions and construct a weighted finite element scheme for the Stokes problem (3), (4). Each node Mk G RQe1 (Nl G Rpres) is associated with a function
9k (x) = pv * (x) • ^ (x), (^xi (x) = p(x) • (x)) ,k = 0,1,..., ( l = 0,1,...),
where Vk G Xh, pk(Mj) = 5kj for k,j = 0,1,... G Zh ,^l(NJ) = 5l], l,j = 0,1,.. ^; Sms is Kronecker delta, v^d ¡i* are real constants.
The sets Vh and Qh for the velocity components and pressure are defined as the linear span of the system of basis functions {9k }k and {^^respectively. In Vh we consider
the subset V0 = {vh E Vh : vh(Mk)\MkeRvei = 0}. Associated with the constructed triangulation, the finite element approximation of the displacement velocity components and pressure have the form
<i(x) = d2k Ok(x), uh ^2(x) = Y; d2k+i Ok(x), ph(x) = J] ei xi(x), (5)
k k i
where dj = p~v* (Mj/2]) dj ,ei = p-M* (A^ e^ The coefficients dj and ei in (5) are defined from the system of equations (see (8)) below.
The corresponding velocity field sets are denoted by Vh = Vh x Vh and V0h = V0h x V)h.
o
Obviously, Vh c Wl,v(Qh,6), Vh CW1 ,v (Qh,6) and Qh C L2,,^(Qh,6).
Definition 2. The approximate Rv-generalized solution of the Stokes problem (3), (4) produced by the weighted finite element method is the pair (u'h(x),p,h(x)) E Vh x Qh such that each component o/u^x) at nodes of the set RTel satisfies the boundary condition (4) and, for an arbitrary pair (vh(x),ph(x)) E V(h x Qh and v > ¡3, we have the equalities
a(uhv, vh) + b(vh,phv) = l(vh), (6)
c(uh,qh) = 0, (7)
where uh = (uh, 1 uh22Wd f E L2 #(Q, 6), g E Wl/¡¡(r,5).
The finite element problem (6), (7) generates a system of linear equations with a saddle point matrix
A B
CT 0
In our case, A is a positive definite square matrix, B and CT are non-square matrices, Z = uh,^ = ph, y = Fh, z = 0.
3. Iterative Method
The system of linear algebraic equations (8) is large and sparse, making direct solutions infeasible. We construct a convergent iterative process [22] of the following form:
1) select an initial guess n°,Z0 to the solution of (8);
2) for k = 0,1, 2,..., until converge do;
3) compute Zk+1 = Zk + A-1(y - A(k - Br/k);
4) compute nk+1 = Vk + S-1(CTZk+1 - z);
5) end do,
where ^^d S are preconditioning mat rices to A and the Schur complement S = CT A-1B respectively.
To construct matrix A, we use an incomplete LU factorization of A - ILU(0) [23], i.e., aA = L • U, where L and U are lower and upper triangular matrices respectively. At each iteration in 3, we solve the problem Aq = x with the left preconditioner A. We use the generalized minimal residual method (GMRES(n)) [23]. The method approximates the solution by the vector in a n-th Krylov subspace with minimal residual. Let r0 = A-1 (x — Aq), then the Arnoldi loop constructs an orthogonal basis of the left preconditioned n-th Krylov subspace: Span{r0, aA-1Ar0,..., (.A-1A)ra-1r0}, n = 10.
С У
п z
(8)
Further, we construct an auxiliary matrix S to S, as a weighted mass matrix of
the pressure space, on each L E Yh :
1
S
(M7*) j = 1 p2^ ^ (x) ^ (x)dx, l,j = 0,1,....
Then, we define the diagonal matrix S = Mp^*, where (Mp^*) = (MIt is
ii k i
well known (see [24]). that such diagonal lumping is a good preconditioner to the initial weighted mass matrix.
It derives from the above that at each iteration in 4 one finds a vector := S-19 as a solution of internal procedure:
1) 0o = 0; 2) 0m = + S-1(e - S 0m_i) (m =1,...,M); 3) ^ = 0m ■
We use restart GMRES(k): (Span{r, S_1Sr,..., ( S-1S)k-1 r}, r = S-1(6-S0m_i),k = 5).
4. Numerical Experiments
In this section we present the results of numerical experiments to illustrate the behaviour of our method applied to the Stokes problem (3), (4) with viscosity S = 1.
Let Q = (-1,1) x (-1,1) \ Di are non-convex polygonal domains with one reentrant corner ui : u1 = = if= on its boundary with the vertex located in the
origin (0, 0). We divide Si into a set of closed triangles {Lm}, where each Lm is a half of a closed square of the size h for corners ui,i = 1, 2; a half of a closed square of the size h in S3 = [-1; 1] x [0; 1] and a half of a closed rectangle with sizes h and | in Q3 \ S3 for a corner u3. Then, each Lm (macro-element) is refined (barycenter refinement) into three triangles Km., their set {Ks} (see Fig. 1).
