CC BY
5
MSC 65N85
DOI: 10.14529/ mmp220304
ANALYSIS OF BIHARMONIC AND HARMONIC MODELS BY THE METHODS OF ITERATIVE EXTENSIONS
A.L. Ushakov1, E.A. Meltsaykin1
1 South Ural State University, Chelyabinsk, Russian Federation E-mail: ushakoval@susu.ru, e.meltsaykin@gmail.com
The article describes the results of recent years on the analysis of biharmonic and harmonic models by the methods of iterative extensions. In mechanics, hydrodynamics and heat engineering, various stationary physical systems are modeled using boundary value problems for inhomogeneous Sophie Germain and Poisson equations. Deflection of plates, flows during fluid flows are described using the biharmonic model, i.e. boundary value problem for the inhomogeneous Sophie Germain equation. Deflection of membranes, stationary temperature distributions near the plates are described using the harmonic model, i.e. boundary value problem for the inhomogeneous Poisson equation. With the help of the developed methods of iterative extensions, efficient algorithms for solving the problems under consideration are obtained.
Keywords: biharmonic and harmonic models; methods of iterative extensions.
Dedicated to the anniversary of Alexander Leonidovich Shestakov
Introduction
First, we consider the biharmonic model, i.e. mixed boundary value problem for the inhomogeneous biharmonic equation
А2 й = f (1)
in a bounded domain on the plane П С R2 with the boundary conditions of four types
U = IГо = 0, й = hit |Г1 = 0,
дй
— = 1\й |n = 0, l\u = l2u |r3 = 0, dn
where
дП = s, s = r„U riU ^U Гз, Г<р| Г = 0, i = j, i,j = 0,1, 2, 3,
1\й = Ай + (1 — o)n1n2Uxy — n2uxx — n1 Uyy,
дАй д 2 2
hи = + (1 - a) — (nin2(uyy - uxx) + (щ - n2)uxy),
n1 = — cos(n,x), n2 = — cos(n,y), a G (0; 1).
The biharmonic model can be formulated as a scalar model, i.e. the problem of representing a functional in the form of a dot product
й G H : [й, v] = F(v) W G H, F G H', (2)
where the Sobolev space is
H = H(Sl) = 1« G W?(ii) : £|roUri = 0, |roUr2 = 0 \ , the bilinear form, i.e. the dot product, is
[u, v] = A(u, v) = J (aAuAv + (1 — a)(uxxvxx + 2uxyvxy + uyyvyy))dQ, a G (0; 1). n
If f is a given function, then the functional
F (v) = (u, v) = J fvdQ.
n
For problem (2), the following assumption ensures the existence and uniqueness of its solution [1, 4]
3ci,c2 G : ci |vlw|(n) < A(v,^) < c2 ||vllwf(n) Vv G H•
Second, we consider the harmonic model, i.e. mixed boundary value problem for the inhomogeneous harmonic equation
—Au = f (3)
in a bounded domain on the plane Q C R2 with the boundary conditions of two types
u |ri = ° , du .
where
3Q = s, s = r^ r2, rif| r2 = 0.
The harmonic model can be formulated as a scalar model, the problem of representing a functional in the form of a dot product
u G H : [u, v] = F(v ) vv G H, F G H', (4)
where the Sobolev space is
H = H(Q) = {v G Wi(Q) : v|ri = 0} ,
the bilinear form, i.e. the dot product, is
|u'e] = A(u'8)^(ux8x +uy 8y )dQ •
n
If f is a given function, then the linear functional
F (v ) = (u, v) = [ fv dQ .
For problem (4), the following assumption ensures the existence and uniqueness of its solution [1, 4]
3ci, c2 G (°;+^) : ci ¡vlW2i(n) < A(v,v ) < c2 llvllW2i(n) vv G H.
