UDC 517.949.2

MATHEMATICS

L. S. FRANK

COERCIVE BOUNDARY-VALUE PROBLEMS

FOR DIFFERENCE OPERATORS

(Presented by Academician G. I. Petrov, 15 X 1969)

1. The concept of ellipticity of difference operators (see \((^{1,2})\)) plays, in the general theory of difference schemes, a role analogous to that of the well-known condition of I. G. Petrovskii for differential operators. It makes it possible to construct difference approximations that preserve on the grid the basic properties inherent in elliptic differential equations. However, the elliptic theory of difference operators cannot be regarded as sufficiently complete without considering difference boundary-value problems and establishing a discrete analogue of the well-known Shapiro–Lopatinskii condition of coercivity of boundary conditions. The necessity of such an algebraic condition is dictated by the needs of applied analysis, since it is important to have a simple criterion for estimating the stability of various difference approximations of elliptic boundary-value problems near the boundary of the domain. The present note is devoted precisely to this question.

2. The space of grid functions \(H_s^*(R_{n,h}^+)\). By \(R_{n,h}\) we shall denote the uniform grid with mesh size \(h\) in the Euclidean space \(R_n\); correspondingly, by \(R_{n,h}^+\) we shall denote the grid half-space:

\[ R_{n,h}^{+}=\{x\mid x\in R_{n,h},\quad x_n\geq 0\}. \tag{1} \]

We shall consider grid functions on \(R_{n,h}\) (respectively on \(R_{n,h}^{+}\)) with values in \(C^1\), depending continuously on the grid parameter \(h\), \(0\leq h\leq h_0\); everywhere in what follows \(h_0\) is fixed and sufficiently small. For the definition and properties of the spaces of grid functions \(H_s^*=H_s^*(R_{n,h})\) for any real \(s\), see \((^2)\). We now define the spaces of grid functions \(H_s^*(R_{n,h}^{+})\). Denote by \(\bar H_s^*\) the closed subspace of functions from \(H_s^*\) that are equal to zero for \(x\in R_{n,h}^{+}\). Put

\[ H_s^*(R_{n,h}^{+})=H_s^*/\bar H_s^*, \tag{2} \]

and \(H_s^*(R_{n,h}^{+})\) is endowed with the topology corresponding to the quotient norm, which we shall denote by \(+\|\cdot\|_s^*\).

1. Factorization of a quasihomogeneous function. In formulating boundary-value problems for difference operators in convolutions (r.o.s.; see \((^2)\)), the problem naturally arises of factorizing, with respect to one of the variables, functions defined on the torus. Let

\[ T^n=\{\omega\mid \omega=(\omega_1,\ldots,\omega_n),\ |1+i\omega_k|=1,\ 1\leq k\leq n\} \]

be the \(n\)-dimensional torus. We shall consider scalar functions \(\tilde a(x;\omega)\), defined on \(\Omega\times T^n\), where \(\Omega\) is a domain in \(R_{n,h}\). We shall assume that:
1) \(\tilde a\) is infinitely differentiable with respect to \(\omega\) for \(\omega\ne 0\);
2) \(\tilde a\) is continuous in the aggregate of its arguments for \(\omega\ne 0\);
3) \(\tilde a\) satisfies the inequalities:

\[ |\tilde a(x;\omega)|\leq c_0|\omega|^\alpha \]

(quasihomogeneity) and

\[ |\tilde a(x;\omega)|\geq c_1|\omega|^\alpha \]

(ellipticity) (see \((^2)\));
4) \(\tilde a(x;\omega)=\tilde a(\infty;\omega)+\tilde a'(x;\omega)\), where \(\tilde a'(x;\omega)\), as a function of the first argument, has compact support and belongs to the space-

space \(S^*\) (the lattice analogue of the Schwartz space \(S\), see \((2)\)). The class of such functions \(\tilde a_\alpha\) will be denoted by \(\mathscr E_\alpha\).

