Full Text
UDC 517.5
MATHEMATICS
P. E. SOBOLEVSKII
ON THE BUBNOV–GALERKIN METHOD FOR PARABOLIC EQUATIONS IN A HILBERT SPACE
(Presented by Academician A. Yu. Ishlinskii, March 27, 1967)
1. In a Hilbert space \(H\) we consider the problem
\[ v' + A(t)v = f(t,v) \quad (0 \leq t \leq T), \qquad v(0)=v_0 . \tag{1} \]
Its generalized solutions were studied in \((^1)\). An approximate solution of problem (1) is the generalized solution of the problem
\[ v_n' + P_n A(t)v_n = P_n f(t,v_n) \quad (0 \leq t \leq T), \qquad v_n(0)=P_n v_0 . \tag{2} \]
Here \(P_n\) is the operator of orthogonal projection onto the linear span \(H_n\) of the first \(n\) elements of the system \(\{e_i\}\).
It is assumed that \(A(t)\), for each \(t \in [0,T]\), is a positive definite self-adjoint operator whose domain \(D\) does not depend on \(t\). It is assumed that the following hold: (a) \(A(t)A^{-1}(0)\) has only discontinuities of the first kind; (b) \(A(t)A^{-1}(0)\) has bounded variation; (c) \(A(t)A^{-1}(0)\) satisfies a Hölder condition with exponent \(\varepsilon > 0\).
It is assumed that all \(e_i \in D\) and (in the convergence theorems) that the system \(\{e_i\}\) is complete in the energy norm, i.e., that the set
\[ \bigcup_{n=1}^{\infty} A^{1/2}(0)H_n \]
is dense in \(H\).
Below we give results on problem (2), obtained in 1957–1958 \((^2)\).
2. Since \(P_n A(t)\) is a positive definite operator in \(H_n\) and \(H_n \subset D\), for any \(0 \leq \alpha \leq 1\), \(0 \leq t \leq T\), \(n=1,2,\ldots\), the functions
\[ \varphi_\alpha=\varphi_\alpha(n,t)=\|A^\alpha(t)[P_n A(t)]^{-\alpha}P_n\|, \]
\[ \psi_\alpha=\psi_\alpha(n,t)=\|[P_n A(t)]^\alpha P_n A^{-\alpha}(t)\| \]
are defined.
Lemma 1. The functions \(\varphi_\alpha\) and \(\psi_\alpha\) are logarithmically convex for \(0 \leq \alpha \leq 1\); \(\varphi_\alpha \leq 1\) for \(0 \leq \alpha \leq 1/2\). If \(\varphi_1 \leq C_1\), then \(\varphi_\alpha \leq C_1^{2\alpha-1}\) for \(1/2 \leq \alpha \leq 1\). If \(\psi_{\alpha_0} \leq D_{\alpha_0}\) for some \(\alpha_0\) from \([0,1]\), then \(\psi_\alpha \leq D_{\alpha_0}^{\alpha/\alpha_0}\) for all \(\alpha\) from \([0,\alpha_0]\).
The proof is based on Heinz inequalities (see \((^{3,4})\) and \((^5)\), p. 205).
The estimate \(\varphi_{1/2}=1\) was used in \((^6)\). The estimate \(\varphi_1 \leq C_1\) holds if there exists an invertible operator \(B\) such that \(D(B)=D\),
\[ C_1 (A(t)v,Bv) \geq \|A(t)v\|\cdot\|Bv\| \quad \text{for } v \in D, \]
and \(H_n\) are invariant with respect to \(B\) \((^7)\). If \(B^{-1}\) is bounded, then the estimate \(\varphi_1 \leq C_1\) means that for the systems \(\{e_i\}\) and \(\{AB^{-1}e_i\}\) the condition (A) of N. I. Pol’skii is satisfied (see \((^{10,8})\), p. 124). The estimate \(\psi_{\alpha_0} \leq D_{\alpha_0}\) in the case \(\alpha_0=1/2,1\) holds if there exists such a bounded invertible operator \(B_{\alpha_0}\) that
\[ D(B_{\alpha_0}) = D(A^{\alpha_0}(0)) = D(A^{\alpha_0}(t)), \]
and \(H_n\) are invariant with respect to \(B_{\alpha_0}\).