Within the framework of the considered direction, such problems were studied by the fictitious domain methods, for example, in works by A.M. Matsokin, S.V. Nepomnyashchikh [3], S.B. Sorokin [5], G.I. Marchuk, Yu.A. Kuznetsov, A.M. Matsokin [2] and others. There are difficulties in solving the above problems. The promising direction of the fictitious domain methods for solving these problems also has difficulties. We use the fact that if the problems considered as systems are similar, then they have similar properties, and the methods for solving these problems are also similar to each other. To develop new efficient methods, we use generalizations of the fictitious domain method, i.e. methods of iterative extensions. In the fictitious domain method, on the example of mechanics, we increase the support reaction and the stiffness of the material on a fictitious continuation, i.e. additionally we use the choice of two parameters. Let us minimize the error in a norm stronger than the energy norm of the emerging problem. We apply the method of minimal residuals with indication of the conditions sufficient for its convergence. With this new approach, the relative errors of the proposed iterative processes are dominated by infinitely decreasing geometric progressions. The main goal of the described works is the development of asymptotically optimal methods for solving the above problems [6-12].
1. Analysis of Biharmonic Model 1.1. Biharmonic Model
Let us present the problem to be solved for u =1 and the fictitious problem for u = II
uw G Hw : A w(uw,vw ) = Fw (vw) vv w G Hw, Fw G HW, (5)
where we use Sobolev spaces
w w I dv
Hu = H„(Sl„) = G : ¿Ur^uiw = °> kouru, = o
in the bounded domains Qw C R2 with the boundaries
dQw = Sw, Sw = rw,0 y rw,iU rw,2 y rw,3, rw,f| rw,j = 0, ifi = = 0,1, 2, 3
nw are outer normals to dQw, bilinear forms at aw G [0; , aw G (0; 1) are
Aw (uwA) = /(aw Auw Avw + (1 — aw )(uwxxvwxx + 2uwxyvwxy + WW) + aw uw vw )dQw •
Each of the problems in (5) has a unique solution under the assumptions [1,4]
3ci,C2 G (0;+ro) : ci I|vw |W2(n„) < Aw (vw A) < c2 ||vw |W2(n„) Vvw G Hw •
If / is a given function, then
(vw) = J / .
In the problem to be solved, with u = 1, a1 = 0, r1;0 = 0. In the fictitious problem with u = II, /u = 0, mii = 0.
1.2. Continued Biharmonic Model and Its Analytical Study
Let us present the continued problem
u G V : A1(u,/1v)+AII(u,v) = F1(/1 V) Vv G V, (6)
where we use the extended solution space
dv
y=y(n)=iiiG^(n): ^IroUTi = ^
=0
ro U r2
We assume that the solution domain of the original problem is complemented to the rectangle
Q 1 U QII = n, ^ p| Q„ = 0, n1, c R2,
and the boundary of the rectangular domain is
dn = s, s = r^ r2, r^ r2 = 0.
We assume that the boundaries of the first domain and the second domain intersect each other
ddQ„ = S, S = ^ f| rn>s = 0,
n is an outer normal to dn. Subspace of solutions of the continued problem is
V = V1 (n) = |v1 G V : v 1 |nQl = 0} .
In the formulation of the continued problem, we use the projection operator
/1 : V ^ V1, V1 = im/1, /1 = /2-
We introduce subspaces
V3 = Vs(n) = {V3 G V : Va|nnn = 0} , V. = V1 © V3, V2 = V2(n) = {v 2 G V : A(V2, Vo) = 0 Wo G V.} ,
V = V1 © V2 © V3 = V1 © Vii, VI = V1 © V2, VII = V2 © V3.
Direct sums are considered using the inner product generated by the bilinear form
A(u, v ) = A1(u, v ) + AII(u, V) Vu, v G V.
54 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2022, vol. 15, no. 3, pp. 51-66
It is assumed that the bilinear form is such that
3C1,C2 > 0 : C1 ||v|W22(n) < A(v, V) < C2 ||v|W22(n) Vv G V.
We use the statement on the possibility to continue the functions
3/ 1 G (0; 1], / G [A; 1] : M^^) < Aii(V2,V2) < M^^) Vv2 G V2. Note that
H) = Vw), u G {1, II} .
Statement 1. [12] The solution to problem (6) u G V coincides with the solution to problem (3) for u =1 on Q1 and equals zero on QII.
