Abstract
Full Text
V. I. LEBEDEV
THE METHOD OF GRIDS FOR EQUATIONS OF S. L. SOBOLEV TYPE
(Presented by Academician S. L. Sobolev on 16 I 1957)
Let the function \(u(x,t)\) be a solution in the domain \(Q=\Omega\times[\Omega,l]\) of the equation
\[ Lu \equiv \frac{\partial^{2}}{\partial t^{2}}L_{0}u+\frac{\partial}{\partial t}L_{1}u+L_{2}u=f(x,t), \tag{1} \]
satisfying the conditions
\[ u\big|_{t=0}=\varphi(x), \qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=\psi(x), \tag{2} \]
\[ u\big|_{S}=0, \tag{3} \]
where \(x=(x_{1},x_{2},\ldots,x_{n})\); \(S\) is the boundary of the domain \(\Omega\);
\[ L_{0}u \equiv \sum_{i,j=1}^{n}\frac{\partial}{\partial x_{i}} \left(A_{ij}\frac{\partial u}{\partial x_{j}}\right)+a_{0}u, \]
\[ L_{s}u \equiv \sum_{i,j=1}^{n}\frac{\partial}{\partial x_{i}} \left(B^{s}_{ij}\frac{\partial u}{\partial x_{j}}\right) +\sum_{i=1}^{n} b^{s}_{i}\frac{\partial u}{\partial x_{i}}+b^{s}_{0}u, \qquad s=1,2. \]
For simplicity of exposition we shall restrict ourselves to the case when the coefficients of equation (1) depend only on \(x\), are bounded and, in addition,
\[ -\sum_{i,j=1}^{n} A_{ij}\xi_{i}\xi_{j}\geq \alpha\sum_{i=1}^{n}\xi_{i}^{2}, \qquad \alpha=\mathrm{const}>0, \qquad a_{0}\geq 0. \]
The existence and uniqueness of a generalized solution of the problem posed have been proved in papers \((^{1,2})\); therefore we shall dwell on the convergence of solutions of the finite-difference analogue of equation (1) to the generalized solution and on the differential properties of the generalized solution in a closed domain. In what follows we shall adhere to the notation of paper \((^{3})\).
We shall call a function \(u\), belonging together with \(\partial u/\partial t\) to the class \(\overset{0}{D}_{1}(Q)\) and satisfying the identity
\[ \iint_{Q}\left[ \sum_{i,j=1}^{n} \left( A_{ij}\frac{\partial^{3}u}{\partial x_{j}\partial t^{2}} +B^{1}_{ij}\frac{\partial^{2}u}{\partial x_{j}\partial t} +B^{2}_{ij}\frac{\partial u}{\partial x_{j}} \right) \frac{\partial\Phi}{\partial x_{j}} +\right. \]
\[ \left. +\left( \sum_{i=1}^{n} b^{1}_{i} \left( \frac{\partial^{2}u}{\partial x_{i}\partial t} +b^{2}_{i}\frac{\partial u}{\partial x_{i}} \right) +a_{0}\frac{\partial^{2}u}{\partial t^{2}} +b^{1}_{0}\frac{\partial u}{\partial t} +b^{2}_{0}u-f \right)\Phi \right]\,dQ=0 \tag{4} \]
for any \(\Phi\) from \(\overset{0}{L}_{1}(Q)\), if, moreover,
\[ \int_{\Omega} \left(\frac{\partial u(x,\Delta t)}{\partial t}-\psi\right)^{2} +\bigl(u(x,\Delta t)-\varphi\bigr)^{2}\,d\Omega \to 0 \qquad \text{as } \Delta t\to 0. \]
Construct a finite-difference analogue of equation (1). Partition the space \(R_{n+1}(x_1, x_2, \ldots, x_n, t)\) by the planes \(x_i=k_i h,\ t=k_0\Delta t,\ h>0,\ \Delta t>0,\ i=1,2,\ldots,n\), where \(k_i\) are integers, into parallelepipeds \(Q_{k_1\ldots k_n k_0}\), the coordinates of whose points satisfy the inequalities \(k_i h\le x_i\le (k_i+1)h,\ i=1,2,\ldots,n,\ k_0\Delta t\le t\le (k_0+1)\Delta t\). Points with coordinates \((k_1h,\ldots,k_nh,k_0\Delta t)\) shall be called vertices or mesh points. Denote by \(\Omega_h\) the domain composed of those cubes \(\Omega_{k_1\ldots k_n}\) \((k_i h\le x_i\le (k_i+1)h,\ i=1,2,\ldots,n)\) which belong to the domain \(\Omega\), and by \(Q_h\) the prism \(\Omega_h\times[0,m\Delta t]\), where \(m=[l/\Delta t]\). The boundary surface of \(\Omega_h\) will be denoted by \(S_h\), and \(S_h\times[0,m\Delta t]=F_h\).
