UDC 517.949.21

MATHEMATICS

V. S. RYABENKII

THE BOUNDARY PROJECTION OPERATOR

(Presented by Academician M. V. Keldysh, 19 VII 1968)

Consider the difference equation

\[ \sum_{k \in K} A_k u_{n+k}=0,\qquad n \in D_0 . \tag{1} \]

Here \(D_0\) and \(K\) are certain finite sets of points \(n=(n_1,n_2,\ldots,n_s)\) and \(k=(k_1,k_2,\ldots,k_s)\) with integer coordinates in a space of finite dimension \(s\); the unknown vector-function \(u=\{u_n\}\) is defined on the set \(D\), which is swept out by the point \(n+k\) when \(n\) and \(k\) independently sweep out \(D_0\) and \(K\), respectively; \(u_n\) is a vector of some finite dimension \(m\); \(A_k,\ k\in K\), are square matrices of the same dimension, independent of \(n\). In the grid set \(D\) we single out the subset \(\Gamma\) of “near-boundary points.” A point \(r,\ r\in D\), will be assigned to \(\Gamma\) if there exists a nonempty subset \(K_r\) of the set \(K\) such that

\[ r-k \in D,\qquad \text{if } k\in K_r . \tag{2} \]

With respect to the coefficients \(A_k\) of equation (1) we shall assume that, for the characteristic matrix \(A(\xi)\), defined by the equality

\[ A(\xi)=\sum_{k\in K} A_k \xi^{-k},\qquad \xi^k=\xi_1^{k_1}\xi_2^{k_2}\ldots \xi_s^{k_s}, \]

the inequality

\[ \det A(\xi)\ne 0 \tag{3} \]

is satisfied for any complex \(\xi_1,\xi_2,\ldots,\xi_s\) satisfying the conditions

\[ |\xi_1|=|\xi_2|=\ldots=|\xi_s|=1. \tag{4} \]

Requirement (3) is usually fulfilled for equations arising on the “upper” time layer when implicit difference schemes are used for solving evolution problems.

For each \(r,\ r\in\Gamma\), introduce the matrix

\[ A_r(\xi)=\sum_{k\in K_r} A_k \xi^{-k}, \]

and construct the matrices \(\beta_n^r,\ n\in D\), by the formulas

\[ \beta_n^r= \frac{1}{(2\pi i)^s} \oint_{|\xi_s|=1}\ldots \oint_{|\xi_1|=1} \frac{A^{-1}(\xi)A_r(\xi)}{\xi^{\,n-r+e}} \,d\xi_1\ldots d\xi_s, \]

where \(e\) is the \(s\)-dimensional vector each component of which is equal to one.

Consider the linear spaces \(U_\Gamma\) and \(U_D\) of grid \(m\)-dimensional vector-functions \(u_n\) of the argument \(n,\ n\in\Gamma\) and \(n\in D\), respectively. Define

onto \(U_\Gamma\) the operator \(P_D\), \(u=P_Dv\), with values in \(U_D\), by setting

\[ u_n=\sum_{r\in\Gamma}\beta_n^r v_r . \tag{5} \]

Theorem 1. The lattice vector-function \(u\), \(u\in U_D\), is a certain solution of equation (1), whatever the lattice function \(v\), \(v\in U_\Gamma\).

Proof. It is enough to verify that \(\beta_n^r v_r\) is a solution:

\[ \sum_{k\in K} A_k\beta_{n+k}^r v_r = \frac{1}{(2\pi i)^s} \left[ \oint\cdots\oint \frac{A(\xi)A^{-1}(\xi)A_r(\xi)}{\xi^{\,n-r+e}} \,d\xi_1\ldots d\xi_s \right]v_r = \]

\[ = \frac{1}{(2\pi i)^s} \left[ \oint\cdots\oint \sum_{k\in K_r} \frac{A_k}{\xi^{\,n-r+k+e}} \,d\xi_1\ldots d\xi_s \right]v_r = \]

\[ = \begin{cases} A_k v_r, & \text{if } n=r-k,\\ 0, & \text{if } n\ne r-k, \end{cases} \qquad k\in K_r . \]

But the point \(n=r-k\) does not belong to \(D_0\) by virtue of (2), and at the points of \(D_0\) equation (1) is satisfied.