The study of the continued biharmonic model is carried out by the modified method of fictitious components [6, 7, 9, 10]:
uk G V : A(uk - uk-1,v ) = -rfc-1(A1(ufc-1,/1v ) + Aii(uk-1,v ) - F\(/1V )) Vv G V,
To = 1, Tk-1 = T = 2/(/ 1 + /2), k G N\ {1} , Vu0 G V1 c V. (7)
Let us introduce the norm
= \JA(v,v).
Theorem 1. [12] There exist the following convergence estimates:
||uk - u\y < e ||u0 - u||v>, k G N,
where
£ = 5iqk~\ S1 = Q<q=02~h)/0i+h)<l.
1.3. Continued Biharmonic Model under Discretization and Its Numerical Analysis
Let us discretize the continued model when
n = (0; 61) x (0; 62), r = {61} x (0; 62) U(0; 61) x {62} ,
r2 = {0} x (0; 62) U(0; 61) x {0} ,61,62 G (0; Let us introduce the grid
(xi; yj) = ((i - 1,5)h1;(j - 1,5)h2),
h1 = 61/(m - 1, 5), h2 = 62/(n - 1, 5), i = 1, 2...,m, j = 1, 2...,n, m - 2,n - 2 G N. We consider grid functions at the grid nodes
v%j = v(xj; yj) G R, i = 1, 2..., m, j = 1, 2..., n, m - 2, n - 2 G N.
Use the completion for the grid functions
(x; y) = (x)^2'j(y), i = 2,..., m - 1, j = 2,...,n - 1, m - 2,n - 2 G N,
^M(x) = [2/i]^(x/h - i + 4) + ^(x/hi - i + 3) - [(i + 1)/m]^(x/hi - i + 1), (y) = [2/jMy/h2 - j + 4) + ^(y/h2 - j + 3) + [(j + 1)/nMy/h2 - j + 1),
tf(z) =
0, 5z2, -z2 + 3z - 1, 5, 0, 5z2 - 3z - 4, 5, 0,
z G [0; 1],
z G [1; 2],
z G [2; 3],
z / (0; 3).
We consider the basis functions to be equal to zero outside the rectangle:
$ij(x; y) = 0, (x; y) / n, i = 2...,m - 1, j = 2...,n - 1, m - 2,n - 2 G N.
Linear combinations of basis functions give a finite-dimensional subspace in the extended space
m— 1n— 1
V = < v = > > v
(x; y) c V.
i=2 j=2
Consider the continued model in the matrix form
> N
u G RN : Bu = f, f G R
N
(8)
under the assumption that the projection operator vanishes the coefficients of the basis functions whose carriers do not belong entirely to the first domain, and the continued matrix and the continued right-hand side of the system are defined by the equalities
(Bu,v) =A1(u,/1 £)+A„(u,v) Vu, v G V, </,v) = F1(/1v) Vv G V,
</,v) = (Z,v)h1h2 = / vh1 h2, v = (v1,v2,...,vN)' g Rn, N = (m - 2)(n - 2).
In this case, we enumerate first the coefficients of the basis functions with carriers that belong entirely to the inside of the first domain. Next, we enumerate the coefficients of the basis functions with the carriers that cross the boundary of both the first and second domains. We finish the enumeration with the coefficients of the basis functions with carriers that belong entirely to the inside of the second domain. Then the vectors have the following structure
v = (v', v2, v3)', u = (u, 0', 0')', / = (/1, 0', 0')'.
The matrix has the structure
A11 A12 0
B = 0 A02 A23
0 A32 A33
We define the matrices
(Aju,v ) =A1(u,v), (Anu,v )
The matrices have the structure
An(û, v) Vu, v G V.
