SIAM J. NUMER. ANAL.Vol. 19, No. 2, April 1982

1982 Society for Industrial and Applied Mathematics

0036-1429/82/1902-0005 $01.00/0

THE NUMERICAL SOLUTION OF LAPLACE’S EQUATIONIN THREE DIMENSIONS*

K. E. ATKINSONS-

Abstract. We consider the Dirichlet problem for Laplace’s equation, on a simply-connected three-dimensional region with a smooth boundary. This problem is easily converted to the solution of a Fredholmintegral equation of the second kind, based on representing the harmonic solution as a double layer potentialfunction. We solve this integral equation formulation by using Galerkin’s method, with spherical harmonics asthe basis functions. This approach leads to small linear systems, and once the Galerkin coefficients for theregion have been calculated, the computation time is small for any particular boundary function. The majordisadvantage of the method is the calculation of the Galerkin coefficients, each of which is a four-fold integralwith a singular integrand. Theoretical and computational details of the method are presented.

1. Introduction. A numerical method will be described and analyzed for thesolution of the interior Dirichlet problem for Laplace’s equation in three dimensions.The problem is first reformulated using a standard integral equation, and this is thensolved using Galerkin’s method.

Let D be an open bounded simply-connected region in R3, and assume itsboundary S is sufficiently smooth. The degree of smoothness will be discussed in 2, butit must be a Lyapunov surface [9, p. 1]. The interior Dirichlet problem is

Au(A) 0, A (x, y, z) E D,(1.1)

u(P)-f(P), PES,

with f a given function.The solution u can be represented as a double layer potential,

II 0 (r_A) dO.(A) AD,(1.2) u(A)= p(O) Ou---s

in which o(O) is called a double layer density function. In the integral roa [rOA[, androA is the vector from 0 to A. The symbol O/Ouo denotes the normal derivative at thepoint O, in the direction of vo, the inner normal to S at O.

By letting A tend to a point P S, we obtain an integral equation for the unknownfunction O 0):

If 0 (r_e) do.(O) f(p), P S.(1.3) 2fro (P)+ p(O) Ou--s

The use of (1.2) and (1.3) is a well-known approach to the existence theory for (1.1); forexample, see [7, p. 334], [9], [10], [13, Chap. 12]. The properties of the solution p and ofthe integral operator in (1.3) will be reviewed in 2. But we note that the kernel of theintegral operator has a singularity of order 1 ! top.

The equations (1.2) and (1.3) have been used as a basis for the numerical solutionof (1.1); for example, see [10], [19] and the references contained therein. But most ofthis work has been of the finite element type. The surface S is approximated bytriangulation, and the approximate surface is usually piecewise linear. The solution p is

* Received by the editors September 25, 1979, and in final revised form April 13, 1981. This researchwas supported in part by the National Science Foundation under grants MCS76-0609 and MCS-8002422.

5- Department of Mathematics, University of Iowa, Iowa City, Iowa 52242.

263

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264 K.E. ATKINSON

approximated by a constant or linear function on each triangular element of the surface.This numerical procedure is very general and quite flexible, but it has the disadvantageof leading to large linear systems of equations. It is also slowly convergent in mostsituations.

We will solve (1.3) using Galerkin’s method, and the basis functions for the methodwill be the spherical harmonics on the unit sphere. When both the surface S and theboundary function f are sufficiently smooth, this numerical method will lead to quitesmall linear systems as well as a rapidly convergent sequence of approximants to p andu. The main disadvantage of the method is in the calculation of the Galerkincoefficients; this is discussed in 5.

The paper begins with a discussion of (1.3) and the harmonic function u of (1.2).This will include results on the solvability of (1.3) and on the smoothness of the functionp. The spherical harmonics are defined in 3, and convergence results on their use in theapproximation of functions are given. The Galerkin method is defined in 4, andconvergence rates are derived. The practical implementation of the numerical methodis covered in 5; numerical examples are given in 6.

2. The double layer potential. A complete discussion of the solution of potentialtheory problems on regions with a smooth boundary and using integral equationtechniques is given in Giinter [9]. We will summarize the results needed here.

