Reports of the Academy of Sciences of the USSR

1. Vol. 163, No. 6

MATHEMATICS

E. G. D’yakonov

ON THE APPLICATION OF SPECTRALLY EQUIVALENT OPERATORS TO THE SOLUTION OF DIFFERENCE ANALOGUES OF STRONGLY ELLIPTIC SYSTEMS

(Presented by Academician S. L. Sobolev on 11 XII 1964)

The present work is a continuation and strengthening of works ((^{1-6})).

1. Let (H_h) be a finite-dimensional Hilbert space of vector-functions (v=(v_1,v_2,\ldots,v_N)), defined on some set (\Omega_h) of mesh points in the space of variables (x_1,x_2,\ldots,x_p), where (h) is a certain vector with positive components, determining (\Omega_h) and (H_h) uniquely. The scalar product in (H_h) of the functions (u_h, v_h) will be denoted by ((u,v)_h). By (|h|) we shall understand the upper bound of the components of (h).

Let (B_h, L_h) be operators mapping (H_h) into (H_h), and let (B_h) be linear. We shall say that the positive definite operator (B_h) is spectrally equivalent to (L_h) and write (B_h \sim L_h), if there exist such numbers (h_0>0,\delta_1\geqslant\delta_0>0,\delta_2\geqslant0,\delta_3>0) that

[
\delta_0(B_h(u-v),u-v)_h \leqslant (L_hu-L_hv,u-v)_h \leqslant
\delta_1(B_h(u-v),u-v)_h,
\tag{1}
]

[
\delta_2(B_h(u-v),u-v)_h \leqslant
(B_h^{-1}(L_hu-L_hv),L_hu-L_hv)_h \leqslant
\delta_3(B_h(u-v),u-v)_h
]

for any (u_h\in H_h,\ v_h\in H_h,\ h\leqslant h_0). The numbers (\delta_0,\delta_1,\delta_2,\delta_3) will be called equivalence estimates.

Introduce the notation: (H(B_h)) is the space with scalar product
[
(u,v)_{B_h}\equiv (B_hu,v)_h;
]
(A^) and (A^+) are the operators adjoint to (A), respectively in (H_h) and (H(B_h)); (\lambda(A)) are the eigenvalues of (A). Then, if (L_h^=L_h) and (1) is fulfilled, then
[
M\equiv B_h^{-1}L_h=M^+
]
and (\lambda(M)\in[\delta_0,\delta_1]). Therefore it is expedient to replace the equation

[
L_hu_h=f_h
\tag{2}
]

by

[
Mu_h=B_h^{-1}f_h^*
\tag{2′}
]

and to seek (u_h), for example, by the iterative method

[
v_h^{\,n+1}=g_n\bigl(v_h^{\,n},\ldots,v_h^{\,0},Mv_h^{\,n},\ldots,Mv_h^{\,0},B_h^{-1}f_h\bigr),
\tag{3}
]

where (v_h^n) is the (n)-th approximation to (u_h); (g_n) depends on the method.

Let, for the equation

[
\widetilde B_h w_h=F_h
\tag{4}
]

there exist an iterative method (we shall call it a (T(\widetilde B))-method) that makes it possible rather easily to obtain the relation

[
w_h-y_h^m=T_m(w_h-y_h^0),\qquad |T_m|<1,
\tag{5}
]

where (y_h^0,y_h^m) are, respectively, the initial and final approximations in the (T(\widetilde B))-method, and (T_m) is a linear operator. Take

[
B_h=\widetilde B_h(E-T_m)^{-1},
\tag{6}
]

where (Ev_h\equiv v_h). Then the solution of the equation (B_hv_h=F_h) will be
[
v_h=B_h^{-1}F_h=y_h^m,
]
where (y_h^m) is the corresponding approximation in the (T(\widetilde B))-method for (4)

* This also applies to the case of a non-self-adjoint and nonlinear operator (L_h).

for (y_h^0=0). Therefore all (Mv^n) and (B_h^{-1}f_h) in ((2')) are easily found.