We use the exact solution (u, p) of the Stokes problem (3), (4), which exhibits corner singularity phenomena at the reentrant corner ui on its boundary with the vertex located in the origin (0, 0). In polar coordinates (r, p) at the origin the exact solution for each ui,i = 1, 2, 3, is given by (see e.g. [25]):
U1(r, p) = rXi • ((1 + Ai) • sin(p) + tf'(p) • cos(p)),
U2(r, p) = rXi • (tf'(p) • sin(p) - (1 + Ai) • cos(p)),
, , A-1 (1 + Ai)2 tf'(p) + tf'''(p)
p(r, p) = -rAi 1 • ----- ,
1 - Ai
sin((l + Ai)p)cos(AiWi) sin((l - AiV)cos(A^i) nW =-1+A--cos((1 + Ai)^)---+ cos((1 - Ai)v).
Ai
sin(A^i) + A si n(^) = 0,
which is Ai « 0,544483 for ^ = f, A2 « 0,673583 for = 5f 'A3 ~ 0,800766 for = . These solutions satisfy the Stokes problem (3), (4), where f = 0. We emphasize that the pair (u, p) is analytical in Qi \ (0, 0), but V^Md p are singular at the origin. Especially, u ^ W;|(n^d p ^ W2(n). This solution reflects perfectly the typical behaviour of the solution of the Stokes problem near a reentrant corner.
Table 1
The error norm ||uh — u||Wi(n) of the generalized solution, v = 0, 5 = 1, v* = ¡1* = 0
Ui N = 80 N = 160 N = 320
3п 2 2,768е-1 1,898е-1 1,302е-1
5п 4 1,538е-1 9,649е-2 6,050е-2
9п 8 6,321е-2 3,627е-2 2,082е-2
Table 2
The influence of parameters 5 and v on the behaviour of the error ||u£ — u||Wi (n) of the ^-generalized solution, v* = ¡1* = Ai — 1
Ui V 8 N = 80 N = 160 N = 320
3п 2 1,5 0,0125 2,619е-4 1,303е-4 6,475е-5
0,015 3.9! Ос-! 1,958е-4 9,744е-5
1,8 0,0125 7,153е-5 3.551 с-5 1,769е-5
0,015 1.1 !!<•-! 5,709е-5 2,824е-5
5п 4 1,5 0,0125 1,362е-4 6,813е-5 3,404е-5
0,015 2,140е-4 1,071е-4 5,332е-5
1,8 0,0125 3,790е-5 1,886е-5 9,399е-6
0,015 6,242е-5 3,107е-5 1,546е-5
9п 8 1,5 0,0125 7,581е-5 3,780е-5 1,880е-5
0,015 9,826е-5 4,891е-5 2,428е-5
1,8 0,0125 2,039е-5 1,017е-5 5,061е-6
0,015 2,827е-5 1,409е-5 7,010е-6
Numerical experiments were carried out on meshes with different step sizes h (numbers N,h = The errors of the numerical approximations to the ^-generalized and generalized ( v = 0, 5 = 1, v* = ¡1* = 0) solutions were computed as the module between
approximate and exact solutions in the points and in the norm of spaces Wlv(Q) and W21(Q) respectively. The results of the numerical experiments are presented in Tables 1, 2. The optimal values of parameters v and 8 were derived numerically.
For the determined approximate ^-generalized and generalized solutions in Table 3 we present the numbers of points (in percentage of their total number), where the errors 8ji = \uj(Mi) — u'hj(Mi)\,j = 1, 2, Mi E ROf1 (for the ^-generalized solution) and 8ji = \uj(Mi) — uh(Mi)\, j = 1, 2, Mi E Rvnel (for the generalized solution) are less than the given limit values △ k. In our experiments, the number of points (in percentage of their total number) for each component of the velocity field u is approximately the same.
Table 3
The number of points (in percentage of their total number), where the errors 81i and 8^ are less than the given limit values △ k
Rv-generalized, v =1, 5 6 = 0, 0125, v* = Ai - 1 generalized, v = 0 6 = 1,v* = 0
Ui A fc N = 80 N = 160 N = 320 N = 80 N = 160 N = 320
3n 2 10"4 36,1% 46,5% 64,2% 31,4% 41,1% 52,3%
10"5 15,7% 18,9% 29,1% 14,9% 15,4% 23,1%
5n 4 10"4 46,2% 59,2% 78,2% 41,3% 55,0% 70,9%
10"5 24,7% 29,8% 40,1% 20,8% 26,3% 35,2%
9n 8 10"4 72,3% 78,5% 89,4% 68,4% 73,7% 84,7%
10"5 38,7% 49,0% 66,2% 33,8% 46,4% 63,2%
□ .3-10~4<b'n<9-10-4
asn<3-io-4
Fig. 2. Distribution of the points Mk with errors for the component u^ of the approximate generalized solution = ,v = 0,6 =1, v* = ¡i* = 0 (left) with N = 160, (right) with N = 320
On Figs. 2, 3 we depict the distribution of the points Mi with err ors 6n and 8'u for the components u% ! and Uh of the approximate ^^-generalized and generalized solutions,
Fig. 3. Distribution of the points Mk with errors 51i for the component u1 of the approximate ^-generalized solu tion u1 = 3f ,v = 1, 5,5 = 0, 0125, v* = ¡i* = A1 — 1, (left) with N = 160, (right) with N = 320
Fig. 4. The dependence of — u||Wi (n) on the degree of v*, for u2 = 51
h
in W1 v(Q) norm on the degree v*(i* = v*) of the weight function p(x). Each minimum on Figs. 4, 5 corresponds to the optimal value v* for the respective v, 5 and u. This research was supported in through computational resources provided by the Shared Facility Center "Data Center of FEB RAS".