Replace equation (1) at the mesh points \(Q_h\) by the difference equations
\[ L_hu\equiv [L_{0h}u]_{\bar t t}+[L_{1h}u]_{t}^{0}+L_{2h}u=f(x,t), \tag{5} \]
where
\[ L_{0h}u\equiv \sum_{i,j=1}^{n}(A_{ij}u_{x_j})_{\bar x_i}+a_0u, \]
\[ L_{sh}u\equiv \sum_{i,j=1}^{n}(B_{ij}^{s}u_{x_j}^{s})_{\bar x_i} +\sum_{i=1}^{n} b_i^{s}u_{x_i}^{s}+b_0^{s}u, \qquad s=1,2, \]
\[ u_t^{0}=\frac12(u_t+u_{\bar t}). \]
Define at the points \(Q\) a function \(u_h\) satisfying inside \(Q_h\) equations (5), and for \(t=0\) and \(t=\Delta t\) the initial conditions
\[ u_h(k_1h,\ldots,k_nh,0)=\varphi(k_1h,\ldots,k_nh), \]
\[ u_h(k_1h,\ldots,k_nh,\Delta t) =\varphi(k_1h,\ldots,k_nh)+\Delta t\,\psi(k_1h,\ldots,k_nh), \]
and equal to zero at the points of \(F_h\) and outside \(Q_h\). The solvability of the system obtained is a consequence of the fundamental inequalities derived below. In what follows, the passage from the functions \(f,\varphi,\psi\) to the averaged functions \(f_h,\varphi_h,\psi_h\), needed in certain limiting passages, is omitted; the functions \(u_h\) are denoted by \(u\).
Sum the equality \(u_t^{0}(L_hu-f)=0\), valid for all mesh points when \(\Delta t\le t\le (m-1)\Delta t\), over all points of the prism \(\Omega_h\times[\Delta t,(p-1)\Delta t]\) \((p<m)\). Then, omitting the summation sign over \(i,j\), we obtain
\[ \Delta t \sum_{\Delta t}^{(p-1)\Delta t} h^n\sum_{\Omega_h} u_t^{0}(L_hu-f)=0; \tag{6} \]
but
\[ \Delta t \sum_{\Delta t}^{(p-1)\Delta t} h^n\sum_{\Omega_h} u_t^{0}[L_{0h}u]_{\bar t t} = \left\{ h^n\sum_{\Omega_h} -A_{ij}u_{x_j\bar t}u_{x_i\bar t} +\frac{a_0}{2}u_{\bar t}^{2} \right\}_{\Delta t}^{p\Delta t}, \]
and from (6), for sufficiently small \(\Delta t\), we obtain the inequality
\[ \left. h^n\sum_{\Omega_h}(u_{x_i\bar t})^{2} \right|_{p\Delta t} \le C_1 e^{C_2p\Delta t} \left[ h^n\sum_{\Omega_h}(\varphi_{x_i}^{2}+\psi_{x_i}^{2}) +\Delta t\sum_{\Delta t}^{p\Delta t}h^n\sum_{\Omega_h}f^2 \right]. \tag{7} \]
Considering now the equality
\[ \Delta t\sum_{\Delta t}^{(p-1)\Delta t} h^n\sum_{\Omega_h} u_{\bar t t}(L_hu-f)=0, \]
we obtain from it, using inequality (7), the inequality
\[ \Delta t \sum_{\Delta t}^{(p-1)(\Delta t)} h^n \sum_{\Omega_h} (u_{\bar t t x_i})^2 \Bigg|_{p\Delta t} \le \]
\[ \le C_3 e^{C_4 p\Delta t} \left[ h^n \sum_{\Omega_h}(\varphi_{x_i}^2+\psi_{x_i}^2) +\Delta t \sum_{\Delta t}^{p\Delta t} h^n \sum_{\Omega_h} f^2 \right]. \tag{8} \]
Carrying out further arguments coinciding with those given in Chapter III of [3], we establish that the sequence \(\{u_h\}\), together with the sequences \(\{u_{ht}\}\), as \(\Delta t,h\to0\), converges weakly in \(W_2^{(1)}(Q)\) to functions \(u,\partial u/\partial t\), from \(\overset{\circ}{D}_1(Q)\); \(u(x,t)\) is a generalized solution of our problem; moreover, \(\{u_{\bar t}\}\) and \(\{u_{ht}\}\) converge, respectively, to \(u\) and \(\partial u/\partial t\) on each plane \(t=\mathrm{const}\) in the norm \(L_2(\Omega)\), uniformly with respect to \(t\in[0,l]\); for the limiting function \(u\) the inequality