Put
\[ D^+=\{\omega_n\mid |1+i\omega_n|>1\},\qquad D^-=\{\omega_n\mid |1+i\omega_n|<1\}, \]
and by \(\Gamma\) we shall denote the boundary of the domains:
\[ \Gamma=\{\omega_n\mid |1+i\omega_n|=1\}. \]
By a quasihomogeneous factorization with respect to \(\omega_n\) of a function \(\tilde a_\alpha\in\mathscr E_\alpha\) we shall mean a representation of \(\tilde a_\alpha\) in the form
\[ \tilde a_\alpha=\tilde a_\chi^+\cdot \tilde a_{\alpha-\chi}^-, \tag{3} \]
where \(\tilde a_\chi^+\in\mathscr E_\chi,\ \tilde a_{\alpha-\chi}^-\in\mathscr E_{\alpha-\chi}\), \(\tilde a_\chi^+\) (respectively, \(\tilde a_{\alpha-\chi}^-\)) is analytic in \(\omega_n\) for \(\omega_n\in D^+\) for all \(\omega'\in T^{n-1}\) (respectively, for \(\omega_n\in D^-\) for all \(\omega'\in T^{n-1}\)); moreover, \(\tilde a_\chi^+\), for arbitrary finite \(\omega_n\in D^+\), satisfies the inequalities
\[ c_1|\omega|^\chi\le |\tilde a_\chi^+|\le c_2|\omega|^\chi,\qquad \forall\,\omega'\in T^{n-1}; \tag{4} \]
respectively, \(\tilde a_{\alpha-\chi}^-\) satisfies the inequalities
\[ c_3|\omega|^{\alpha-\chi}\le |\tilde a_{\alpha-\chi}^-|\le c_4|\omega|^{\alpha-\chi},\qquad \forall\,\omega_n\in D^-,\quad \forall\,\omega'\in T^{n-1}. \tag{5} \]

Theorem 1. If \(\tilde a_\alpha\in\mathscr E_\alpha\), then \(\tilde a_\alpha\) admits a quasihomogeneous factorization \((3)\) with respect to \(\omega_n\). The quasihomogeneous factorization \((3)\) for \(\tilde a_\alpha\) is unique up to a factor depending only on \(\omega'\) and different from zero for all \(\omega'\in T^{n-1}\).

Let us note that for \(\omega'\in T^{n-1}\setminus\{0\}\), for \(\tilde a_\alpha\in\mathscr E_\alpha\) there is defined
\[ \operatorname{ind}\tilde a_\alpha=(2\pi)^{-1}[\arg \tilde a_\alpha]_\Gamma, \tag{6} \]
which, by virtue of the connectedness of the set \(T^{n-1}\setminus\{0\}\), does not depend on \(\omega'\). It is obvious that for all \(\omega'\in T^{n-1}\setminus\{0\}\)
\[ \operatorname{ind}\tilde a_\alpha=\operatorname{ind}\tilde a_\chi^+=m, \tag{7} \]
but, generally speaking, the integer \(m\) does not coincide with the order of quasihomogeneity \(\chi\) of the function \(\tilde a_\chi^+\). We shall assume that \(m\) and \(\chi\) do not depend on \(x\in\Omega\).

4. Smooth operators. In what follows, an essential role will be played by the following

Condition (*). A quasihomogeneous function \(\tilde a_\alpha\) of order \(\alpha\) satisfies condition \((*)\) if, uniformly with respect to \(x\in\Omega\) and for all integers \(k,\ k\ge 0\), the relations
\[ \int_\Gamma \omega_n^k\tilde a_\alpha\,d\omega_n = O\bigl(|\omega'|^{\alpha+k+1}\bigr) \quad \text{as }\omega'\to 0,\quad \omega'\in T^{n-1}\setminus\{0\}. \tag{8} \]
hold.

Of course, \((8)\) can be rewritten in terms of equality to zero of the integrals over \(\Gamma\) of the product of \(\omega_n^k\) by the corresponding derivatives with respect to \(\omega'\) of \(\tilde a_\alpha\) at the point \(\omega'=0\).

Theorem 2. Let \(\tilde a_\alpha\in\mathscr E_\alpha\), and let \((3)\) be the corresponding quasihomogeneous factorization of the function \(\tilde a_\alpha\). Suppose that \(\tilde a_\alpha\) and \(\tilde a_\alpha^{-1}\) satisfy condition \((*)\). Then
\[ m=\operatorname{ind}\tilde a_\chi^+=\chi,\qquad \forall\,\omega'\in T^{n-1}\setminus\{0\}. \tag{9} \]

To the quasihomogeneous function \(\tilde a_\alpha(x;\omega)\) one can assign a generalized homogeneous function of \((h,\zeta,\bar\zeta)\) of order \(\alpha\):
\[ \tilde a_\alpha(x;h,\zeta,\bar\zeta) = h^{-\alpha}\tilde a_\alpha(x;h\xi),\qquad \zeta=(\zeta_1,\ldots,\zeta_n),\quad \zeta_k=(e^{ih\xi_k}-1)/ih. \tag{10} \]
The function \(\tilde a_\alpha(x;h,\zeta,\bar\zeta)\) defines on lattice functions in \(R_{n,h}\) a homogeneous difference operator in convolutions (h.d.o.c.) \(A_\alpha\), acting continuously from \(H_s^*\) to \(H_{s-\alpha}^*\) for any \(s\in R^1\) (see \((2)\)).