- Denote by \(U_n(t,s)\) \((0\leq s\leq t\leq T)\) the operator such that the function \(v_n=U_n(t,s)v_{0n}\) is the solution of the homogeneous problem
\[ v_n' + P_n A(t)v_n=0\quad (s\leq t\leq T),\qquad v_n(s)=v_{0n}\in H_n . \tag{3} \]
With the aid of the methods developed in [1] and the estimates of item 2, the following is established:
Lemma 2. Suppose that (a) is satisfied and \(\varphi_1\leq C\). Then for every \(v\in H_n\) the estimates
\[ \max\left\{\left\|[P_nA(0)]^{1/2}U_n(t,s)v\right\|, \left(\int_0^t\left\|P_nA(\tau)U_n(\tau,s)v\right\|^2\,d\tau\right)^{1/2}\right\} \leq \]
\[ \leq K\left\|[P_nA(0)]^{1/2}v\right\| \tag{4} \]
are valid.
Lemma 3. The estimates
\[ \left\|[P_nA(t)]^\beta U_n(t,s)[P_nA(s)]^{-\alpha}P_n\right\| \leq K|t-s|^{\alpha-\beta} \tag{5} \]
hold for \(0\leq \alpha\leq\beta\leq 1/2\), if (b) is satisfied; for \(0\leq \alpha\leq\beta<1/2+\varepsilon\), if (c) is satisfied; for \(0\leq \alpha\leq 1\), \(0\leq\beta<1+\varepsilon\), \(\alpha\leq\beta\), if (c) is satisfied and \(\varphi_1\leq C_1\).
- The uniform (in \(n\) and \(t\)) estimates of items 2 and 3 make it possible to establish the convergence of the solutions of problem (3) (for \(v_{0n}=P_nv_0\)) to the solution \(v(t)=U(t,s)v_0\) of the problem
\[ v' + A(t)v=0\quad (s\leq t\leq T),\qquad v(s)=v_0 . \tag{6} \]
First of all, the limiting relation
\[ \lim_{n\to\infty}\ \sup_{0\leq s<t\leq T} \left\|U_n(t,s)v_m-U(t,s)v_m\right\|=0 \tag{7} \]
is established. For this purpose it is first shown that \(U_n(t,s)x_m\) converges weakly to \(U(t,s)x_m\). The proof of this fact is based on the inequalities
\[ \left\|A^{1/2}(0)U_n(t,s)v_m\right\| \leq K\left\|A^{1/2}(0)v_m\right\|, \]
\[ \left\|[U_n(t+\Delta t,s+\Delta s)-U(t,s)]v_m\right\| \leq \bigl[|\Delta t|^{1/2}+|\Delta s|^{1/2}\bigr] K\left\|A^{1/2}(0)v_m\right\|, \tag{8} \]
which follow from Lemmas 1–3. Then the convergence of \(\|U_n(t,s)x_m\|\) to \(\|U(t,s)x_m\|\) is proved. For this one uses the weak convergence of \(\widetilde U_n(2t-s,s)v_m\) to \(\widetilde U(2t-s,s)v_m\), constructed from the operator
\[ \widetilde A(\tau)= \begin{cases} A(\tau), & s\leq \tau\leq t,\\ A(2t-\tau), & t\leq \tau\leq 2t, \end{cases} \tag{9} \]
and the formulas \(\widetilde U_n(2t-s,s)=U_n^*(t,s)U_n(t,s)\), \(\widetilde U(2t-s,s)=U^*(t,s)U(t,s)\). Further, by means of (7), Lemmas 1–3, and the moment inequality (9), the following are established:
Theorem 1. For every \(v_0\in H\) the limiting relation
\[ \lim_{n\to\infty}\ \sup_{0\leq s<t\leq T} |t-s|^\alpha \left\|A^\alpha(0)\bigl[U_n(t,s)P_nv_0-U(t,s)v_0\bigr]\right\|=0 \tag{10} \]
holds for \(\alpha=0\), if (a) is satisfied and \(\varphi_1\leq C_1\); for \(0\leq\alpha<1/2\), if (b) is satisfied; for \(0\leq\alpha\leq 1/2\), if (c) is satisfied; for \(0\leq\alpha<1\), if (b) or (c) is satisfied and \(\varphi_1\leq C_1\).