\[ \int_0^t\!\!\int_{\Omega} \sum_{i=1}^n \left[ \left(\frac{\partial^2 u}{\partial x_i \partial t}\right)^2 + \left(\frac{\partial^3 u}{\partial x_i \partial t^2}\right)^2 \right]\,d\Omega\,dt \le \]
\[ \le C_5 e^{C_6 t} \left\{ \int_{\Omega}\sum_{i=1}^n \left[ \left(\frac{\partial\psi}{\partial x_i}\right)^2 + \left(\frac{\partial\psi}{\partial x_i}\right)^2 \right]\,d\Omega + \int_0^t\!\!\int_{\Omega} f^2\,d\Omega\,dt \right\}. \tag{9} \]
Let us investigate the differential properties of the solution obtained and determine under what conditions, imposed on the data functions in the problem and on the smoothness of the boundary \(S\) of the domain \(\Omega\), the generalized solution found belongs to \(W_2^{(k)}(Q)\). We assume that the domain \(\Omega\) can be covered by a finite number of overlapping canonical domains \(\Omega_1,\ldots,\Omega_N\) [3] in such a way that the sum \(\Omega^1,\ldots,\Omega^N\) gives all of \(\Omega\). Introduce in \(\overline{\Omega}\) new coordinates \(y_1,y_2,\ldots,y_n\) by means of functions \(y_i=y_i(x_1,x_2,\ldots,x_n)\) \((i=1,2,\ldots,n)\), \(k+1\) times continuously differentiable in \(\overline{\Omega}\). Let the canonical domain \(\Omega_1\) be mapped onto the cube \(D_1\), and the whole domain \(\Omega\) onto a domain \(D\); let
\[ J=\frac{D(x_1,x_2,\ldots,x_n)}{D(y_1,y_2,\ldots,y_n)}>0 \quad\text{in }\overline{\Omega}. \]
Since \(L_0u\) is a self-adjoint operator, in the new coordinates \(y_1,y_2,\ldots,y_n\) the self-adjoint operator will be \(L_0'u=JL_0u\). Taking this into account, instead of equation (1) we consider in the new coordinates the equation
\[ JLu=Jf. \tag{10} \]
This equation is of the same form as equation (1), and therefore we shall retain for it the former notation of coefficients and independent variables. For the solutions of the difference analogue of equation (10), the estimates (7), (8) will be valid. Suppose that the coefficients of equation (1) have continuous derivatives up to order \(k-1\) in the cylinder \(\overline{Q}\); \(f\in W_2^{(k-1)}(Q)\); \(\varphi,\psi\in W_2^{(k)}(\Omega)\); the boundary functions \(z_n=\omega(z_1,z_2,\ldots,z_{n-1})\) are assumed continuously differentiable with respect to \(z_1,z_2,\ldots,z_{n-1}\) up to order \(k+1\). Then the difference quotients with respect to \(t\) of the solution of (10) are easily estimated, since the function \(u_t\) satisfies the equation
\[ L_n u_t=f_t \]
and the conditions
\[ u_t\big|_{F_h}=0,\qquad u_t\big|_{t=0}=\psi,\qquad u_{\bar t t}\big|_{t=0}=\psi_1. \]
Having estimated \(u_t\), we estimate \(u_{tt},u_{ttt}\), etc., up to order \(k+1\).