A11 A12 0 0 0 0
Ai = A21 A20 0 , Aii = 0 A02 A23
0 0 0 0 A32 A33
Л11 Л12 0 Л11 Л12 0 0 0 0
Л21 Л22 Л23 = Л21 Л20 0 + 0 Л02 Л23
0 Л32 Л33 0 0 0 0 Л32 Л33
Define the extended matrix A = Ai + An =
We introduce the corresponding subspaces
0 = {V = (v', /2, /3)' G RN : v2 = 0, v3 = 0
V3 = {0 = (oi, v2, v3)' G RN : Vi = 0, V2 = 0} , Vo = Vi © V3,
V2 = IV = (v',02,v3)' G Rn : AiiVi + Ai2V2 = 0, A32/2 + A33V3 = 0 There exist decompositions
Rn = Vi © V2 © V3 = Vi © Vii, Vi = Vi © V2, V„ = V2 © V3. Let us present the assumptions about the continuation in the matrix form
3ft G (0; ft G [ft; : ft (A02,02) < (Aii/2,02) < ftA (Av2,/2) VV2 G V>. The matrix form of the continued biharmonic model is
Л11 Л12 0
Bu = / 0 Л02 Л23
0 Л32 Л33
" u1 " /1 "
0 = 0
0 0
The original problem in the matrix form and the fictitious problem in the matrix form are
Aii ui = /i, a02 A23
A32 A33
U2 0 U2 0
U3 0 , U3 0
When studying the continued biharmonic model in the matrix form, we define the extended matrix in a new way as follows:
C = Ai + y An,
C11 C12 0 C21 C22 C23 0 C32 C33
Лц Л12 0 Л21 Л20 0 0 0 0
0 0 0
0 Л02 Л23 0 Л32 Л33
Y G (0;
We use the fulfilment of the statements about the continuation of functions in the following form:
3yi G (0; Y2 G [yi; : Y2 (C02, CV2) < (AA11V2, An/2) < Y^ (CV2, CV2) VV2 G V2,
3a G (0; : (Ai02, A1V2) < a2 (An02, A11O2) VV2 G V2.
To solve problem (8), as a generalization of the modified method of fictitious components, we apply the method of iterative extensions [8, 9, 11,12]:
uk G RN : C(uk - uk-i) = -rfc-i(BUfc-i - /), k G N, (9)
Vu0 G Vi, Y > a, To = 1, Tk-i = (rfc-i,nfc-i)/(nfc-i,nfc-i), k G N {1} ,
where residuals, corrections, and equivalent residuals are respectively calculated ffc-1 = Buk-1 - f, wk-1 = C-1ffc-1, n k-1 = Bwk-1, k G N. Let us define the norm
||t)||C2 = V(C2v,v) Vv G Rw. Theorem 2. [14] Process (9) has the following estimate
||ufc - u||C 2 < e ||u° - u||c2 , e = 2(72/71)(a/7)fc-1, k G N.
Let us present an algorithmic implementation of the method of iterative extensions for the biharmonic model. We use the method of minimal residuals to solve problem (8).
I. Set the initial approximation and the iterative parameter
Vu° G V, to = 1.
II. Calculate the residual
ffc-1 = Buk-1 - f, k G N.
III. Calculate the absolute error norm squared
Efc-1 = (ffc-1,ffc-1> , k G N.
IV. Find the correction
wk-1 : Cwk-1 = ffc-1, k G N.
V. Calculate the equivalent residual
nk-1 = Bwk-1, k G N {1} .
VI. Calculate the iteration parameter
Tfc-1 = (ffc-1, nk-1> / (nk-1, nk-1>, k G N {1} .
VII. Calculate the next approximation
uk = uk-1 - Tfc-1wfc-1, k G N.
VIII. Check the iteration stop criterion
Efc-1 < E°E2, k G N {1} , E G (0; 1).
