ON THE APPLICATION OF THE BUBNOV–GALERKIN METHOD TO THE SOLUTION OF BOUNDARY VALUE PROBLEMS FOR REGIONS OF COMPLICATED FORM

V. L. RVACHEV, L. I. SHKLYAROV

Many problems of mathematical physics lead to the determination of a function \(u(x,y)\) satisfying, in a given region \((s)\), some elliptic partial differential equation \(L[u(x,y)]=0\) and vanishing on its boundary.

One of the effective methods for solving such problems is the Bubnov–Galerkin method [1, 2]. According to this method, an approximate solution of the problem is sought in the form

\[ u^*(x,y)=\sum_{i=1}^{n} c_i \varphi_i(x,y), \tag{1} \]

where the functions \(\varphi_i(x,y)\) are linearly independent and constitute the first \(n\) functions of some system of functions \(\{\varphi_i(x,y)\}\) \((i=1,2,\ldots)\), relatively complete in the region \((s)\) [3], while \(c_1,c_2,\ldots,c_n\) are undetermined coefficients. The requirement of relative completeness of the system of functions \(\{\varphi_i(x,y)\}\) \((i=1,2,\ldots)\) is of fundamental importance: violation of this requirement may lead to a gross error in applying the Bubnov–Galerkin method.

In the well-known book [3] it is shown that, as a relatively complete system of functions, one may choose the sequence

\[ \varphi_1=\omega;\quad \varphi_2=\omega x;\quad \varphi_3=\omega y;\quad \varphi_4=\omega x^2;\ \ldots, \tag{2} \]

where \(\omega(x,y)\) is a continuous function having bounded and continuous derivatives \(\dfrac{\partial \omega}{\partial x}\), \(\dfrac{\partial \omega}{\partial y}\) inside the region \((s)\), and satisfying the conditions \(\omega(x,y)>0\) inside \((s)\), \(\omega(x,y)=0\) on the boundary of \((s)\). Then formula (1) is transformed into the form

\[ u^*(x,y)=\omega(x,y)\sum_{i=1}^{n} c_i \psi_i(x,y), \tag{3} \]

where \(\psi_1=1;\ \psi_2=x;\ \psi_3=y;\ \psi_4=x^2;\ \ldots,\) and the problem is reduced to finding the function \(\omega(x,y)\) with the properties described above.

The construction of the function \(\omega(x,y)\) for certain regions (circle, ellipse, convex polygon, etc.) is not difficult. A number of recommendations in this regard is contained in [3]. Thus, for example, for the region \((s_1)\) shown in Fig. 1, \(a\), the function \(\omega(x,y)\) can be obtained by multiplying the left-hand sides of the equations of the lines from which the contour is composed:

\[ \omega_1(x,y)=(y+x^2-1)(y-x^2)(-1.4x-0.9y+4.44)(2x-2.6y+9.2), \]

and for the region \((s_2)\), shown in Fig. 1, b, in the form

\[ \omega_2(x,y)=(25-x^2-y^2)\left[y^2+(x+2.5)^2-1\right](-5.8x-5.8y+22.04). \]

Fig. 1.

However, for the regions \((s_3)\), \((s_4)\) (Fig. 2, a, b) a similar approach to constructing the function \(\omega(x,y)\) proves impossible. Indeed, if one takes, for example, for the region \((s_3)\) (Fig. 2, a)

\[ \omega(x,y)=\left[R^2-(x-a)^2-y^2\right]\left[R^2-(x+a)^2-y^2\right], \]

then the condition of strict positivity of the function \(\omega(x,y)\) inside the region \((s_3)\) will be violated, since on the arcs \(MnN\) and \(MmN\) the function \(\omega(x,y)=0\).

Fig. 2.