Lemma. If (\widetilde B_h\sim L_h) with estimates (\widetilde\delta_0,\widetilde\delta_1,\widetilde\delta_2,\widetilde\delta_3) and (T_m=T_m^), (|T_m|=q<1), (T_m\widetilde B_h=\widetilde B_hT_m), then
[
B_h\equiv \widetilde B_h(E-T_m)^{-1}\sim L_h
]
with estimates
[
\delta_0=\widetilde\delta_0(1-q),\quad
\delta_1=\widetilde\delta_1(1+q),\quad
\delta_2=\widetilde\delta_2(1-q)^2,\quad
\delta_3=\widetilde\delta_3(1+q)^2;
]
if (L_h=L_h^), then (\widetilde\delta_2\geqslant \widetilde\delta_0^2,\ \widetilde\delta_3\leqslant \widetilde\delta_1^2).

We note that the idea of replacing (2) by ((2')) for the case of differential operators is contained in ((^7)); for difference problems, operators (B_h) of the form (6) were used in ((^{1-6})).

2. We restrict ourselves to consideration of the method (3) corresponding to the Richardson method for ((2')):
[
B_hu_h^{\,n+1}=B_hu_h^{\,n}-\gamma_n(L_hu_h^{\,n}-f_h).
\tag{7}
]
Here (u_h^n) is the (n)-th approximation to (u_h), the solution of (2); (\gamma_n) is a parameter.

Theorem 1. Under condition (1), the solution of (2) exists, and if (\gamma_n=\gamma^0), where (\gamma^0) corresponds to
[
\inf_{\gamma>0}\ \sup_{\delta_0\leqslant \lambda\leqslant \delta_1 \atop \delta_2\leqslant \xi\leqslant \delta_3}
\left|1-2\gamma\lambda+\gamma^2\xi\right|
\equiv \rho<1,
]
then
[
|u_h-u_h^{\,n+1}|{B_h}^2\leqslant \rho\,|u_h-u_h^{\,n}|^2.
\tag{8}
]

Theorem 2. If (B_h\sim L_h,\ L_h=L_h^*),
[
\gamma_r=\frac12\left[\delta_1+\delta_0+(\delta_1-\delta_0)\cos\frac{2r+1}{2n}\pi\right]^{-1},\quad r=0,1,\ldots,n,
]
then
[
|u_h-u_h^{\,n+1}|{B_h}
\leqslant
\left[T_n\left(\frac{\delta_1+\delta_0}{\delta_1-\delta_0}\right)\right]^{-1}
|u_h-u_h^0|,
]
where
[
T_n(\lambda)=\frac12\left[(\lambda+\sqrt{\lambda^2-1})^n+(\lambda-\sqrt{\lambda^2-1})^n\right].
]

Theorem 3. If (B_h) is defined by (6), then ({u_h^n}), obtained by (7), coincides with ({v_h^n}) for (v_h^0=u_h^0), where ({v_h^n}) is constructed according to the rules: 1) from known (v_h^n) one computes
[
F_h^n=\widetilde B_hv^n-\gamma_n(L_hv^n-f_h);
]
2) (y_h^0) in the (T(\widetilde B))-method for solving (\widetilde B_hv_h=F_h^n) is taken equal to (v_h^n), and iterations are carried out by the (T(\widetilde B))-method to obtain (5); 3) (y_h^m=v_h^{n+1}).