The surface S will be a Lyapunov surface, with H61der exponent A, 0 < A --< 1, forthe first derivatives of the local surface representations. This is denoted by S e Ll,a. Ifthe kth order derivatives of the surface representations are H61der continuous withexponent A, denote it by S e Lk,. Finally, assume that the region D enclosed by S issimply connected.

The standard Banach spaces for work on S are L2(S) and C(S), with the usual innerproduct norm I]" 112 and uniform norm II’ I[, respectively. If a function f is k timescontinuously differentiable on S and if the kth order derivatives are H61der continuouswith exponent A, we say f Hk, (S).

Let the integral operator in (1.1) be denoted by YL Then (1.3) is written as

(2.1) (2r + Y{)p f.

From [5, p. 359], Y{" is a compact operator on L2(S) to L2(S), and from results in [9, p.49], it is straightforward to show Y{" is compact on C(S) to C(S). The Fredholmalternative theorem leads directly to a proof of the unique solvability of (2.1) for everyboundary function f, with respect to both L2(S) and C(S). In addition, (2r + y{)-i is abounded operator with respect to both of these spaces.

Smoothness of the double layer. From [9, p. 49],

A <1,(2.2) p C(S), S LI,X implies Y{OHoa,(S), A 1,

with 0 < A’ < 1 arbitrary. Since

1p (f- Y{’P),

we have that

oES,

(2.3) f e Ho,x (S), S s Ll,x

with A’ chosen as in (2.2).

implies O Ho.’(S),

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NUMERICAL SOLUTION OF LAPLACE’S EQUATION 265

If f and S have greater smoothness, then so does O. From [9, p. 312],

(2.4) p e Hk,a(S), S e Lk+2,x implies

with 0 < h’< , arbitrary, k >_-0. By simple induction,

(2.5) f H,x (S), S e L+l,X implies p H,x,(S).

For results on the ditterentiability of u on D D tO S, see [9, p. 102].

3. Spherical harmonies. Let U denote the unit sphere in R3,U {(x, y, z)lx 2 + y2 + z 2 1}.

The spherical harmonics can be defined in a number of ways, and a very extensivediscussion of the classical theory is given in MacRobert [12]. If the homogeneousharmonic polynomials of degree n in R are restricted to U, we obtain the sphericalharmonics of degree n. If any other polynomial is restricted to U, then the restrictionwill be called a spherical polynomial. It can be proven that every spherical polynomial ofdegree <_- n can be written as a linear combination of spherical harmonics of degree <_- nsee [12, p. 128].

There is a standard basis for the spherical harmonics, and it is orthogonal in L(U).Let P,,(u) and P(u) denote the Legendre polynomials and associated Legendrefunctions on [-1, 1], n_->0, 1-<_m-<n; see [1, Chap. 8] or [12, Chaps. 5, 7]. For(x, y, z) U, let

(x, y, z) (cos sin 0, sin sin 0, cos 0)

for 0 <_- & -<_ 27r, 0 <_- 0 -<_ 7r. If p is defined on U, then p(&, 0) and p(x, y, z) will be usedinterchangeably.

If p is a spherical polynomial of degree N, then

(3.1) p(, 0)= Y AnPn(cos 0)+ [A, cos (m)+B sin (m)]P’ (cos 0)n=O m=l

The basis functions

(3.2) Pn (cos 0), P (cos 0) cos (me), P(cos 0) sin (me), 1 -< m -< n

are spherical harmonics of degree n. For 0 _-< n <_- N, the total number of basis functions isd (N) (N + 1)2.

Using the orthogonality of the spherical harmonics of (3.2), we have

An2n + 1 If= I0=--4---- p(, O)Pn (cos 0) sin 0 dO de,

(3.3)

[A] 2n+l.(n m)_____ 2=

[B,ml 2zr (n + m)!p(, 0)

sin m P (cos 0) sin 0 dO d.

The functions P and P, can be evaluated very efficiently with recursion relations; see[1, p. 334]. The same is also, of course, true for sin (m) and cos (mO).