Let a lattice function \(u_+\in H_s^*(R_{n,h}^+)\); denote by \(l_0u_+\) the lattice function in \(R_{n,h}\), equal to \(u_+\) for \(x\in R_{n,h}^+\) and to zero for \(x\notin R_{n,h}^+\), and, so-

respectively, by \(+H_s^*\) the space of all such extensions. Conversely, if a mesh function \(u\) is given on \(R_{n,h}\), then we denote by \(R_+\) the operator of restriction of \(u\) to \(R_{n,h}^+\): \(R_+u=u_+\). Naturally, the question arises as to when
\(R_+A_\alpha l_0u_+\in H_{s-a}^*(R_{n,h}^+)\)? As simple examples show (for example, the shift operator with respect to \(x_n\)), such an inclusion does not always hold. However, the following is true.

Theorem 3. Let the quasi-homogeneous function \(\tilde a_\alpha\) satisfy condition \((*)\). Then the homogeneous d.o. \(A_\alpha\) corresponding to it acts continuously from \(+H_s^*\) into \(H_{s-\alpha}(R_{n,h})\).

5. Formulation of the boundary-value problem in the case of an integer \(\varkappa\ge 0\). Let \(A\) be a d.o. of order \(a\) with complete symbol

\[ \sigma(A)\sim\sum_{k=0}^{\infty}\tilde a_k(x;h,\xi,\bar\xi), \tag{11} \]

satisfying the ellipticity condition. Suppose that the number \(\varkappa\) in the factorization (3) of the function

\[ \tilde a(x;\omega)=\tilde a_0(x;1,\omega,\bar\omega) \tag{12} \]

is an integer and \(\varkappa\ge 0\). Let, moreover, \(\tilde a_k(x;1,\omega,\bar\omega)\), \(k\ge 0\), and \(\tilde a^{-\tau}(x;\omega)\) satisfy condition \((*)\). We prescribe \(\varkappa\) d.o.’s \(B_j\) of orders \(\alpha_j\), \(1\le j\le \varkappa\), with complete symbols

\[ \sigma(B_j)\sim\sum_{k=0}^{\infty}\tilde b_{kj}(x;h,\xi,\bar\xi). \tag{13} \]

Suppose that the functions \(\tilde b_{kj}(x;1,\omega,\bar\omega)\) satisfy condition \((*)\), and set

\[ \tilde c_{jl}(x;\omega')=(2\pi i)^{-1}\int_{\Gamma}\tilde b_{0j}\omega_n^{\,l-1}(\tilde a_x^+)^{-1}\,d\omega_n,\qquad 1\le j,l\le \varkappa,\quad \omega'\in T^{n-1}\setminus\{0\}, \tag{14} \]

where \(\tilde a_x^+\) is the factor analytic for \(\omega_n\in D^+\) in the factorization (3) of the function \(\tilde a(x;\omega)\) from (12). It is not difficult to verify that \(\tilde c_{jl}(x;\omega')\) is a quasi-homogeneous function of \(\omega'\) of order \(\alpha_j+l-\varkappa\). On the matrix \(\|\tilde c_{jl}\|\) we impose the following basic condition (an analogue of the Shapiro–Lopatinskii condition for differential elliptic boundary-value problems):

Condition \((**)\). The matrix \(\|\tilde c_{jl}\|\) satisfies the inequality

\[ \left|\det\|\tilde c_{jl}\|\right|\ge c|\omega'|^{p_0},\qquad \omega'\in T^{n-1}\setminus\{0\},\quad x\in R_{n,h}^+, \tag{15} \]

where

\[ p_0=\sum_{j=1}^{\varkappa}\alpha_j-\varkappa(\varkappa-1)/2, \]

and \(c\) is a constant independent of \(x\) and \(\omega'\).

