Abstract Generated abstract
The paper develops an iterative method for solving a class of integral equations in potential theory when the usual Neumann series is obstructed by a nearest eigenvalue. A general theorem is proved for completely continuous operators in a Hilbert space, showing convergence of an iteration on the orthogonal complement of the adjoint eigenspace under a simple-pole condition for the resolvent. The result is applied to electrostatic boundary integral equations for the surface charge density on a perfectly conducting body, both in an external field and for a prescribed total charge, yielding convergent schemes and a justification of an empirical capacitance approximation. The paper also derives approximate formulas for electric, magnetic, and dielectric polarizability tensors of bodies of arbitrary shape, with errors expressed in terms of the next eigenvalue.
Full Text
UDC 517.94:538.3:59.55
MATHEMATICAL PHYSICS
A. G. RAMM
AN ITERATIVE METHOD FOR SOLVING AN INTEGRAL EQUATION OF POTENTIAL THEORY
(Presented by Academician V. A. Fock on 3 October 1968)
The main results of the present paper are formulated in Theorems 1, 2, 3. Theorem 1 is of a general character. Theorems 2 and 3 represent an application of Theorem 1 to two principal problems of electrostatics, namely to the determination of the density of surface charges in two cases: (a) a conductor in a given external field and (b) a conductor with a given total charge.
§ 1. Let \(A\) be a completely continuous linear operator. It is known that the series
\[ (I-\lambda A)^{-1}=\sum_{n=0}^{\infty}\lambda^n A^n \tag{1} \]
converges in the uniform operator topology if \(|\lambda|<r=\) \(=1/\lim\limits_{n\to\infty}\sqrt[n]{\|A^n\|}\), where \(r\) is the distance from the origin to the nearest eigenvalue of the operator \(A\) \({}^{1}\). Here eigenvalues are understood in the sense adopted in the theory of integral equations: \(\varphi_n=\lambda_n A\varphi_n\). Suppose that for \(\lambda=\lambda_1\), \(|\lambda_1|=r\), the operator \(A\) has an eigenvalue to which there corresponds a finite-dimensional eigensubspace \(H_1\). We shall assume that the operator \(A\) acts in a Hilbert space \(H\), although the arguments, with minor changes, remain valid also in the case when \(A\) acts in a Banach space.
Denote by \(G_1\) the eigensubspace of the adjoint equation
\[ (I-\overline{\lambda}_1 A^*)\psi_j=0,\qquad G_1=\{\psi_j\}. \tag{2} \]
It is known that, for the solvability of the equation
\[ (I-\lambda_1 A)\varphi=f \tag{3} \]
it is necessary and sufficient that \(f\subset G_1^{\perp}\). By \(G_1^{\perp}\) is denoted the orthogonal complement in \(H\) to the subspace \(G_1\). The subspace \(G_1^{\perp}\) is invariant with respect to the operator \(A\). Suppose that the solution of equation (3) belonging to \(G_1^{\perp}\) is unique. This assumption means that the eigensubspace corresponding to the number \(\lambda_1\) coincides with the root subspace corresponding to the number \(\lambda_1\) \({}^{2}\). An equivalent assumption is that \(\lambda_1\) is a simple pole of the resolvent \((I-\lambda A)^{-1}\). We shall show that then the operator \(A\) has in \(G_1^{\perp}\) a first eigenvalue outside the circle \(|\lambda|=r\), and therefore the series (1) converges in the sense of uniform convergence of operators in \(G_1^{\perp}\). Since \(G_1^{\perp}\subset H\), it is enough to show that \(\lambda=\lambda_1\) is not an eigenvalue of the operator \(A\), considered as an operator in \(G_1^{\perp}\). Assuming the contrary, suppose that \((I-\lambda_1 A)\varphi=0\), \(\varphi\subset G_1^{\perp}\). By the assumption made, \(\varphi=0\), as was required to be shown. We formulate the preceding observation as a theorem.
Theorem 1. Let \(A\) be a completely continuous operator acting in a Hilbert space \(H\); let \(\lambda_1\) be its eigenvalue of smallest modulus; let \(G_1\) be the subspace of zeros of the operator \(I-\overline{\lambda}_1 A^*\); let \(G_1^{\perp}\) be its
orthogonal complement in \(H\). If the proper subspace of the operator \(A_1\) corresponding to the eigenvalue \(\lambda_1\) coincides with the root subspace of the operator \(A\) corresponding to this same number, then the operator \(A_1\), considered in the invariant subspace \(G_1^\perp\), has eigenvalues outside the circle of radius \(r=1/\lim\limits_{n\to\infty}\sqrt[n]{\|A^n\|}\), and therefore equation (3), considered as an equation in \(G_1^\perp\)*, is solvable, and in fact uniquely so, by the iteration method:
\[ \varphi_{n+1}=\lambda_1 A\varphi_n+f;\qquad \varphi_0=\Phi \tag{4} \]
for any \(f\subset G_1^\perp\) and any \(\Phi\subset G_1^\perp\), and the solution does not depend on the initial approximation \(\Phi\subset G_1^\perp\).