Approximation by spherical polynomials. From Gronwall [8] and Ragozin [14], wehave the following result for approximation of continuous functions on U. If OHla (U), then there is a sequence of spherical polynomials Tu of degree NN for which

(3 4) IlP- Tu[ <gl

NI+, N 1.

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266 I,:. E. ATKINSON

The spherical polynomials are dense in both C(U) and L2(U). The expansion ofpLz(U) is called the Laplace series; it is given by (3.1), (3.3) with N =. Letdenote the partial sum of the Laplace series of p, restricted to terms of degree <= N. OnLZ(u), N is orthogonal, and thus [liNI] 1. On C(U), it was shown by Gronwall[8, 2] that

with 6N--> 0. Using this, it is easy to show that

(3.6) p Hl,, (U) implies lip upl[--< NI+X-1/2

The potentialproblem on the sphere. The spherical harmonics were designed for theanalysis of three-dimensional potential theory problems, particularly on the sphere.One of the first interesting results we note is that spherical harmonics furnish theeigenfunctions for the operator Y{ of (1.3) in the case S U. In particular, we then have

27"g(3.7) Y{h h, n >- O,

2n+lfor any spherical harmonic h of degree n. Since there are 2n + 1 linearly independentfunctions h of degree n, of which (3.2) is a basis, we have that h 27r/(2n + 1) is aneigenvalue of multiplicity 2n + 1. This furnishes an interesting example of the integralequation eigenvalue problem.

We can use (3.7) to solve (1.3) when S U. Write p in its Laplace expansion, withunknown coefficients. Substituting into (1.3), expanding f in its Laplace expansion, andmatching coefficients, we are led to the solution O of (1.3):

p(b, 0)= Y A,Pn(cos 0)+ [AT cos (mck)+B sin (m4)]PT(cos 0)n=0 m=l

(3.8)2n+l 2n+l 2n+l

47r(n + 1)47r(n +1)An 4zr(n+l)an’A B fl

where {an, a’, 3m} are the Laplace expansion coefficients of f. The correspondingsolution of (1.1) is given by

(3.9)

47r(n+1) { i }u(A)= Y r AnPn(cos 0)+ [A cos (m4))+B sin (mb)]P(cos 0)n=0 2n+l m=l

where A (r cos b sin 0, r sin b sin 0, r cos 0) in spherical coordinates.

4. The Galerkin method. We change the variable of integration in (1.3), convert-ing it to a new integral equation defined on U. The Galerkin method is applied to thisnew equation, using spherical polynomials to define the approximating subspaces.

1--1We assume there is a mapping :U----- S for which the following property is

ontosatisfied"

(4.1)

where

f Hk,x (S), S Lk+l.h implies )e H,x (U),

(4.2) )(0) =--f((O)), 0 e U.

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NUMERICAL SOLUTION OF LAPLACE’S EQUATION 267

All of our numerical examples have been for regions D starlike with respect to theorigin, but the numerical method is not restricted to such regions. For starlike regions,we assume that a general point, P (O), of $ is given by

(4.3) P R (Q). (A, Brl, Cr), Q (, r/, r) U.

The constants A, B, C are positive, and the function R is a continuous positive functionon U. If R Hk/l.x (U), then (4.1) is satisfied.

Change the variable of integration on (1.3), to obtain the new equation over U,

(4.4) (27r + {’)t3 =/ e C(U).

The notation will denote the change of variable from S to U, as in (4.2). The operator(2r + {)-1 exists and is bounded on C(U) and L2(U).

The theoretical framework for the Galerkin method is that given in [3, p. 62]. LetL2(U), and let the approximating subspace of spherical polynomials of degree <- N

be denoted by N. The dimension of N is dN= (N + 1)2, and we will let {h 1, , ha}denote the basis of spherical harmonics given in (3.2).

Galerkin’s method for solving (4.4) is given by

(4.5) (2r + NC)t3N N)The solution is given by

d

(4.6)/=1

d

27toni(hi, hi)+ E cei(hi, f(hi)= (Y, hi),/’=1

i=1,... ,d.

A convergence theorem follows immediately from [3, Thm. 2, p. 51] and the resultsin3.

