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MATHEMATICS
M. A. VELIEV
INVESTIGATION OF THE STABILITY OF THE BUBNOV–GALERKIN METHOD FOR NONSTATIONARY PROBLEMS
(Presented by Academician V. I. Smirnov, 3 January 1964)
In the present note we use, without reservation, the notation, terminology, and results of the monograph of S. G. Mikhlin \((^1)\). The paper studies the stability of the Bubnov–Galerkin method for equations of parabolic and hyperbolic types, as well as for equations with operator coefficients containing the second and first derivatives with respect to time. Here the “coordinate system” is not chosen arbitrarily.
S. G. Mikhlin \((^2, ^3)\) considered the equation \(Au=f\) in a Hilbert space \(H\) with a positive definite self-adjoint operator \(A\) and proved that the Ritz method for this equation is stable with respect to small changes of the coefficients and the free terms of the Ritz system, if the coordinate system is strongly minimal \((^4)\) in the space \(H_A\). These works served as the starting point for our investigation. Let us also note the article \((^5)\), in which the stability of the Galerkin–Petrov method for stationary problems is studied.
Consider the problem
\[ \mathcal{L}u \equiv A\frac{d^2u(t)}{dt^2}+B\frac{du(t)}{dt}+Cu(t)=f(t), \tag{1} \]
\[ u(t)\big|_{t=0}=\varphi,\qquad \frac{du(t)}{dt}\bigg|_{t=0}=\psi, \tag{2} \]
where \(f(t)\) is a given function and \(u(t)\) the unknown function with values in \(H\); \(A,B,C\) are self-adjoint, time-independent operators acting in a certain Hilbert space \(H\), such that they are nonnegative and at least two of them are positive definite in \(H\), while \(\varphi\) and \(\psi\) are given elements of the space \(H\).
We shall assume that the domains of definition \(D(A), D(B)\), and \(D(C)\) intersect in a set \(\mathfrak{M}\), everywhere dense simultaneously in all three spaces \(H_A, H_B\), and \(H_C\). On the set \(\mathfrak{M}\) we define a new scalar product by putting
\[ [u,v]_0=(Au,v)+(Bu,v)+(Cu,v);\qquad u\in\mathfrak{M},\ v\in\mathfrak{M}. \]
By virtue of this definition, \(\mathfrak{M}\) becomes a new Hilbert space, which we shall denote by \(H_0\). The norm in \(H_0\) will be denoted by
\[ \|u\|_0^2=\|u\|_A^2+\|u\|_B^2+\|u\|_C^2;\qquad u\in\mathfrak{M}. \]
If it turns out that the space \(H_0\) is incomplete, we complete it in the usual way. Obviously, \(\mathfrak{M}\) will be dense in \(H_0\). We shall assume that \(\varphi\) and \(\psi\) belong to \(H_0\).