In applications to partial differential equations, (10) means convergence of the solutions together with their derivatives with respect to the spatial variables.
Theorem 2. For every \(v_0\in H\) the limiting relation
\[ \lim_{n\to\infty}\ \sup_{0\leq s<t\leq T} |t-s| \left\| \frac{\partial}{\partial t}U_n(t,s)P_nv_0 - \frac{\partial}{\partial t}U(t,s)v_0 \right\|=0, \tag{11} \]
holds, if (c) is satisfied and \(\varphi_1\leq C_1\).
If \(v_0\in D(A^\gamma(0))\), then the preceding assertions can be sharpened.
Theorem 3. For every \(v_0 \in D(A^{1/2}(0))\) the limit relation
\[ \lim_{n\to\infty}\ \sup_{0\le s<t\le T} \left\|A^\alpha(0)\,[U_n(t,s)P_n v_0-U(t,s)v_0]\right\|=0, \tag{12} \]
holds, if (a) is satisfied, \(\varphi_1\le C_1\) and \(\psi_{1/2}\le D_{1/2}\).
Theorem 4. For every \(v_0\in D(A^\gamma(0))\) the limit relation
\[ \lim_{n\to\infty}\ \sup_{0\le s<t\le T} |t-s|^{\alpha-\gamma} \left\|A^\alpha(0)[U_n(t,s)P_n v_0-U(t,s)v_0]\right\|=0 \tag{13} \]
holds for \(0\le\gamma\le\alpha<1\), if (b) or (c) is satisfied, \(\varphi_1\le C_1\) and \(\psi_\gamma\le D_\gamma\).
Theorem 5. For every \(v_0\in D(A^\gamma(0))\) the limit relation
\[ \lim_{n\to\infty}\ \sup_{0\le s<t\le T} |t-s|^{1-\gamma} \left\| \frac{\partial}{\partial t}U_n(t,s)P_n v_0 - \frac{\partial}{\partial t}U(t,s)v_0 \right\|=0 \tag{14} \]
holds for \(0\le\gamma\le1\), if (c) is satisfied, \(\varphi_1\le C_1\) and \(\psi_\gamma\le D_\gamma\).
Finally, we formulate conditions under which not only the solutions \(v_n\) of problems (3) (with \(v_{0n}=P_n v_0\)) converge to the solution \(v\) of problem (6), and not only the derivatives \(v_n'\to v'\) (Theorems 2 and 5), but also the residuals in equation (6) tend to zero.
Theorem 6. For every \(v_0\in D(A^\gamma(0))\) the limit relation
\[ \lim_{n\to\infty}\ \sup_{0\le s<t\le T} |t-s|^{1-\gamma} \left\|A(t)[U_n(t,s)P_n v_0-U(t,s)v_0]\right\|=0 \tag{15} \]
holds for \(0\le\gamma\le1\), if (c) is satisfied, \(\varphi_1\le C_1\), and there exists an operator \(B\) having an inverse such that \(D(B)=D\) and the \(H_n\) are invariant with respect to \(B\).
Recall (Section 2) that \(\varphi_1\le C_1\), if \(C_1(A(t)v,Bv)\ge \|A(t)v\|\cdot\|Bv\|\) for \(v\in D\).