Boundary operators satisfying conditions \((*)\) and \((**)\) will be called coercive. To the operators \(A\) and \(\{B_j\}\), \(1\le j\le \varkappa\), we associate the boundary-value problem in \(R_{n,h}^+\):

\[ R_+Au_+=f,\qquad f\in H_{s-a}^*(R_{n,h}^+), \tag{16} \]

\[ R_+B_ju_+\big|_{x_n=0}=\varphi_j,\qquad \varphi_j\in H_{s-\alpha_j-1/2}^*(R_{n-1,h}),\qquad 1\le j\le \varkappa. \tag{17} \]

Theorem 4. Let all the conditions listed above on the operators \(A\) and \(B_j\), \(1\le j\le \varkappa\), be satisfied. Then for a solution \(u_+\) of the boundary-value problem (16), (17) the a priori estimate

\[ {}^{+}\|u_+\|_s^* \le c\left({}^{+}\|f\|_{s-a}^* +\sum_{j=1}^{\varkappa}{}'\|\varphi_j\|_{s-\alpha_j-1/2}^* +{}^{+}\|u_+\|_{s-1}^*\right), \tag{18} \]

holds, where \({}'\|g\|_s^*\) is the norm of \(g(x')\) in \(H_s^*(R_{n-1,h})\), and \(c\) is independent of \(h\).

A converse assertion is also true in a certain sense.

Theorem 5. If, for the boundary-value problem (16), (17), for some \(s\) the estimate (18) holds for arbitrary \(f \in \dot H^{*}_{s-\alpha}(R^{+}_{n,h})\) and \(\varphi_j \in \dot H^{*}_{s-\alpha_j-1/2}(R_{n-1,h})\), and the operators \(A, B_j,\ 1 \le j \le \chi\), satisfy condition \((*)\), then the difference operator \(A\) is elliptic, and the boundary difference operators \(B_j,\ 1 \le j \le \chi\), satisfy the coercivity condition (15).

6. Formulation of the boundary-value problem in the case of an integer \(\chi<0\). In this case every solution of equation (16) with smooth right-hand sides will contain a nonsmooth part of the form

\[ \sum_{j=1}^{|\chi|}\rho_j(x')D_n^{j-1}\delta(x_n), \tag{19} \]

where \(\delta(x_n)\) is the lattice \(\delta\)-function, \(\delta(x_n)=h^{-1}\) for \(x_n=0\) and \(0\) for \(x_n\ne0\). It is natural to divide the problem of finding a solution of equation (16) into the problem of finding the part \(u_+\), smooth up to the boundary, and the nonsmooth summands \(\rho_j(x')D_n^{j-1}\delta(x_n)\), concentrated in the boundary strip \(0\le x_n\le (j-1)h,\ 1\le j\le |\chi|\). Therefore, for \(\chi<0\), the following problem is posed: to find a solution \(u_+\), smooth up to the boundary, and potentials \(\rho_j,\ 1\le j\le |\chi|\), satisfying the equation

\[ R_+\left(Au_+ + \sum_{j=1}^{|\chi|}G_j\rho_j\right)=f, \tag{20} \]

where \(G_j\) are difference operators of orders \(\alpha_j\) satisfying condition \((*)\), as well as a coercivity condition analogous to \((**)\). In this case there is also a corresponding a priori estimate for \(u_+(x)\) and the densities \(\rho_j(x')\).

In conclusion, we note that, for reasonable difference approximations of elliptic boundary-value problems for differential operators, condition \((*)\) is always satisfied (possibly after applying some power of the shift operator in the normal-to-the-boundary variable, which means merely another writing of the difference equations in an equivalent form). However, condition (15) does not always hold; in that case, as was already said above, an a priori estimate of type (18) is invalid, and near the boundary of the domain, as a rule, instability arises in numerical experiments. Therefore the algebraic condition (15) is in practice an effective way of checking the soundness of a difference approximation near the boundary of an elliptic boundary-value problem, just as the ellipticity condition for a difference operator is a criterion of a good approximation of an elliptic differential operator inside the domain.

Institute for Space Research
Academy of Sciences of the USSR
Moscow

Received
9 IX 1969

REFERENCES

1. V. Thomée, B. Westergren, Numer. Math., 11, No. 3, 196 (1968).
2. L. S. Frank, Dokl. Akad. Nauk SSSR, 181, No. 2 (1968).