THEOREM 1. Assume that f Hk,x (S), S e L+1,, and that the mapping satisfies(4.1) for some k >= O. Then for all sufficiently large N, the inverses (27r + N{’)-1 exist andare uniformly bounded, and

C(4.7) I1 0,, [1= --< N+,where 0 < A’< A is arbitrary. The constant c depends on k, p, and A’.

Convergence in C(U). To prove uniform convergence of fin to is slightly moredifficult. The main problem is that there are t; in C(U) for whichN does not convergeto ; convergence for all would imply uniform boundedness of IINII, contradicting(3.5).

THEOREM 2. Assume that S L,x with > 1/2, and thatl satisfies (4.1) with k O.Then considering f{ as an operator on C(U),

(4.8)

This implies the existence and uniform boundedness on C(U) of (27r + Nf{’)-a ]:or allsufficiently large N.

If (2 rr + {’)t3 1 and (2 rr + NYg)t3N N] and if f Ho,x (S), then N convergesuniformly to . Moreover, if S eL+.x and f eH,x(S), k +A > 1/2, then

C(4.9)

with 0 < , < , arbitrary. The constant c depends on , k, and ’.

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268 K.E. ATKINSON

Proof. To prove (4.8), we must show N - I uniformly on the set

Looking at 3fi on U is equivalent to looking at Y{p with p C(S) and [[pllo_-< 1. Pick1/2 < h’< h. It is shown in [9, Thm. 1, p. 49] that the set of all such Y{p have a uniformH61der constant.

For each $ 5, let TN(O) be the spherical polynomial approximation of degree_-< N which is referred to in (3.4) with -0. Then

(4.10) ][0- TN(O)[Ioo < Ko=N,, N_->I, 05.

From the derivation in [8], the constant K0 is a simple multiple of the H61der constant oftO, and is otherwise independent of 0. Since there is a uniform H61der constant forelements of S;, this shows the convergence of Ts(0) to 0 is uniform on

Using uTs(0) T,(O), we have

Combined with (4.10),

N>_-l, 0e5,

for some c independent of 0. Thus

C(4.11) IIW#- f#ll NX’-x/2.

The remainder of the proof is straightforward from (3.4) and [3, Thm. 2, p. 51].From the identity

(2rr + sf{’)(t3 flu) 2r (t3 ut3),

it is easy to see that ON -’> P if and only if )NP "-> P. Thus the existence of e C(U) forwhich ufi does not converge to will also furnish an example in which flu does notconverge uniformly to 3. But Theorem I still assures us that we will have convergence inL2(U) of flu to fi, for any t3 e C(U).

The approximate harmonic function. For pu, define

(4.12) UN(A)-- ON(Q) c3p---s

Then uu(A) is harmonic in D. Using the maximum principle, we have

(4.13) max [u (a)- UN(A)[- max lu(P)- UN(P)[,AeI) PeS

with respect to the true solution u of (1.2). Let A - P S in (4.2), obtaining

us (P) 2ZrpN (P) + Y{PN (P), P e S.

Subtracting from (1.3) gives

(4.14) [u(P)-uN(P)[<-(2rr+lVdl[)llo-oNll, PeS.

For convex regions D, the kernel of Y( is positive, and then II2)’(11 2 zr. In any case, theconvergence results for ON lead directly to similar uniform bounds for us(A).

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NUMERICAL SOLUTION OF LAPLACE’S EQUATION 269

For an example of the present method on which all calculations can be carried outexplicitly, refer to (3.8) and (3.9) for formulas on the Sphere. In that case, ON and UN arejust the appropriate partial sums in (3.8) and (3.9), respectively, chosen to lie in thesubspace N.

5. Implementation of Galerkin’s method. Most of the work of this method is inthe setup of the linear system (4.6) and the evaluation of uN. And in both cases, the mostcostly step is the numerical integration of surface integrals over U. We have used theproduct Gaussian quadrature method with fairly good success; it will be discussedbelow. Once the linear system has been formed, it is relatively inexpensive to solve. TheparameterN in (4.6) can usually be kept fairly small, say N =< 7, and the approximationsON and/AN will be sufficiently accurate. Since d (N + 1)2 is the order of the system, it isa relatively small linear system and ordinary elimination can be used.