The Bubnov–Galerkin method for nonstationary problems \((^6)\) consists in the following: one chooses a linearly independent complete coordinate system of elements \(\{\varphi_k\}\) belonging to the space \(H_0\). The approximate solution of problem (1), (2) is sought in the form
\[ u_n(t)=\sum_{k=1}^{n} C_k^{(n)}(t)\varphi_k, \tag{3} \]
where the coefficients \(C_k^{(n)}(t)\) are determined from the system of ordinary differential equations
\[ \sum_{k=1}^{n}\frac{d^2 C_k^{(n)}(t)}{dt^2}[\varphi_k,\varphi_j]_A + \sum_{k=1}^{n}\frac{d C_k^{(n)}(t)}{dt}[\varphi_k,\varphi_j]_B + \sum_{n=1}^{n} C_k^{(n)}(t)[\varphi_k,\varphi_j]_C = (f,\varphi_j) \]
\[ (j=1,2,\ldots,n) \tag{4} \]
under the initial conditions
\[ C_k^{(n)}(t)\big|_{t=0}=C_k^{(n)},\qquad \frac{d C_k^{(n)}(t)}{dt}\bigg|_{t=0}=\dot C_k^{(n)}. \tag{4'} \]
The values \(C_k^{(n)}\) and \(\dot C_k^{(n)}\) may be prescribed to a considerable extent arbitrarily (see (6)), but we shall determine them in the following way: let \(\varphi^{(n)}\) and \(\psi^{(n)}\) denote the projections of the elements \(\varphi\) and \(\psi\) in the space \(H_0\) onto the subspace of the elements \(\varphi_1,\varphi_2,\ldots,\varphi_n\). It is obvious that \(\varphi^{(n)}\) and \(\psi^{(n)}\) are linear combinations of the elements \(\varphi_1,\varphi_2,\ldots,\varphi_n\); we shall take the coefficients of these combinations as \(C_k^{(n)}\) and \(\dot C_k^{(n)}\). Obviously, these coefficients are determined from the conditions
\[ \|\varphi-\varphi^{(n)}\|_0^2 = \left\|\varphi-\sum_{k=1}^{n} C_k^{(n)}\varphi_k\right\|_0^2 = \min, \]
\[ \|\psi-\psi^{(n)}\|_0^2 = \left\|\psi-\sum_{k=1}^{n} \dot C_k^{(n)}\varphi_k\right\|_0^2 = \min. \]
Finding the minima of these functionals leads to the solution of systems of algebraic equations
\[ \sum_{k=1}^{n}[\varphi_k,\varphi_j]_0\, C_k^{(n)} = [\varphi,\varphi_j]_0, \]
\[ \sum_{k=1}^{n}[\varphi_k,\varphi_j]_0\, \dot C_k^{(n)} = [\varphi,\psi_j]_0 \qquad (j=1,2,\ldots,n). \]
Obviously, these systems are solvable.
Suppose that the coefficients \([\varphi_k,\varphi_j]_A\), \([\varphi_k,\varphi_j]_B\), \([\varphi_k,\varphi_j]_C\) and the free terms \((f,\varphi_j)\) are computed with small errors, respectively \(a_{kj}=\bar a_{jk}\), \(\beta_{kj}=\bar\beta_{jk}\), \(\gamma_{kj}=\bar\gamma_{jk}\), and \(\Delta_j^{(n)}(t)\). In addition, we shall assume that the initial values \(C_j^{(n)}\) and \(\dot C_j^{(n)}\) are also computed with small errors \(\delta_j^0\) and \(\delta_j\), respectively. Then, instead of the Bubnov–Galerkin system (4), we obtain the system
\[ \sum_{k=1}^{n}\frac{d^2 \widetilde C_k^{(n)}(t)}{dt^2} \{[\varphi_k,\varphi_j]_A+a_{kj}\} + \sum_{k=1}^{n}\frac{d\widetilde C_k^{(n)}(t)}{dt} \{[\varphi_k,\varphi_j]_B+\beta_{kj}\} + \]
\[ + \sum_{k=1}^{n}\widetilde C_k^{(n)}(t) \{[\varphi_k,\varphi_j]_C+\gamma_{kj}\} = (f,\varphi_j) \qquad (j=1,2,\ldots,n), \tag{5} \]
where \(\widetilde C_k^{(n)}(t)\) is the solution with error.
Denote by \(\Gamma_{n0},\Gamma_{n1},\Gamma_{n2}\) the matrices with elements \(a_{kj}=\bar a_{jk}\), \(\beta_{kj}=\bar\beta_{jk}\), \(\gamma_{kj}=\bar\gamma_{jk}\), and by \(\Delta_n(t)\), \(\delta_{n0}\), \(\delta_{n1}\), \(C_n(t)\), and \(\widetilde C_n(t)\) the \(n\)-dimensional vectors with components \((\Delta_1^{(n)}(t),\ldots,\Delta_n^{(n)}(t))\), \((\delta_1^0,\ldots,\delta_n^0)\), \((\delta_1,\ldots,\delta_n)\), \((C_1^{(n)}(t),\ldots,C_n^{(n)}(t))\), and \((\widetilde C_1^{(n)}(t),\ldots,\widetilde C_n^{(n)}(t))\).