- Since (see (1)) the solution of problem (2) is a solution of the equation
\[ v_n(t)=U_n(t,0)P_n v_0+\int_0^t U_n(t,s)P_n f[s,v_n(s)]\,ds, \tag{16} \]
the assertions given in Section 4 make it possible to study the convergence of the solutions of problem (6) to the solution of problem (1).
We first consider the case of a linear equation. We shall state only the theorem on the convergence of the residuals to zero.
Theorem 7. Suppose that the conditions of Theorem 6 are satisfied. Then the limit relation
\[ \lim_{n\to\infty}\ \sup_{0\le t\le T} \left\|(I-P_n)A\int_0^t U_n(t,s)P_n f(s)\,ds\right\|=0 \tag{17} \]
holds for any function \(f(t)\) satisfying the Hölder condition.
We now proceed to the consideration of the nonlinear equation (16).
Theorem 8. There exists a segment \([0,h]\subset[0,T]\), independent of \(n\), such that equation (16) has at least one solution \(v_n(t)\), defined on \([0,h]\). The family \(A^\alpha(0)v_n(t)\) is compact in \(C([0,T],H)\). Its limit points have the form \(A^\alpha(0)v(t)\), where \(v(t)\) are generalized solutions of problem (1). If problem (1) has a unique solution \(v(t)\), then
\[ \lim_{n\to\infty}\ \sup_{0\le t\le h} \left\|A^\alpha(0)[v_n(t)-v(t)]\right\|=0. \tag{18} \]
These assertions hold for some \(\alpha \in [0,\,1/2)\), if (a) is satisfied, the operator \(A^{-1}(0)\) is completely continuous, the operator \(f(t,A^{-\alpha}(0)w(t))\) acts boundedly and continuously from \(C([0,T],H)\) into \(B_2([0,T],H)\), \(v_0 \in D(A^{1/2}(0))\), \(\varphi_1 \leq C_1\), and \(\psi_{1/2}\leq D_{1/2}\); and for some \(\alpha \in [0,1)\), if (b) or (c) is satisfied, the operator \(A^{-1}(0)\) is completely continuous, the operator \(f(t,A^{-\alpha}(0)w(t))\) acts boundedly and continuously from \(C([0,T],H)\) into \(B_p([0,T],H)\) for some \(p>1/(1-\alpha)\), \(v_0\in D(A^\gamma(0))\) for some \(\alpha<\gamma\leq 1\), \(\varphi_1\leq C_1\), and, if \(\alpha\geq 1/2\), \(\psi_\gamma\leq C_\gamma\).
Finally, if the hypotheses of Theorem 6 are satisfied and the nonlinear operator \(f(t,v)\) is sufficiently smooth, then the residuals converge to zero (uniformly in \(t\)).
- In the preceding sections, the convergence, uniform in \(t\), of the solutions of problem (2) to the solution of problem (1) was investigated. The use of the estimates from (1) makes it possible to study convergence in the space \(B_2([0,T],H)\), and in some cases also in an arbitrary space \(B_p([0,T],H)\).
The author expresses his gratitude to M. N. Gitenskii for discussing the results of the work.
Voronezh Agricultural Institute
Received
17 III 1967
REFERENCES
- P. E. Sobolevskii, DAN, 122, No. 6 (1958).
- P. E. Sobolevskii, On differential equations of parabolic type in Hilbert space and their approximate solution by the Bubnov–Galerkin method, Dissertation, Leningrad, 1958.
- P. E. Sobolevskii, DAN, 128, No. 1 (1959).
- P. E. Sobolevskii, DAN, 155, No. 1 (1964).
- M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, P. E. Sobolevskii, Integral operators in spaces of summable functions, “Nauka,” 1966.
- P. E. Sobolevskii, DAN, 115, No. 2 (1957).
- P. E. Sobolevskii, DAN, 116, No. 5 (1957).
- G. G. Mikhlin, Numerical implementation of variational methods, “Nauka,” 1956.
- M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, 1956.
- N. I. Pol’skii, DAN, 143, No. 4 (1962).