In the present article we consider the question of constructing a function \(\omega(x,y)\) for regions whose boundary is composed of arcs of smooth curves with equations

\[ \varphi_1(x,y)=0,\quad \varphi_2(x,y)=0,\ldots,\quad \varphi_n(x,y)=0, \]

where the functions \(\varphi_i(x,y)\) have continuous and bounded derivatives

\[ \frac{\partial \varphi_i}{\partial x},\quad \frac{\partial \varphi_i}{\partial y}\quad (i=1,2,\ldots,n). \]

1. The method set forth below is based on the use of the apparatus of \(R\)-functions [4—8].

In the paper the following \(R\)-operations will be used:

1. \(R\)-conjunction
   \[ f_1 \Lambda_\alpha f_2=\frac{1}{2}\left(f_1+f_2-\sqrt{f_1^2+f_2^2-2\alpha f_1 f_2}\right), \]

2. \(R\)-disjunction
   \[ f_1 \vee_\alpha f_2=\frac{1}{2}\left(f_1+f_2+\sqrt{f_1^2+f_2^2-2\alpha f_1 f_2}\right), \]

3. \(R\)-negation
   \[ \bar f=-f. \]

These operations, in some of their properties, resemble the operations of the algebra of logic \(X\wedge Y\) (conjunction), \(X\vee Y\) (disjunction), \(\bar X\) (negation), where \(X,Y\) are binary Boolean variables [10].

In particular, the following properties hold [2]:

\(1^0.\ \bar{\bar f}=f;\)

\(2^0.\ f_1\Lambda_\alpha f_2=f_2\Lambda_\alpha f_1;\)

\(3^0.\ f_1\vee_\alpha f_2=f_2\vee_\alpha f_1;\)

\(4^0.\ \bar f_1\Lambda_\alpha \bar f_2=\overline{f_1\vee_\alpha f_2};\)

\(5^0.\ \bar f_1\vee_\alpha \bar f_2=\bar f_1\Lambda_\alpha f_2;\)

\(6^0.\ f_1\Lambda_\alpha f_2=0\) if and only if \(f_1=0,\ f_2\geqslant 0\) or \(f_2=0,\ f_1\geqslant 0;\)

\(7^0.\ f_1\vee_\alpha f_2=0\) if and only if \(f_1\leqslant 0,\ f_2=0\) or \(f_1=0,\ f_2\leqslant 0;\)

\(8^0.\ f_1\Lambda_\alpha f_2>0\), if \(f_1>0\) and \(f_2>0;\)

\(9^0.\ f_1\vee_\alpha f_2<0\), if \(f_1<0\) and \(f_2<0\).

For \(\alpha=1\) formulas \((1^0\text{--}9^0)\) simplify, and the operations \(f_1\Lambda_1 f_2,\ f_1\vee_1 f_2\) acquire a number of additional properties [6]. However, for \(\alpha=1\) the functions \(f_1\Lambda_1 f_2,\ f_1\vee_1 f_2\) are not differentiable at points where \(f_1=f_2\), which makes them unsuitable for constructing the function \(\omega(x,y)\).

II. Let \((s_i)\) be domains defined respectively by the inequalities \(f_i>0\) \((i=1,2,\ldots,n)\). We shall agree to denote the predicate \(p_i(x,y)\), which takes the truth value equal to 1 in the domain \((s_i)\) and the false value equal to 0 outside it, as follows:
\[ p_i(x,y)\equiv (f_i(x,y)>0)\qquad (i=1,2,\ldots,n). \]

Suppose that the domain \((s)\) is composed of the domains \((s_1),(s_2),\ldots,(s_n)\) in such a way that it corresponds to the predicate
\[ p(x,y)\equiv F[p_1(x,y);\ p_2(x,y);\ \ldots;\ p_n(x,y)], \]
where \(F(X_1,X_2,\ldots,X_n)\) is some Boolean function. It is known that every Boolean function can be represented in a form containing only the operations \(X\wedge Y,\ X\vee Y,\ \bar X\) [10]. The following theorem holds [6, 8].

Theorem 1. Let the open domain \((s)\) be defined by means of the predicate
\[ p(x,y)\equiv F[p_1(x,y);\ p_2(x,y);\ \ldots;\ p_n(x,y)], \]
where \(p_i(x,y)\equiv (f_i(x,y)>0);\ f_i(x,y)\ (i=1,2,\ldots,n)\) are functions defined everywhere and continuous in the plane \(XOY\), and \(F(p_1,p_2,\ldots,p_n)\) is an arbitrary Boolean function written by means of the operations of conjunction, disjunction, and negation, observing a strict parenthesized notation (i.e. each binary operation is placed in parentheses). Then, if in the formula \(F(p_1p_2,\ldots,p_n)\) one makes the formal replacement of \(p_i\) by \(f_i\ (i=1,2,\ldots,n)\), and of the signs of disjunction, conjunction, and negation respectively by the signs of \(R\)-disjunction, \(R\)-conjunction, and \(R\)-negation, then the function obtained:

\(\omega(x,y)\) will be defined everywhere and continuous, positive inside the region \((s)\) and negative outside this region.