3. Let in (\Omega={x=(x_1,x_2,\ldots,x_p):\ 0<x_s<1;\ s=1,2,\ldots,p}) one seek the solution of the system of equations
[
\sum_{l=1}^{N}\ \sum_{\substack{|\alpha|\leqslant m_r\ |\beta|\leqslant m_l}}
(-1)^{|\alpha|}D^\alpha\left(a_{rl}^{\alpha\beta}(x)D^\beta z_l\right)
=f_r(x),\quad r=1,2,\ldots,N,
\tag{9}
]
satisfying the boundary conditions
[
D^\alpha z_l\big|{\Gamma}=0,\quad |\alpha|\leqslant m_l-1,\quad l=1,2,\ldots,N,
\tag{10}
]
where (z(x)\equiv(z_1,z_2,\ldots,z_N)) is the desired solution:
[
\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_p);\quad
D^\alpha=D_1^{\alpha_1}D_2^{\alpha_2}\cdots D_p^{\alpha_p};\quad
D_s=\partial/\partial x_s;\quad
|\alpha|=\sum\alpha_s,}^{p
]
[
|\alpha|0=\max_s \alpha_s;\quad
\beta=(\beta_1,\beta_2,\ldots,\beta_p);\quad
m_l\geqslant 1;\quad l=1,2,\ldots,N;
]
[
a|\beta|>m_r;}^{\alpha\beta}=0,\quad \text{if
\tag{11}
]
[
\sum_{l,r=1}^{N}\ \sum_{\substack{|\alpha|=m_r\ |\beta|=m_l}}
a_{rl}^{\alpha\beta}\xi_l^{(\beta)}\xi_r^{(\alpha)}
\geqslant
\delta\sum_{l=1}^{N}\sum_{|\alpha|=m_l}(\xi_l^{(\alpha)})^2,
\quad \delta>0,
\tag{12}
]
for any real (N)-dimensional vectors
[
\xi^{(\alpha)}=(\xi_1^{(\alpha)},\xi_2^{(\alpha)},\ldots,\xi_N^{(\alpha)}).
]
On the grid (x_i=ih), where (h=(h_1,h_2,\ldots,h_p)), (h_s=1/N_s) is the step in (x_s), (i=(i_1,i_2,\ldots,i_p)), define
[
\Omega_h^r={x_i:\ m_r\leqslant i_s\leqslant N_s-m_r,\ s=1,2,\ldots,p}.
]
Then the difference analogue of problem (9), (10) will be the problem of finding—

of finding the grid vector-function (u_h=(u_1,u_2,\ldots,u_N)) satisfying the system

[
L_h^r u_h \equiv \sum_{l=1}^{N}\sum_{\substack{|\alpha|\le m_r\ |\beta|\le m_l}}
(-1)^{|\alpha|}\bar{\Delta}^{\alpha}\bigl(a_{rl}^{\alpha\beta}\Delta^{\beta}u_l(x_i)\bigr)
= f_r(x_i)\quad \text{for } x_i\in \Omega_h^r,
]

[
r=1,2,\ldots,N,
\tag{13}
]

and the boundary conditions

[
u_l(x_i)=0,\quad \text{if } x_i\in \bar{\Omega}_h^l,\quad l=1,2,\ldots,N.
\tag{14}
]

Here (\bar{\Delta}_s,\Delta_s) are respectively the left and right differences with respect to (x_s),
(\bar{\Delta}^{\alpha}=\bar{\Delta}_1^{\alpha_1}\bar{\Delta}_2^{\alpha_2}\cdots \bar{\Delta}_p^{\alpha_p}),
(\Delta^{\beta}=\Delta_1^{\beta_1}\Delta_2^{\beta_2}\cdots \Delta_p^{\beta_p}).

In the space of grid functions (u_h), whose (l)-th component is defined on (\Omega_h^l), we introduce the scalar product

[
(u_h,v_h)h=\sumh_1h_2\cdots h_p}^{N
\sum_{x_i\in \Omega_h^l}u_l(x_i)v_l(x_i).
]

Along with the operator (L_h), defined by formula (13), and
(L_hu_h\equiv (L_h^1u_h,L_h^2u_h,\ldots,L_h^Nu_h)), consider the operators
(\tilde{B}_h,\ T_m\ \bigl(m=(M_1,M_2,\ldots,M_N)\bigr)):

[
\tilde{B}_h u_h \equiv
(\tilde{B}_h^1u_1,\tilde{B}_h^2u_2,\ldots,\tilde{B}_h^Nu_N),
\quad
T_m\equiv (T^1u_1,T^2u_2,\ldots,T^Nu_N);
\tag{15}
]

[
\tilde{B}h^r u_r \equiv (-1)^{m_r}\sum}^{p}\bar{\Deltas^{m_r}\Delta_s^{m_r}u_r,
\quad
T^r u_r \equiv \prod}^{M_r-1
\bigl(E^r-\tau_k^r(A_{h,k}^r)^{-1}\tilde{B}_h^r\bigr)u_r;
\tag{16}
]

[
E^r u_r\equiv u_r,\quad
A_{h,k}^r=\prod_{s=1}^{p}
\left[E^r+\sigma_{s,k}^r\left((-1)^{m_r}\bar{\Delta}s^{m_r}\Delta_s^{m_r}
+\mu^rE^r\right)\right].
\tag{17}
]