2. Analysis of Harmonic Model 2.1. Harmonic Model
Let us present the problem to be solved for u = 1 and the fictitious problem for u = II
uu G Hu : Au (uu, Hu ) = Fu (Hu) VH u G Hu , Fu G hu , (10)
58 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2022, vol. 15, no. 3, pp. 51-66
where we use the Sobolev spaces
= (Qw) = {vw G W^w) : vw|rw1 = 0} in the bounded domains Qw C R2 with the boundaries
dQw = 0w, Sw = rw>2, rW)i P| rw>2 = 0,
nw are outer normals todQw, bilinear forms at Kw G [0; are
AW (UW , VW ) J'' (uwxvwx + uwy vwy + Kwuwvw )dQw •
Each of the problems in (10) has a unique solution under the assumptions [1, 4]
3ci,c2 G (0;+ro) : ci ||tw||W2i(nw) < Aw(vw,vw) < C2 ||#w||W2i(nw) VVw G , If fw is a given function, then
Fw ) = J fwVw dQw •
In the problem to be solved with u =1, K' = 0, ri,i = 0. In the fictitious problem with u = II, fII = 0, uII = 0.
2.2. Continued Harmonic Model and Its Analytical Study
Let us present the continued problem
u G V: a'(u,/iv) + AII(uft) = f'(i'v) Vv G V (11)
where we use the extended solution space
V = V(n) = {v G W2'(n) : v|ri = 0} .
We assume that the solution domain of the original problem is complemented to the rectangle
Q^QII = n, ^ p| Qn = 0, Qn c R2,
and the boundary of the rectangular domain is
dn = 0, s = r^ r2, rip| r2 = 0.
We assume that the boundaries of the first domain and the second domain intersect each other
5QiQ dQ„ = ft S = ri,^ rn>2 = 0,
n is an outer normal to dn. The subspace of solutions to the continued problem is
V1 = V1(n) = {v1 G V: V1|nQl = 0} .
In the formulation of the continued problem, we use the projection operator
/1 : V ^ V1, H = im/1, /1 = /?.
We introduce subspaces
V3 = Va(n) = {H3 G H : = 0} , V° = V1 © V3,
V2 = V2(n) = |h2 G H : A(H2,h°) = 0 VH° G V°} ,
H = V © V2 © V3 = V © vii, hi = V © V2, vii = V2 © V3.
Direct sums are considered using the dot product generated by the bilinear form
A(u,H) = A1(u,H) + AII(u,H) Vu, H G V. It is assumed that the bilinear form is such that
3c1 ,C2 > 0 : C1 ||H|W2i(n) < A(H,H) < C2 ||H|W2i(n) VH G V. We use the statement on the possibility of continuing the functions
3/51 G (0; 1], H G 1] : M^,^) < A„(H,#2) < M^,^) VH2 G V2 Note that
HHu(Qu) = Vu(Qu), UG {1, II} .
Statement 2. [12] The solution to the problem (11) u G V1 coincides with the solution to problem (10) foru = 1 on Q1 and equals to zero on QII.
The study of the continued harmonic model is carried out by the modified method of fictitious components [6, 8, 9]:
uk G V : A(uk - uk-1,H) = -Tfc-1(A1(ufc-1,/1H) + Aii(uk-1,H) - ^(/1H)) VH G V,
t° = 1, Tfc-1 = t = 2/(/51 + /^2), kG N\ {1} , Vu°G V1 C V. (12)
Let us introduce the norm
= y/A(v,v) .
Theorem 3. [12] There exist the following convergence estimates:
||uk - u||^ < e ||u° - u|, k G N,
where
£ = S1qk~1, 61 = ^\\I1\\l-l, O<q=02-Pi)/0i+P2)<l.
2.3. Continued Harmonic Model under Discretization and Its Numerical Analysis
Let us discretize the continued model when
n = (0; bi) x (0; ), r = {bi} x (0; ^(0; bi) x |b2} ,
r2 = {0} x (0; b2^(0; bi) x {0} , bi,62 G (0; Let us introduce the grid
(xi; Vj) = ((i - 1> 5)hi;(j - 1> 5)h2),
hi = bi/(m - 1, 5), h2 = b2/(n - 1, 5), i = 1, 2...,m, j = 1, 2...,n, m - 2,n - 2 G N. We consider the grid functions at the grid nodes
Vij = v(x^ Vj) G R, i = 1, 2..., m, j = 1, 2..., n, m - 2, n - 2 G N. Use completion for the grid functions
$i>j(x; y) = (x)^2'j(y), i = 2..., m - 1, j = 2...,n - 1, m - 2,n - 2 G N,
(x) = [2/i]^(x/hi - i + 3, 5) + ^(x/hi - i + 2, 5), (y) = [2/jMy/h2 - j + 3, 5) + tf(y/h2 - j + 2, 5),
f z, z G [0; 1], *(z) = S 2 - z, z G [1;2],
I 0, z/ (0; 2).