It follows from Theorem 1 that the rule for constructing the function \(\omega(x,y)\) for the region \((s)\) is as follows. First one must construct a predicate \(p(x,y)\) that takes the truth value inside the region \((s)\). Then, in the predicate \(p(x,y)\), the operations \(X \wedge Y, X \vee Y, \overline X\) are to be replaced by the \(R\)-operations \(X \wedge_{\alpha} Y, X \vee_{\alpha} Y, \overline X\), and instead of the predicates \(p_i(x,y) \equiv (f_i(x,y)>0)\) one writes the functions \(f_i(x,y)\) \((i=1,2,\ldots,n)\), respectively.

Fig. 3.

Example 1. Suppose it is necessary to construct the function \(\omega(x,y)\) for the region \((s_3)\) (Fig. 2, \(a\)).

The region \((s_3)\) may be regarded as the union of two circles of radius \(R\) with centers at the points \(o_1(a,0)\) and \(o_2(-a,0)\), defined by the inequalities

\[ f_1 = R^2 - (x-a)^2 - y^2 > 0,\quad f_2 = R^2 - (x+a)^2 - y^2 > 0. \]

Then the region \((s_3)\) can be specified by the predicate

\[ p(x,y) \equiv ((f_1>0)\vee(f_2>0)). \]

Following Theorem 1, we obtain the function \(\omega(x,y)\):

\[ \omega(x,y)=[R^2-(x-a)^2-y^2]\vee_{\alpha}[R^2-(x+a)^2-y^2]=R^2-x^2-a^2-y^2+ \]

\[ +\frac{1}{2}\{[R^2-(x-a)^2-y^2]^2+[R^2-(x+a)^2-y^2]^2 -2\alpha [R^2-(x-a)^2-y^2]\times \]

\[ \times [R^2-(x+a)^2-y^2]\}^{\frac{1}{2}}. \]

Example 2. Consider a more complicated region \((s)\) (Fig. 3). Let us construct the function \(\omega(x,y)\) for it.

The region \((s)\) is composed of parts of the circles \((p_1)\), \((p_2)\), \((p_3)\), \((p_4)\) with centers at the points \(o_1(a,0)\), \(o_2(-a,0)\), \(o_3(0,b)\), \(o_4(0,-b)\), defined by the predicates

\[ p_1(x,y)\equiv (R^2-(x-a)^2-y^2>0), \]

\[ p_2(x,y)\equiv (R^2-(x+a)^2-y^2>0), \]

\[ p_3(x,y)\equiv (R^2-x^2-(y-b)^2>0), \]

\[ p_4(x,y)\equiv (R^2-x^2-(y+b)^2>0) \]

respectively. The region \((s)\) can be divided into regions (I), (II), (III), (IV), (V). If we take an arbitrary point \((x',y')\) in region (I), then we obtain

\[ p_1(x',y')=0,\quad p_2(x',y')=1,\quad p_3(x',y')=1,\quad p_4(x',y')=0. \]

Consequently, region (I) can be defined by the predicate

\[ p_{\mathrm I}\equiv(\overline{p_1}\wedge p_2\wedge p_3\wedge \overline{p_4}). \tag{4} \]

Similarly, regions (II), (III), (IV), (V) can be defined by the predicates

\[ p_{\mathrm{II}} \equiv (p_1 \land \overline{p}_2 \land p_3 \land \overline{p}_4), \]

\[ p_{\mathrm{III}} \equiv (p_1 \land p_2 \land p_3 \land \overline{p}_4), \]

\[ p_{\mathrm{IV}} \equiv (p_1 \land p_2 \land p_3 \land p_4), \]

\[ p_{\mathrm{V}} \equiv (\overline{p}_1 \land p_2 \land p_3 \land p_4), \]

Therefore the region \((s)\) can be defined by the predicate

\[ p \equiv (\overline{p}_1 \land p_2 \land p_3 \land \overline{p}_4) \vee (p_1 \land \overline{p}_2 \land p_3 \land \overline{p}_4) \vee \]

\[ \vee (p_1 \land p_2 \land p_3 \land \overline{p}_4) \vee (p_1 \land p_2 \land p_3 \land p_4) \vee (\overline{p}_1 \land p_2 \land p_3 \land p_4). \tag{5} \]