The operator (T_m) is connected with the iterative method (T(\tilde{B})) for solving the equation (\tilde{B}_h w_h=F_h) by means of the iterations

[
A_{h,k}^r(y_r^{k+1}-y_r^k)
=-\tau_k^r(\tilde{B}_h^r y_r^k-F_r),
\quad r=1,2,\ldots,N,
\tag{18}
]

where (\tau_k^r,\ \sigma_{s,k}^r,\ \mu_{s,k}^r) are iteration parameters. Method (18) was proposed in ((^2,^3,^9)) and may be regarded as a natural generalization of the alternating-direction methods ((^{10},^{11})). We shall use the known fact that, applying method (18) with a suitable set of iteration parameters
(\tau_k^r,\sigma_{s,k}^r,\mu_{s,k}^r,\ k=0,1,\ldots,M_r-1;\ M_r \asymp |\ln h_r|,\ r=1,2,\ldots,N), one can obtain
(|T^r|=q_r<1), and hence

[
|T_m|=q<1,
\tag{19}
]

where (q_r) is an arbitrary number in ((0;1)), with the expenditure of only
(\asymp \dfrac{|\ln h\,\ln q'|}{h_1h_2\cdots h_p}) arithmetic operations,
(q=\max\limits_r q_r,\ q'=\min\limits_r q_r).

Theorem 4. (\tilde{B}_h=\tilde{B}_h^,\ T_m=T_m^,\ \tilde{B}_hT_m=T_m\tilde{B}_h,\ \tilde{B}_h>0.)

Theorem 5. If conditions (11), (12) are fulfilled, (a_{rl}^{\alpha\beta}(x)) are bounded in (\bar{\Omega}), and (a_{rr}^{00}) are sufficiently large, then (\tilde{B}h\sim L_h); if, moreover, (a), then (L_h=L_h^}^{\alpha\beta}=a_{lr}^{\beta\alpha).

Theorem 6. If the conditions of Theorem 5 are fulfilled, and (u_h^0) is an initial approximation to (u_h), the solution of (13), (14), then, carrying out iterations by method (7) with the operator (B_h) defined by formulas (6), (15)—(17), the estimate

[
|u_h-u_h^n|{\tilde{B}_h}
\le
\varepsilon |u_h-u_h^0|}_h
\tag{20}
]

can be obtained with the expenditure of only
(\asymp |\ln\varepsilon\cdot \ln h|/h_1h_2\cdots h_p) arithmetic operations; (n\asymp |\ln\varepsilon|).

* If, for example, all (a_{rl}^{\alpha\beta}=0) for (1\le |\alpha|+|\beta|<m_r+m_l), then it is sufficient that (a_{rr}^{00}\ge 0).

1. The proposed method is also successfully applied to other boundary conditions. For brevity, we restrict ourselves to the self-adjoint problem 2:

[
-\sum_{s=1}^{p} D_s(a_s(x)D_s z)+c(x)u=f,\qquad
0\le x_s\le 1,\quad s=1,2,\ldots,p;
\tag{21}
]

[
[-(-1)^{x_s}a_s(x)D_s z+b_s(x)z]_{x_s=0;\,1}=0,\qquad
s=1,2,\ldots,p,
\tag{22}
]

where (a_s,c,b_s) are symmetric matrices of order (N),

[
a_s\ge \gamma E,\quad b_s\ge bE,\quad \gamma>0,\quad b\ge 0,\quad
c\ge dE,\quad d\ge 0,\quad b^2+d^2>0,
\tag{23}
]

(E) is the identity matrix; (\gamma,b,d) are constants.

The difference analogue of problem 2 will be

[
L_h\bar u\equiv -\sum_{s=1}^{p}\bar\Delta_s(a_s\Delta_s\bar u(x_i))+\bar c\bar u(x_i)=\bar f(x_i)
\quad \text{for } x_i\in\Omega_h;
\tag{24}
]

[
-(-1)^{x_s}a_s\Delta_s\bar u+\bar b_s\bar u=0
\quad \text{for } x_i\in\Gamma_s,\qquad s=1,2,\ldots,p;
\tag{25}
]

[
\Omega_h={x_i:\ 1\le i_s\le N_s-1,\ s=1,2,\ldots,p,\ N_s=1/h_s};
\quad \Gamma_s={x_i:\ i_s=0;\ N_s;\ i_l=1,2,\ldots,N_l-1,\ l\ne s};
]
(\bar\Delta_s\bar u=\Delta_s\bar u) for (x_s<1), (\bar\Delta_s\bar u=\Delta_s\bar u) for (x_s=1); (\Gamma_h=\bigcup\Gamma_s).