Fig. 5. The dependence of ||uh — u||Wi (V) on the degree of v*, for u3 =
Conclusions
The results of numerical experiments leads to the following conclusions:
• The approximate Rv-generalized solution (velocity field) of the Stokes problem (3), (4) converges to the exact one with the rate O(h) in the W1, v (Q) norm for all corners
= 1, 2, 3 (see Table 2), while the approximate generalized solution by classical FEM has an O(h0 ' 55) rate of convergence for a corner u1 = O(h0' 67) — for a corner U2 = 51, O(h0'8) — for a corner u3 = 91 in the W^Q) norm (see Table 1) (the so-called pollution effect [6]);
•
the difference between the approximate and exact solutions are less than the given limit values, more for the proposed weighted method in comparison with the classical FEM (see Table 3 and Figs. 2, 3);
• For all degrees v*(¡1* = v*) of the weight function p(x), that lie between Xi — 1 and 0, and parameters v, 8 close to optimal, the approximate Rv-generalized solution converges to the exact one with the rate O(h) in the W1, v(Q) norm (see Figs. 4, 5).
Acknowledgement. This paper is connected with problematic of the project no. 18-1100021 of Russian Science Foundation.
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Received December 20, 2017
УДК 519.63+532.51 DOI: 10.14529/mmpl80109
НОВЫЙ ПРИБЛИЖЕННЫЙ МЕТОД РЕШЕНИЯ ЗАДАЧИ СТОКСА В ОБЛАСТИ С УГЛОВОЙ СИНГУЛЯРНОСТЬЮ
В.А. Рукавишников1, А.В. Рукавишников2
1 Вычислительный центр Дальневосточного отделения Российской академии наук, г. Хабаровск, Российская Федерация
2Хабаровское отделение Института прикладной математики, г. Хабаровск, Российская Федерация
В статье определено понятие Д^-обобщенного решения задачи Стокеа с сингулярностью в двумерной невыпуклой многоугольной области с одним входящим углом на
границе области в специальных весовых множествах. Построено новое приближенное решение задачи с помощью весового метода конечных элементов. Предложен итерационный процесс решения полученной системы линейных алгебраических уравнений с блочным переобуславливанием ее матрицы на основе неполного алгоритма Удзавы и обобщенного метода минимальных невязок. Результаты численных экспериментов показали, что скорость сходимости приближенного Rv-обобщенного решения к точному решению задачи не зависит от величины входящего угла на границе области и равна первой степени по шагу сетки h в норме весового пространства (Q) для компонент вектора скоростей, в отличие от стандартных конечно-элементных и конечно-разностных схем, приближенное решение которых сходится к точному решению задачи не быстрее чем со скоростью O(ha) в норме пространства W^Q) для компонент вектора скоростей, где а < 1 и степень а зависит от величины входящего угла.
Ключевые слова: угловая сингулярность; весовой метод конечных элементов; предобуславливатель.
Литература
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22. Bramble, J.H. Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems / J.H. Bramble, J.E. Pasciak, A.T. Vassilev // SIAM Journal on Numerical Analysis. - 1997. -V. 34. - P. 1072-1092.
23. Saad, Y. Iterative Methods for Sparse Linear Systems / Y. Saad. - Minneapolis: University of Minnesota, 2003.
24. Olshanskii, M.A. Analysis of a Stokes Interface Problem / M.A. Olshanskii, A. Reusken // Numerische Mathematik. - 2006. - V. 103. - P. 129-149.
25. Verffirth, R. A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques / R. Verffirth. - Chichester; Stuttgart: Wiley-Teubner, 1996.
Виктор Анатольевич Рукавишников, доктор физико-математических наук, профессор, заведующий лабораторией, Вычислительный центр Дальневосточного отделения Российской академии наук (г. Хабаровск, Российская Федерация), [email protected].
Алексей Викторович Рукавишников, кандидат физико-математических наук, ведущий научный сотрудник, Хабаровское отделение Института прикладной математики Дальневосточного отделения Российской академии наук (г. Хабаровск, Российская Федерация), [email protected].
Поступила в редакцию 20 декабря 2017 г.