Remark. If the value \(|\lambda_2|\) is known, where \(\lambda_2\) is the second eigenvalue of the operator \(A\) (it is assumed that the eigenvalues are arranged in increasing order of modulus), then the process (4) converges no more slowly than a geometric progression with \(q=|\lambda_1/\lambda_2|\). (If \(\Gamma\) is a sphere, then \(\lambda_2=3\), \(q=1/3\) for the operator \(A\) defined by formula (5), see below.)
§ 2. Let us consider the application of Theorem 1 to a problem of potential theory that is of interest in electrostatics. If an ideally conducting body with piecewise-smooth surface \(\Gamma\) is placed in an arbitrary external electrostatic field \(E_0=-\nabla\varphi\), then the density \(\sigma\) of the charges induced on the surface of the conductor satisfies the integral equation
\[ \sigma(s)=-2\frac{\partial\varphi}{\partial n_s} -\int_\Gamma \frac{\partial}{\partial n_s}\frac{1}{2\pi r_{st}}\sigma(t)\,dt \equiv f-A\sigma, \tag{5} \]
where \(n\) is the exterior normal to the surface \(\Gamma\); \(dt\) is the area element of this surface. The operator \(A\) in equation (5) is completely continuous in \(H=L_2(\Gamma)\) and, as is known \(\left({}^{4}\right)\), has no eigenvalues inside the unit circle. The number \(\lambda_1=-1\) is an eigenvalue of the operator \(A\). Equation (2) in our case has the form
\[ \psi(s)+\int_\Gamma \frac{\partial}{\partial n_t}\frac{1}{2\pi r_{st}}\psi(t)\,dt=0 \tag{6} \]
and, as is well known, has the single eigenfunction \(\psi=1\) \(\left({}^{4},\ \text{p. }624\right)\). Consequently, the subspace \(G_1^\perp\) in the problem under consideration consists of functions \(\sigma\) satisfying the condition
\[ \int_\Gamma \sigma(t)\,dt=0. \tag{7} \]
Let us note at once that
\[ \int_\Gamma \frac{\partial\varphi}{\partial n_s}\,ds=0, \tag{8} \]
since the potential of the external electrostatic field is a harmonic function. The homogeneous equation (5), as is known \(\left({}^{4},\ \text{p. }625\right)\), has one and only one linearly independent solution \(\varphi_0\) (the solution of Robin’s problem). It is known, moreover, that this solution satisfies the condition \(\left({}^{4},\ \text{p. }625\right)\)
\[ \int_\Gamma \varphi_0(s)\,ds\ne 0. \tag{9} \]
This means that every solution of the homogeneous equation (5) belonging to \(G_1^\perp\) (i.e., satisfying condition (7)) is identically zero. All the conditions of Theorem 1 are fulfilled. Therefore the following is true.
Theorem 2. The solution of the problem of finding the distribution of charges induced on an ideally conducting body placed in an electrosta-
\[ \text{* For this equation to be regarded as an equation in }G_1^\perp,\text{ it is necessary and sufficient that } f\subset G_1^\perp,\text{ since the subspace }G_1^\perp\text{ is invariant with respect to the operator }A. \]
static external field \(E_0=-\nabla\varphi\), where \(\varphi\) is the potential of the external field, can be obtained by means of the convergent iterative process according to the scheme
\[ \sigma_{n+1}(s)=-\int_\Gamma \frac{\partial}{\partial n_s}\frac{1}{2\pi r_{st}}\,\sigma_n(t)\,dt-2\frac{\partial\varphi(s)}{\partial n_s}. \tag{10} \]
As the initial distribution of charges one may take any distribution satisfying condition (7), for example \(\sigma_0(s)=-2\partial\varphi/\partial n_s\).
Theorem 2 gives a convergent iterative method for solving certain problems of electrostatics (for example, the problem of a cylinder in an external field), which were previously solved by a numerical method based on solving systems of algebraic equations arising in the investigation of a certain integral equation of the first kind (5).