We define the basis functions to be equal to zero outside the rectangle:
$i>j(x; y) = 0, (x; y) / n, i = 2..., m - 1, j = 2...,n - 1, m - 2,n - 2 G N.
Linear combinations of basis functions give a finite-dimensional subspace in the extended space
!m— in— i ^
* = £ i,j$i,j (x; v) c V. i=2 j=2 J
Consider the continued model in the matrix form
u G Rn : Bu = f, f G RN , (13)
under the assumption that the projection operator vanishes the coefficients of the basis functions whose carriers do not belong entirely to the first domain, and the continued matrix and the continued right-hand side of the system are defined by the equalities
(Bu,V) =Ai (U, 1iV) + Aii (U,v) VU, v G Ù, (f, V> = Fi(1i v) VV G Ù,
</,V> =(f,V)hih2 = f Vhih2, V=(vi,V2,...,VN )' G Rn , N = (m - 2)(n - 2).
In this case, we enumerate first the coefficients of the basis functions with carriers that belong entirely to the inside of the first domain. Next, we enumerate the coefficients of the basis functions with the carriers that cross the boundary of both the first and second domains. We finish the enumeration with the coefficients of the basis functions with carriers that belong entirely to the inside of the second domain. Then the vectors have the following structure
V = (V'i, v2, V3)', u = (u'i, 0', 0')', / = (/i, 0', 0')'.
The matrix has the structure
A11 A12 0
B = 0 Ao2 A23
0 A32 A33
We define the matrices
(Aiu,v ) = A1(U,v), (AuU,V ) The matrices have the structure
AII(U, v) VU, û G V.
A11 A12 0 0 0 0
AI= A21 A2o 0 , AII = 0 Ao2 A23
0 0 0 0 A32 A33
Define the extended matrix
A11 A12 0 A11 A12 0 0 0 0
A = Ai + Aii = A21 A22 A23 = A21 A2o 0 + 0 Ao2 A23
0 A32 A33 0 0 0 0 A32 A33
We introduce the corresponding subspaces
V1 = {v = (v 1,4,4)' G Rn : v2 = 0, U3 = 0}
V3
v = (v1, v2, v3)' G Rn : v i = 0, V2 = 0 ¡> , V0 = V © Fa,
V> = |v = (v 1, v 2,4)' G RN : A11 u 1 + A12V2 = 0, A32H2 + A33V3 = 0 j . There exist decompositions
Rn = V © V2 © V3 = V © Vii , VI = V © V2, VII = V2 © V3. Let us present the assumptions about the continuation in the matrix form
3ft G (0; ft G [ft; : ft (Av2,£2) < <Anv2,£2) < ft (Av2,£2) Vv2 G V2.
The matrix form of the continued harmonic model is
A11 A12 0
Bu = /, 0 Ao2 A23
0 A32 A33
u1 " /1 "
0 = 0
0 0
The original problem in the matrix form and the fictitious problem in the matrix form are A11u1 = /h a02 A23
A32 A33
U2 0 U2 0
U3 0 , U3 0
When studying the continued harmonic model in the matrix form, we define the extended matrix in a new way as follows:
C = Ai+y An,
C11 C12 0 C21 C22 C23 0 C32 C33
A11 A12 0 A21 A2o 0 0 0 0
0 0 0
0 Ao2 A23 0 A32 A33
, Y G (0; +œ).
We use the fulfilment of the statements about the continuation of functions in the following form:
371 G (0; 72 G [71; : 72 (Cv2, C^) < (An^, An^) < (Cv2, C^) Vv2 G V2,
3a G (0; : (A^, A1V2) < a2 (A„v2, An^) W2 G V2.