We now estimate the other difference quotients. For this purpose we introduce into consideration, as in ยง 3 of Ch. III of the work \({}^{(3)}\), the domains \(D', D'', D''', D_1\) and the function \(\zeta\), and also, as in the derivation of inequalities (7), (8), by the method developed in \({}^{(3)}\), from the equalities
\[ (L_h u)_{x_k}=f_{x_k},\qquad (L_hu)_{\bar x_k}=f_{\bar x_k} \]
we obtain that
\[ \Delta t \sum_{\Delta t}^{(p-1)\Delta t} h^n \sum_{D'} \sum_{i,j=1}^{n} \left(u_{x_jx_it}\right)^2+\left(u_{x_j\bar x_i tt}\right)^2<C_7 . \]
By the same method one proves the boundedness of sums of the form
\[ \Delta t \sum_{\Delta t}^{(p-1)\Delta t} h^n \sum_{D_1} \left\{ u_{ht}+\sum_{s=1}^{m}\sum_{\alpha_0,\ldots,\alpha_s=0}^{n} \left( \frac{\Delta^s u_{ht}}{\Delta t^{\alpha_0}\Delta x_{\alpha_1}\cdots \Delta x_{\alpha_s}} \right)^2 \right\}, \qquad m=3,\ldots,k. \]
As \(\Delta t,h\to 0\), \(u_{ht}\to \partial u/\partial t\) weakly in \(W_2^{(k)}(Q)\), and for the function \(u\) we obtain the theorem:
Theorem. Suppose that the coefficients of equation (1) have continuous derivatives up to order \(k-1\) in the cylinder \(\bar Q\); \(f\in W_2^{(k-1)}(Q)\); \(\varphi,\psi\in W_2^{(k)}(\Omega)\); the boundary \(S\) of the domain \(\Omega\) is continuously differentiable \(k+1\) times and \(u|_S=0\).
Then the generalized solution \(u\) of the mixed problem for equation (1) exists and, together with \(\partial u/\partial t\), belongs to the space \(W_2^{(k)}(Q)\). For it the inequality
\[ \left\|\frac{\partial u}{\partial t}\right\|_{W_2^{(k)}(Q)} \leq C_8\left( \|\varphi\|_{W_2^{(k)}(\Omega)} + \|\psi\|_{W_2^{(k)}(\Omega)} + \|f\|_{W_2^{(k-1)}(Q)} \right) \]
holds.
This theorem makes it possible to establish differential properties of the solutions of the system
\[ \frac{\partial \bar U}{\partial t} = A\bar U-\operatorname{grad}p+\bar F,\qquad \operatorname{div}\bar U=0, \]
which was considered by the author in \({}^{(4)}\), for the case when the matrix \(A\) is constant and \(p|_S=0\) (for notation see \({}^{(4)}\)), since in this case the function \(p\) satisfies an equation of type (1), while
\[ \left\|\frac{\partial \bar U}{\partial t}\right\|_{W_2^{(k)}(Q)} \leq C_9\left( \|\bar U_0\|_{W_2^{(k)}(\Omega)} + \|\bar F\|_{W_2^{(k)}(Q)} + \|\operatorname{grad}p\|_{W_2^{(k)}(Q)} \right). \]
Moscow State University
named after M. V. Lomonosov
Received
11 I 1957
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