Throughout the calculations, it is very important to arrange the calculations quitecarefully so as to avoid repeating calculations unnecessarily. By doing so, we haveobtained computer programs which are much faster than might have been expected.Because the Galerkin coefficients (hi, f{hi) depend only on the surface S, we are able tosave time by calculating them separately, say for N _-< Nmax, and they are stored on disk,in a form for rapid retrieval by the main program used in solving (1.1).

Numerical integration on the unit sphere. Let

271"

I(f)= II f(O) do-= i fo f(da, O)sinOdOdd.u

Approximate it by the product Gaussian formula

2M M

(5.1) E Z w f(6 ,i=1j=1

or its midpoint variant

2M M

(5.2) Iv(f) =6i=1/=1

Here 6 ,r/M, 6i i6, ci (i -.5)6; and {wi}, {cos 0i} are the Gauss-Legendre weightsand nodes on [- 1, 1]. The degree of precision of both formulas is 2M- 1; a proof for(5.1) is given in Stroud [16, p. 40], and a similar one can be given for (5.2).

THEOREM 3. For any f C(U),

It(f)- I(f) asM oo.

Iff e Hl,x U), then

(5.3) [I(f)-It(f)l <c

(2M_ 1)+x, M->I,

for some c > O. The same results are also true for I’ (f).Proof. As a linear functional on C(U),

and all the weights in (5.1) are positive. Since the spherical polynomials are dense inC(U), it is straightforward to show I (f) converges to I(f), for any f e C(U).

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270 K.E. ATKINSON

Let E,(f) =- I(f) IM(f), and let T2M- denote any spherical polynomial of degree2M- 1. Since the degree of precision of IM is 2M- 1,

E(f) Elvl(f T2,u_l.),

iEM(f)[ I]EM[I I]f T2M-11]oo 8"lTllfTo complete the proof, let T2M_ be the polynomial in (3.4). ]

It is possible to prove convergence for other integrands. We will considerintegrands which are bounded and which are continuous except at a finite number ofpoints on U. Convergence can be shown; but we omit the proof since we cannot obtainany usable result on the rate of convergence.

The integral for us(A) is evaluated using It. In (4.12), the integrand is increasinglypeaked as A approaches the boundary. Let P e S be chosen near to A. Use the identity

A D,

to write

(5.4) UN(A)=4rcp(P)+ [PN(Q)--PN(P)] OV-----S

This will.result in a substantial increase in the accuracy of the numerical integration,without any significant increase in computation time.

Calculation of the Galerkin coefficients. To simplify the writing of the formulas, let/(P, Q) denote the kernal of t’. The coefficients (h, 2h) are fourfold integrals with asingular integrand. To decrease the effect of the singularity in computing 2hi(P), usethe identity

to write s

(5.5) f{hi(P) 2rhi(P)+ [ [ [hi(O)- hi(P)]ff2(P, O) &r.u

The new integrand contains a bounded discontinuity at O P. We numerically evaluatethe integral in (5.5) by applying IM to it, with respect to the spherical coordinatesrepresentation O O(& ’, O’), &-<O’<_-27r+O, 0 <_- 0’ _-< Tr, where P---P(&,O). Thisresults in some decrease in the integration error, since the point of discontinuity occursat a boundary point of the integration region.

Calculate the entire coefficient by

(5.6) (hi, (hi) IM(hi (hi),and approximate .{hi as above. Denote the complete integration rule by ,-M(hi,Because 2(hi should be quite smooth, by (2.4), the integration in (5.6) should be quiteaccurate with low M. Thus the major error will be in integrating (5.5). Empirically,

(5.7) (hi, ?fdhi)- (hi, d{hi)= 0 --and this has been used for extrapolation. The behaviour of the integration rule is still notas regular as one would like, and we are continuing to look for better methods forcalculating these coefficients.