In the space of (u_h) defined on (\Omega_h), we define the operator (L_h) by formula (24), where (\bar u=u_h) for (x_i\in\Omega_h), and on (\Gamma_h) it is extended by the conditions (25); the operator (\widetilde B_h) is specified by the formula

[
\widetilde B_h u_h\equiv -(\Delta_h\tilde u_1,\Delta_h\tilde u_2,\ldots,\Delta_h\tilde u_N)+d u_h,
\tag{26}
]

where (\Delta_h\tilde u_r\equiv \sum_{s=1}^{p}\bar\Delta_s\Delta_s\tilde u_r); (\tilde u=u_h) for (x_i\in\Omega_h), and on (\Gamma_h) (\tilde u) is extended from the conditions

[
-(-1)^{x_s}\Delta_s\tilde u+\tilde b\,\tilde u=0
\quad \text{for } x_i\in\Gamma_s,\qquad s=1,2,\ldots,p.
]

Theorem 7. Suppose (23) is satisfied; (a_s,b_s,c) are bounded in (\bar\Omega). Then
[
L_h=L_h^,\qquad \widetilde B_h=\widetilde B_h^,\qquad \widetilde B_h\sim L_h,
]
where (L_h,\widetilde B_h) are defined by (24)—(26).

Define the operator (T_m) by (15), (16), understanding by (A_{h,k}^r u=A_{n,k}^r u),

[
A_{h,k}^r=A_{h,k}\equiv \prod_{s=1}^{p}\left(E-\sigma_{s,k}(\bar\Delta_s\Delta_s-\mu_{s,k}E)\right).
]

If the conditions of Theorem 7 are satisfied, then the assertions of Theorems 4 and 6 also hold, with (L_h,\widetilde B_h,T_m,B_h) understood in them as the operators indicated in item 4 and with (u_h) regarded as the solution of (24), (25).

1. It is useful to bear in mind that if the computation by (18) proceeds with a cyclic repetition of (\tau,\sigma,\mu) leading in one cycle to the estimate (|T^r|=\tilde q_r<1), where (\tilde q_r) does not depend on (h), then such a transition from one iteration cycle to a new cycle is equivalent to applying (7) with (L_h=\widetilde B_h), (\gamma_n=1), (B_h=(E-T_m)^{-1}\widetilde B_h), (B_h\sim \widetilde B_h).

Remarks. 1) Theorems 5 and 7 have also been obtained for more general domains (\Omega). 2) Nonhomogeneous boundary conditions (14), (22) do not cause any essential changes in the indicated method. 3) The results of the work, as is clear from Theorem 1, admit generalization also to the case of some nonlinear strongly elliptic systems and systems of integro-differential equations.

Moscow State University
named after M. V. Lomonosov

Received
26 XI 1964

REFERENCES

1. E. G. Dyakonov, DAN, 138, No. 3 (1961).
2. E. G. Dyakonov, Candidate’s dissertation, V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 1962.
3. E. G. Dyakonov, in the collection Computational Methods and Programming, vol. 3, Moscow, 1965.
4. E. G. Dyakonov, UMN, 19, No. 5 (1964).
5. J. E. Gunn, Numer. Math., 6, No. 3 (1964).
6. E. L. Wachspress, J. Soc. Indust. Appl. Math., 11, No. 3 (1963).
7. L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, ch. XV, Moscow, 1959.
8. M. Sh. Birman, UMN, 5, No. 3 (1950).
9. E. G. Dyakonov, Journal of Computational Mathematics and Mathematical Physics, 2, No. 4 (1962).
10. D. W. Peaceman, H. H. Rachford, J. Soc. Indust. Appl. Math., 3, No. 1 (1955).
11. J. Douglas, H. H. Rachford, Trans. Am. Math. Soc., 82, No. 2 (1956).