To solve problem (13), as a generalization of the modified method of fictitious components, we apply the method of iterative extensions [8, 9, 11,12]:
uk G RN : C(uk - uk-1) = —Tfc_i(BUfc-1 - f), k G N, (14)
Vu° G Vi, y > a, to = 1, Tfc_i = <ffc_1,nfc_1)/<nfc_1,nfc_1>, kG N {1} , where residuals, corrections, and equivalent residuals are respectively calculated as
rk-1 = Buk-1 - f , wk-1 = C-1rk-1, n k-1 = Bwk-1, k G N.
r
Let us define the norm
\\v\\c2 = ^{C2v,v) Vv G Rn. Theorem 4. [15] Process (14) has the following estimate:
||uk — u||C 2 < e ||u° — u||c2 , e = 2(Y2/Y1)(a/Y)k_1, k G N.
Let us present an algorithmic implementation of the method of iterative extensions for the biharmonic model. We use the method of minimal residuals to solve problem (13).
I. Set the initial approximation and the iterative parameter
Vu° G V1, t° = 1.
II. Calculate the residual
ffc_1 = Buk_1 — f, k G N.
III. Calculate the absolute error norm squared
Efc_1 = <ffc_1,ffc_1> , k G N.
^fc
IV. Find the correction
wk-1 : Cwk-1 = rk-1, k G N.
V. Calculate the equivalent residual
Пк-1 = Bwk-1, k G N {1} .
VI. Calculate the iteration parameter
Tk-1 = <rfc-1,n fc-1)/<n k-1,r fc-1>, kG N {1} .
VII. Calculate the next approximation
= uk-1 - rfc-1wfc-1, k G N.
VIII. Check the iteration stop criterion
Efc-1 < EoE2, k G N {1} , E G (0; 1).
Conclusion
The biharmonic and harmonic problems considered as models and systems are similar, have similar properties and similar methods for their solution. With necessary changes, the corresponding results for the biharmonic and harmonic models hold for the scalar model.
Acknowledgements. The authors express their gratitude to Alexander Leonidovich Shestakov as the head of the South Ural State University, which provides ample opportunities for scientific research in the South Urals.
References
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Received 17 March, 2022
УДК 519.63 БЭТ: 10.14529/mmp220304
АНАЛИЗ БИГАРМОНИЧЕСКИХ И ГАРМОНИЧЕСКИХ МОДЕЛЕЙ МЕТОДАМИ ИТЕРАЦИОННЫХ РАСШИРЕНИЙ
А.Л. Ушаков1, Е.А. Мельцайкин1
1Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
В статье приводится описание результатов за последние годы по анализу бигармо-нических и гармонических моделей методами итерационных расширений. Различные стационарные физические системы в механике, гидродинамике, теплотехнике моделируются с помощью краевых задач для неоднородных уравнений Софи Жермен и Пуассона. Используя бигармоническую модель, т.е. краевую задачу для неоднородного уравнения Софи Жермен, описывают прогибание пластин, потоки при течениях жидкостей. Используя гармоническую модель, т.е. краевую задачу для неоднородного уравнения Пуассона, описывают прогибания мембран, стационарные распределения температур у пластин. С помощью разработанных методов итерационных расширений получаются эффективные алгоритмы решения рассматриваемых задач.
Ключевые слова: бигармонические и гармонические модели; методы итерационных расширений.
Литература
1. Aubin, J.-P. Approximation of Elliptic Boundary-Value Problems / J.-P. Aubin. - New York: Wiley-Interscience, 1972.
2. Bank, R.E. Marching Algorithms for Elliptic Boundary Value Problems. I: the Constant Coefficient Case / R.E. Bank, D.J. Rose // SIAM Journal on Numerical Analysis. - 1977. -V. 14, № 5. - P 792-829.
3. Marchuk, G.I. Fictitious Domain and Domain Decomposion Methods / G.I. Marchuk, Yu.A. Kuznetsov, A.M. Matsokin // Russian Journal Numerical Analysis and Mathematical Modelling. - 1986. - V. 1, № 1. - P. 3-35.