§ 3. The problem of the distribution of charge \(q\) on the surface of a perfectly conducting body leads to the equations
\[ \sigma(s)=-A\sigma;\qquad \int_\Gamma \sigma(t)\,dt=q, \tag{11} \]
where \(\sigma\) is the surface charge density, and the operator \(A\) is defined in equality (5). Solving problem (11) is equivalent to solving the Robin problem \((^4)\). We shall give a solution of problem (11) by means of a convergent iterative procedure. Introduce the notation: \(H=L_2(\Gamma)\); \(G_1\) is the one-dimensional subspace whose elements are functions that are constant on the surface \(\Gamma\). In accordance with the formula \(H=G_1\oplus G_1^\perp\), represent the function \(\sigma\) in the form
\[ \sigma=\sigma_1+\sigma_2;\qquad \sigma_2\subset G_1^\perp;\qquad \sigma_1\subset G_1. \tag{12} \]
Since \(\int_\Gamma \sigma_2dt=0\), \(\sigma_1=\operatorname{const}\), from the second condition (11) we obtain
\[ \sigma_1=q/S,\qquad S\equiv \operatorname{mes}\Gamma. \tag{13} \]
Consequently,
\[ \sigma(t)=q/S+\sigma_2(t);\qquad \int_\Gamma \sigma_2(t)\,dt=0. \tag{14} \]
Equation (11), after substituting expression (14) for \(\sigma(t)\), will take the form
\[ \sigma_2=-A\sigma_2+f;\qquad f\equiv-\frac{q}{S}-\frac{q}{S}\int_\Gamma \frac{\partial}{\partial n_s}\frac{1}{2\pi r_{st}}\,dt. \tag{15} \]
We note that
\[ \int_\Gamma f\,ds=-\frac{q}{S}\left[S+\int_\Gamma ds\int_\Gamma \frac{\partial}{\partial n_s}\frac{1}{2\pi r_{st}}\,dt\right] \]
\[ =-\frac{q}{S}\left[S+\int_\Gamma dt\int_\Gamma ds\,\frac{\partial}{\partial n_s}\frac{1}{2\pi r_{st}}\right] =-\frac{q}{S}[S-S]=0. \tag{16} \]
As was already noted above, the homogeneous equation (15) has in \(G_1^\perp\) only the zero solution. According to formula (16), \(f\subset G_1^\perp\). Considering equation (15) in \(G_1^\perp\) and applying Theorem 1, all conditions of which are fulfilled, we obtain the theorem:
Theorem 3. The distribution of surface charge density on a perfectly conducting body with piecewise-smooth surface \(\Gamma\) can be found by formula (14), where the function \(\sigma_2\) is uniquely determined by means of the convergent iterative process
\[ \sigma_2^{(n+1)}=-A\sigma_2^{(n)}+f;\qquad \sigma_2^{(0)}=f, \tag{17} \]
where the function \(f\) is defined by formula (15).
Remark. Theorem 3 contains, in particular, a theoretical justification of the well-known empirical Houw method \((^3)\) for calculating the capacitance of conductors.
In fact, according to Hoy, the capacitance is computed approximately by the formula
\[ C=\frac{q}{V}, \qquad \text{where } \quad V=\frac{1}{S}\int_{\Gamma} ds \int_{\Gamma}\frac{q}{S}\frac{dt}{4\pi r_{st}} =\frac{q}{S^2}\int_{\Gamma} ds \int_{\Gamma}\frac{dt}{4\pi r_{st}} . \]
The indicated expression for \(V\) is obtained if, in formula (14), the term \(\sigma_2\) is neglected and the potential produced by the part \(\sigma_1\) of the surface-charge distribution is averaged over the surface. It follows from this that Hoy’s method is applicable in those cases in which the potential produced on the surface of the body by charges distributed with surface density \(\sigma_2\) is noticeably smaller than the potential produced by charges distributed with surface density \(\sigma_1=q/S\). In conclusion, let us note that, although the literature devoted to the method of iterations for the solution of functional equations is extensive (see, for example, paper \({}^{6}\) and the references cited therein), the simple idea constituting the content of Theorem 1 has, as far as the author knows, not been encountered before.
§ 4. From the results of § 2, the definition of the dipole moment
\[ P_i=\int_{\Gamma}s_i\sigma(s)\,ds \]
and of the polarizability tensor \(\alpha_{ij}\), \(P_i=\alpha_{ij}VE_{0j}\), there follows an approximate formula for computing this tensor for a body of arbitrary shape placed in a homogeneous electrostatic field \(E_0\):
\[ \alpha_{ij}=4\delta_{ij}-\frac{1}{\pi V}\iint_{\Gamma\Gamma}\frac{n_i(s)n_j(t)}{r_{st}}\,ds\,dt; \qquad n_i(s)=\cos(n(s),x_i). \tag{18} \]
An analogous formula for the tensor \(\beta_{ij}\) of magnetic polarizability \((M_i=\beta_{ij}VH_{0j})\) has the form
\[ \beta_{ij}=-\frac{1}{\pi V}\iint_{\Gamma\Gamma}\frac{n_i(s)n_j(t)}{r_{st}}\,ds\,dt . \tag{19} \]
Both formulas are valid with error \(|1/\lambda_2|^2\), since \(\lambda_1=1\) in the cases under consideration.
For a dielectric body with constant \(\varepsilon_i\), situated in a medium with constant \(\varepsilon_e\), the polarizability tensor in a homogeneous electrostatic field \(E_0\) is computed by the approximate formula
\[ \alpha_{ij}=\delta_{ij}(2\varepsilon+2\varepsilon^2) -\frac{\varepsilon^2}{\pi V}\iint_{\Gamma\Gamma}\frac{n_i(s)n_j(t)}{r_{st}}\,ds\,dt; \qquad \varepsilon \equiv \frac{\varepsilon_i-\varepsilon_e}{\varepsilon_i+\varepsilon_e}. \tag{20} \]
The error of this formula is \(|\varepsilon/\lambda_2|^2\). Formulas (18), (19) are obtained from formula (20), respectively, for \(\varepsilon_i=\infty\), \(\varepsilon_i=0\) (a conductor in electrostatic and a superconductor in magnetostatic fields).
The author expresses sincere gratitude to V. A. Fock and G. A. Grinberg for their interest in this work.
Received
25 IX 1968
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