Since the coefficients, the free terms of the Bubnov–Galerkin system, and also the initial values are computed approximately, the question of stability under small changes of the coefficients, free terms, and initial values naturally arises. In this connection we introduce a definition.
Definition. We shall call the Bubnov—Galerkin method stable on the finite interval \(0 \leq t \leq l\) if there exist constants \(p_i\) \((i=0,1,2,3,4,5)\), independent of \(n\), such that, for sufficiently small norms of the matrices \(\|\Gamma_{n0}\|\), \(\|\Gamma_{n1}\|\), \(\|\Gamma_{n2}\|\) and the following norms of the vectors \(\|\Delta_n(t)\|_{L_2[0,l]}\), \(\|\delta_{n0}\|\), \(\|\delta_{n1}\|\), the inequality
\[
\|\widetilde C_n(t)-C_n(t)\|\leq
p_0\|\delta_{n0}\|+p_1\|\delta_{n1}\|+p_2\|\Gamma_{n0}\|+
+p_3\|\Gamma_{n1}\|+p_4\|\Gamma_{n2}\|+p_5\|\Delta_n(t)\|_{L_2[0,l]},
\tag{6}
\]
holds for \(0 \leq t \leq l\), where the norm \(\|\cdot\|\) of a vector is defined by
\[
\|C_n(t)\|=\left(\sum_{k=1}^{n}|C_k^{(n)}(t)|^2\right)^{1/2}.
\]
Stability on the infinite interval \(0 \leq t < +\infty\) is defined in an analogous way.
Theorem 1. Suppose \(A=0\), \(B=I\), and the coordinate system is normal\(^4\) in the space \(H\).
Then the Bubnov—Galerkin method is stable on the infinite interval of time.
Theorem 2. Suppose \(A=I\), \(B=aI\), where \(a=\mathrm{const}>0\), and suppose the coordinate system \(\{\varphi_k\}\) is normal in \(H\).
Then the Bubnov—Galerkin method is stable on the infinite interval of time.
Theorem 3. Suppose \(A=I\), \(B=0\), and the right-hand side \(f(t)\) is such that
\[
\int_{0}^{\infty} t^2\|f(t)\|\,dt<+\infty.
\]
Suppose, moreover, that the coordinate system \(\{\varphi_k\}\) is normal in \(H\).
Then the Bubnov—Galerkin method is stable on the infinite interval of time.
Theorem 4. Assume that the operators \(A\), \(B\), \(C\) are self-adjoint and positive definite in \(H\), and that the spaces \(H_A\), \(H_B\), \(H_C\) consist of the same elements. Suppose the coordinate system \(\{\varphi_k\}\) is normal in \(H_A\).
Then the Bubnov—Galerkin method is stable on a finite interval of time.
Theorem 5. Suppose \(A=0\), and the operators \(B\) and \(C\) are self-adjoint and positive definite in \(H\), and suppose \(H_C \subset H_B\). If the coordinate system is normal in \(H_B\), then the Bubnov—Galerkin method is stable on the infinite interval of time.
In conclusion I express my sincere gratitude to Prof. S. G. Mikhlin for suggesting the topic and for supervising the work.
Azerbaijan State University
named after S. M. Kirov
Received
24 I 1964
CITED LITERATURE
- S. G. Mikhlin, Variational Methods in Mathematical Physics, Moscow, 1957.
- S. G. Mikhlin, DAN, 135, No. 1, 16 (1960).
- S. G. Mikhlin, Vestn. LGU, No. 13, issue 3 (1961).
- A. T. Taldykin, Matem. sborn., 29 (71), No. 1, 79 (1951).
- G. N. Yaskova, M. N. Yakovlev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66 (1962).
- M. I. Vishik, Matem. sborn., 39 (81), No. 1, 51 (1956).