Using methods for simplifying perfect disjunctive normal forms [10] (for example, the Veitch diagram method [11]), formula (5) can be rewritten in the form

\[ p \equiv (p_2 \land p_3) \vee ((p_1 \land p_3) \land \overline{p}_4). \]

Then, on the basis of Theorem 1, we obtain

\[ \omega(x,y) = [f_2(x,y) \land_\alpha f_3(x,y)] \vee_\alpha [(f_1(x,y) \land_\alpha f_3(x,y)) \land_\alpha \overline{f_4(x,y)}]. \]

III. As was indicated above, the function \(\omega(x,y)\) inside the region \((s)\) must have bounded and continuous derivatives \(\dfrac{\partial \omega}{\partial x}\), \(\dfrac{\partial \omega}{\partial y}\).

The proof that the function \(\omega(x,y)\) constructed by the algorithm described will possess these properties follows from the following theorem.

Theorem 2. Let the region \((s)\) be defined by means of the predicate

\[ p(x,y) \equiv F[p_1(x,y);\ p_2(x,y);\ \ldots;\ p_n(x,y)], \]

where \(p_i(x,y) \equiv (f_i(x,y)>0)\) \((i=1,2,\ldots,n)\), and the functions \(f_i(x,y)\) \((i=1,2,\ldots,n)\) have, inside the region \((s)\), continuous and bounded partial derivatives \(\dfrac{\partial f_i}{\partial x}\) and \(\dfrac{\partial f_i}{\partial y}\) \((i=1,2,\ldots,n)\). Then the function \(\omega(x,y)\), constructed according to Theorem 1 for the predicate \(p(x,y)\), has continuous and bounded derivatives in the region \((s)\).

Proof. From formula 1 of item I we find

\[ \frac{\partial}{\partial l}(f_1 \land_\alpha f_2) = \frac{1}{2} \left( 1-\frac{f_1-\alpha f_2}{\sqrt{(f_1-\alpha f_2)^2+(1-\alpha^2)f_2^2}} \right) \frac{\partial f_1}{\partial l} + \]

\[ + \frac{1}{2} \left( 1-\frac{f_2-\alpha f_1}{\sqrt{(f_2-\alpha f_1)^2+(1-\alpha^2)f_1^2}} \right) \frac{\partial f_2}{\partial l}. \]

Since the coefficients of \(\dfrac{\partial f_1}{\partial l}\) and \(\dfrac{\partial f_2}{\partial l}\) do not exceed 1, it follows from the boundedness of the derivatives \(\dfrac{\partial f_1}{\partial l}\) and \(\dfrac{\partial f_2}{\partial l}\) that the function

\[ \frac{\partial}{\partial l}(f_1 \land_\alpha f_2) \]

is bounded. It is obvious that for \(-1<\alpha<1\), from the continuity

from

\[ \frac{\partial f_1}{\partial l} \quad\text{and}\quad \frac{\partial f_2}{\partial l} \]

there follows the continuity of

\[ \frac{\partial}{\partial l}(f_1 \Lambda_\alpha f_2) \]

inside the domain defined by the predicate \(p(x,y)\equiv(p_1\Lambda p_2)\).

Similarly, we verify the boundedness and continuity of the function

\[ \frac{\partial}{\partial l}(f_1 V_\alpha f_2). \]

Since the function \(\omega(x,y)\) is a composite function constructed by applying a finite number of operations of \(R\)-disjunction, \(R\)-conjunction, and \(R\)-negation, we arrive at the conclusion that its partial derivatives inside the domain \((s)\) will also be continuous and bounded.

IV. Let us consider the application of the method described above to the solution of Poisson’s equation for the cross-shaped domain \((s_4)\), shown in Fig. 2, б. This domain can be constructed by taking as the initial domains the domains \((s_{13})\), \((s_{23})\), \((s_{33})\), and \((s_{43})\), defined by the predicates

\[ p_{13}(x,y)\equiv(b^2-x^2>0), \qquad p_{23}(x,y)\equiv(c^2-y^2>0), \]

\[ p_{33}(x,y)\equiv(d^2-y^2>0), \qquad p_{43}(x,y)\equiv(a^2-x^2>0), \]

respectively.