Theorem 2. Let \(u_\Gamma\), \(u_\Gamma\in U_\Gamma\), be a lattice function formed by the boundary values of some solution \(u\) of equation (1). The solution \(u\) that coincides with \(u_\Gamma\) on \(\Gamma\) is unique and is representable in the form

\[ u_n=\sum_{r\in\Gamma}\beta_n^r u_r . \]

Proof. Let \(u\), \(u=\{u_n\}\), be a solution of equation (1). Construct the generating polynomial \(U(\xi)\) by setting

\[ U(\xi)=\sum_{n\in D} u_n\xi^n . \]

Multiply equation (1) by \(\xi^n\) and sum over all \(n\), \(n\in D_0\). We obtain

\[ 0= \sum_{n\in D}\sum_{k\in K} A_k\xi^{-k}\left(u_{n+k}\xi^{n+k}\right) = \sum_{n\in D}\left[\sum_{k\in K} A_k\xi^{-k}\right]u_n\xi^n - \]

\[ - \sum_{r\in\Gamma} \left[\sum_{k\in K_r} A_k\xi^{-k}\right]u_r\xi^r = A(\xi)U(\xi)-\sum_{r\in\Gamma}A_r(\xi)u_r\xi^r . \]

Hence

\[ u_n= \frac{1}{(2\pi i)^s} \oint_{|\xi_s|=1}\cdots\oint_{|\xi_1|=1} \frac{U(\xi)}{\xi^{\,n+e}}\,d\xi_1\ldots d\xi_s = \]

\[ = \sum_{r\in\Gamma} \left[ \frac{1}{(2\pi i)^s} \oint\cdots\oint \frac{A_r(\xi)u_r}{\xi^{\,n-r+e}} \,d\xi_1\ldots d\xi_s \right] = \sum_{r\in\Gamma}\beta_n^r u_r . \]

Definition. We shall call the operator \(P_\Gamma\) the boundary projection operator if, to each vector-function \(v\) from \(U_\Gamma\), it assigns the vector-function \(u=P_\Gamma v\) from \(U_\Gamma\) by formulas (5), where \(n\) runs through the points of the set \(\Gamma\).

Theorem 3. The operator \(P_\Gamma\) maps \(U_\Gamma\) onto the subspace of those vector-functions \(U_\Gamma'\) each of which is formed by the values of some solution of equation (1) at the points of the set \(\Gamma\). Each vector-function from this subspace is mapped by \(P_\Gamma\) into itself.

Proof. The validity of the first assertion follows from Theorem 1, and of the second—from Theorem 2.

Corollary. The grid function \(u_\Gamma,\ u_\Gamma \in U_\Gamma\), coincides with the values of some solution \(u\) of equation (1) at the points of the set \(\Gamma\) if and only if its values satisfy the system of equations

\[ u_n=\sum_{r\in\Gamma}\beta_n^r u_r,\qquad n\in\Gamma . \tag{6} \]

Theorem 4. There exist numbers \(C,\ C>0\), and \(q,\ 0<q<1\), independent of the domain \(D_0\), such that

\[ \|\beta_n^r\|<Cq^{|n-r|},\qquad r\in\Gamma,\quad n\in D, \tag{7} \]

where

\[ |n-r|=|n_1-r_1|+|n_2-r_2|+\cdots+|n_s-r_s|. \]

Proof. We deform the circles \(|\xi_j|=1\), replacing them by the circles

\[ |\xi_j|=q^{\operatorname{sgn}(r_j-n_j)},\qquad j=1,2,\ldots,s. \tag{8} \]

Here \(q,\ q<1\), can be chosen so close to unity that, on the manifold (8), as well as on the manifold (4), inequality (3) holds, and so that in the formula given above for \(\beta_n^r\) one may pass from integration over the circles (4) to integration over the circles (8). Then

\[ \|\beta_n^r\|= \frac{1}{(2\pi)^s} \left\| \cdots \oint_{|\xi_j|=q^{\operatorname{sgn}(r_j-n_j)}} \frac{A^{-1}(\xi)A_r(\xi)}{\xi^{\,n-r+e}} \,d\xi_1\cdots d\xi_s \right\| \le \]

\[ \le \frac{\sup_{\xi}\|A^{-1}(\xi)A_r(\xi)\|}{(2\pi)^s} \prod_j \oint \left| \frac{1}{\xi^{\,n_j-r_j}} \right| \left\| \frac{d\xi}{\xi} \right\| = \sup_{\xi}\|A^{-1}(\xi)A_r(\xi)\|q^{|n-r|}. \]

Taking into account that

\[ \|A_r(\xi)\|\le M\sup_{k,\xi}\|A_k\xi^{-k}\|, \]

where \(M\) is the number of points forming the set \(K\), one may take for \(C\) in (7)

\[ C=M\sup_{\xi}\|A^{-1}(\xi)\|\sup_{k,\xi}\|A_k\xi^{-k}\|, \]

where \(\xi\) is a point on the manifold (8).

Thanks to Theorem 4, equations (6) can be approximated by a system of difference equations of fixed order, independent of the dimensions of the domain \(D\) (independent of the degree of refinement of the grid). This can be used for computing solutions of evolutionary implicit difference equations on graphs and polyhedra, for computing the spectrum of a family of difference operators on functions in multidimensional domains and on polyhedra, and also in certain other questions.

Received
16 VII 1968