4. Swarztrauber, P.N. A Direct Method for Discrete Solution of Separable Elliptic Equations / P.N Swarztrauber // SIAM Journal on Numerical Analysis. - 1974. - V. 11, № 6. -P. 1136-1150.
5. Swarztrauber, P.N. The Method of Cyclic Reduction, Fourier Analysis and FACR Algorithms for the Discrete Solution of Poisson's Equations on a Rectangle / P.N Swarztrauber // SIAM Review. - 1977. - V. 19, № 3. - P. 490-501.
6. Manteuffel, T. An Incomlete Factorization Technigue for Positive Definite Linear Systems / T. Manteuffel // Mathematics of Computation. - 1980. - V. 38, № 1. - P. 114-123.
7. Matsokin, A.M. The Fictitious-Domain Method and Explicit Continuation Operators / A.M. Matsokin, S.V. Nepomnyaschikh // Computational Mathematics and Mathematical Physics. - 1993. - V. 33, № 1. - P. 52-68.
8. Mukanova, B. Numerical Reconstruction of Unknown Boundary Data in the Cauchy Problem for Laplace's Equation / B. Mukanova // Inverse Problems in Science and Engineering. -2012. - V. 21, № 8. - P. 1255-1267.
9. Оганесян, Л.А. Вариационно-разностные методы решения эллиптических уравнений / Л.А. Оганесян, Л.А. Руховец. - Ереван: Издательство АН Армянской ССР, 1979.
10. Sorokin, S.B. Analytical Solution of Generalized Spectral Problem in the Method of Recalculating Boundary Conditions for a Biharmonic Equation / S.B. Sorokin // Siberian Journal Numerical Mathematics. - 2013. - V. 16, № 3. - P. 267-274.
11. Сорокин, С.Б. Экономичный прямой метод численного решения задачи Коши для уравнения Лапласа / С.Б. Сорокин // Сибирский журнал вычислительной математики. -2019. - Т. 12, № 12. - C. 87-103.
12. Ушаков, А.Л. О моделировании деформаций пластин / А.Л. Ушаков // Вестник ЮУр-ГУ. Серия: Математическое моделирование и программирование. - 2015. - Т. 8, № 2. -С. 138-142.
13. Ushakov, A.L. Investigation of a Mixed Boundary Value Problem for the Poisson Equation / A.L. Ushakov // International Russian Automation Conference. - 2020. - Article ID: 9208198. - 6 c.
14. Ushakov, A.L. Numerical Anallysis of the Mixed Boundary Value Problem for the Sophie Germain Equation / A.L. Ushakov // Journal of Computational and Engineering Mathematicsю - 2021. - V. 8, № 1. - P. 46-59.
15. Ушаков, А.Л. Анализ смешанной краевой задачи для уравнения Пуассона / А.Л. Ушаков // Вестник ЮУрГУ. Серия: Математика, Механика, Физика. - 2021. - Т. 13, № 1. -C. 29-40.
16. Ushakov, A.L. Research of the Boundary Value Problem for the Sophie Germain Equationinin in a Cyber-Physical System / A.L. Ushakov // Studies in Systems, Decision and Control. -2021. - V. 338. - P. 51-63.
17. Ushakov, A.L. Аnalysis of the Boundary Value Problem for the Poisson Equation / A.L. Ushakov // Вестник ЮУрГУ. Серия: Математика. Механика. Физика. - 2022. -Т. 14, № 1. - С. 64-76.
18. Ushakov, A.L. Anallysis of the Problem for the Biharmonic Equation / A.L. Ushakov // Journal of Computational and Engineering Mathematics. - 2022. - V. 9, № 1. - P. 43-58.
Андрей Леонидович Ушаков, кандидат физико-математических наук, доцент, кафедра математического и компьютерного моделирования, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), ushakoval@susu.ru.
Евгений Андреевич Мельцайкин, студент, кафедра математического и компьютерного моделирования, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), e.meltsaykin@gmail.com.
Поступила в редакцию 17 марта 2022 г.