We construct the Boolean function \(F(p_{13},p_{23},p_{33},p_{43})\) defining the domain \((s_4)\):

\[ F(p_{13},p_{23},p_{33},p_{43}) \equiv (\overline{p}_{13}\Lambda p_{23}\Lambda p_{33}\Lambda p_{43}) \vee (\overline{p}_{13}\Lambda \overline{p}_{23}\Lambda p_{33}\Lambda p_{43}) \vee \]

\[ \vee\,(p_{13}\Lambda p_{23}\Lambda p_{33}\Lambda p_{43}). \]

Or, after simplification,

\[ F(p_{13},p_{23},p_{33},p_{43}) \equiv (p_{13}\Lambda p_{33})\vee(p_{23}\Lambda p_{43}). \]

According to Theorem 1, the function \(\omega(x,y)\) can be written in the form

\[ \omega(x,y)= [(b^2-x^2)\Lambda_\alpha(d^2-y^2)] V_\alpha [(c^2-y^2)\Lambda_\alpha(a^2-x^2)]. \]

We seek the approximate solution of the problem in the form

\[ u^*(x,y)= \{[(b^2-x^2)\Lambda_\alpha(d^2-y^2)] V_\alpha [(c^2-y^2)\Lambda_\alpha(a^2-x^2)]\} \times \]

\[ \times \sum_{i=1}^{n} c_i \psi_i(x,y), \]

where, in view of the symmetry of the domain,

\[ \psi_1=1,\quad \psi_2=x^2,\quad \psi_3=y^2,\quad \psi_4=x^4,\quad \psi_5=x^2y^2,\ldots . \]

This problem was solved on the Ural-2 electronic computer. The coefficients \(c_i\) \((i=1,2,\ldots,n)\) were found in the usual way [3]. In particular, for \(a=d=1\), \(b=c=0.25\), \(n=6\), the approximate solution has the form

\[ u^*(x,y)= \{[(b^2-x^2)\Lambda_\alpha(d^2-y^2)] V_\alpha [(c^2-y^2)\Lambda_\alpha(a^2-x^2)]\} [-1.08483+ \]

\[ +\,1.05706(x^2+y^2)-0.91999(x^4+y^4)-0.2881x^2y^2]. \]

The solution obtained was used to determine the torsional rigidity of a bar with a cross-shaped cross section by the formula [12]:

\[ c= \frac{ 4\left(\iint\limits_{(s)} u\,dx\,dy\right)^2 }{ \iint\limits_{(s)} \left[ \left(\frac{\partial u}{\partial x}\right)^2+ \left(\frac{\partial u}{\partial y}\right)^2 \right]dx\,dy }. \]

In the case considered, the value \(c = 0.1579\) was obtained, which is 0.0009 less than the value of the same quantity computed by another method in [9].

References

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2. Galerkin B. G. Rods and plates. Vestnik inzhenerov, 1, 19, 897—908, 1915.

3. Kantorovich L. V., Krylov V. I. Approximate Methods of Higher Analysis. L., Fizmatgiz, 1962.

4. Rvachev V. L. On the analytic description of certain geometric objects. Collection of Works of the Institute of Cybernetics, Academy of Sciences of the Ukrainian SSR, No. 1, 1963.

5. Rvachev V. L. Doklady Akademii Nauk SSSR, 153, No. 4, 1963.

6. Rvachev V. L., Yushchenko E. L. On a class of functions convenient for the analytic description of geometric images. Cybernetics and Computing Technology. Kiev, 1964.

7. Rvachev V. L., Yushchenko E. L. Doklady Akademii Nauk Ukrainian SSR, No. 2, 1965.

8. Rvachev V. L., Yushchenko E. L. Some questions of the analytic description of geometric objects of complex logical structure. Materials of Scientific Seminars on Theoretical and Applied Problems of Cybernetics. Kiev, 1965.

9. Arutyunyan N. Kh., Abramyan B. L. Torsion of Elastic Bodies. M., Fizmatgiz, 1963.

10. Glushkov V. M. Introduction to Cybernetics. Publishing House of the Academy of Sciences of the Ukrainian SSR. Kiev, 1964.

11. Vavilov E. I., Portnoy G. P. Synthesis of Circuits of Electronic Digital Machines. M., 1963.

12. Pólya G., Szegő G. Isoperimetric Inequalities in Mathematical Physics. M., Fizmatgiz, 1962.

Received by the editors
May 27, 1965

Kharkov Institute of Mining Machinery,
Automation and Computer Engineering