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NUMERICAL SOLUTION OF LAPLACE’S EQUATION 271

The cost of evaluating all of the Galerkin coefficients of degree <_- N is proportionalto M4. Empirically, we have found the total cost in time is about

N

although the fraction - should be replaced by a slowly decreasing function of N. On aCYBER 72, with any ellipsoidal region $, c 0012 seconds. Also note that we usuallyneed only N -<_ 7 for quite adequate accuracy in solving (1.1).

6. Numerical examples. We begin with two calculations involving the sphereS U, and then we give examples for three other surfaces. The sphere is a good case inwhich to test the approximations at different stages of the method because we know theexact answers in most cases, based on (3.7)-(3.9).

Example 1. We consider the calculation of the Galerkin coefficients (hi, {hi).Using (3.7), (hi, Y{hi) 0 for j, and we give some of the cases

(hi, if{hi)kii

(hi, hi)

In Table 1, if m 0, then the coefficient kii is for hi P, (cos 0), and if m > 0, it is forhi P(cos 0) cos (mO). ow14 and ,16 are defined following (5.6), and ow1"6 denotes theRichardson extrapolate of ,-14 and ow16, based on (5.7). The results show the potentialvalue of using extrapolation.

n m kii

0 2.09439512 0 1.25663713 0 .89759794 0 .6981317

2.09439512 1 1.25663713 2 .8975979

TABLE

Error in kii using indicated integration

z14

-6.47E-4-1.90E-3-3.84E-3-6.44E-33.20E-4

-5.16E-44.16E-4

-4.39E-4-1.29E-3-2.60E-3-4.35E-32.16E-4

-3.51E-42.77E-4

-1.53E-5-4.40E-5-7.87E-5-1.09E-43.81E-6

-1.44E-5-6.07E-6

Example 2. We compare the calculation of UN(A) for S U using differentamounts of approximation in the solution process. The four options given in Table 2 areas follows; in all cases, N 5.

(1) Evaluate us(A) using the exact formula (3.9), with the coefficients (f, hi)evaluated using 116.

(2) Use (3.8) to evaluate pN, but evaluate us(A) using (5.4) to cancel partially thepeaked behavior of the integrand when A is near to U. Use 116 to evaluateus(A) and the coefficients (f, hi).

(3) Use the linear system (4.6) to solve for pv, with the Galerkin coefficientsobtained using a*6. Use I8 to evaluate us(A) (as in (5.4)) and (f, hi).

(4) The same as (3), but replace I8 with 116.The true solution is

(6.1) u=e cos(y)+e sin(x).

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272 I. E. ATKINSON

TABLE 2

Option

Error in us(A) for given A =(x, y, z)

(.1, .1, .1) (.25, .25, .25) (.5, .5, .5) (.55, .55, .55)

1.11E-8 -2.71E-6 1.72E-4 -3.04E-4-1.11E-8 -2.71E-6 -2.91E-4 -1.05E-3-1.08E-7 -3.27E-6 -4.87E-4 1.39E- 31.08E-7 -2.86E-6 -2.90E-4 1.05E- 3

It will also be used in the remaining examples. It is fairly illustrative of the behavior ofthe method, being neither the best nor worst of the examples computed.

The additional inaccuracy of options 3 and 4, in comparison with option 1, is due totwo causes. For inner points A, the Galerkin coefficients have not been computed withsufficient accuracy; and for the outer points A, the integration of UN(A) is notsufficiently accurate. In all cases, if we want more accuracy near the boundary, we willhave to increase the degree N of the approximation. The series (3.9) converges morerapidly for smaller values of r-IAI, and thus the worst error in uN(A) is near theboundary.

Example 3. Let S be the ellipsoidal surface given by

x + + =1.

The solution u is given by (6.1). We solve (4.6) for PN, the Galerkin coefficients arecalculated using 51"6, and both us(A) and the coefficients (/, hi) are calculated using 116.In Table 3, a is the number for which (1/a)(x, y, z) is on the boundary S; it is ameasure of the closeness of (x, y, z) to the boundary.

TABLE 3

x y z

0 0 0.1 .1 .1.25 .25 .25.5 .5 .5

0 .25 .25-.1 -.2 .20 0 1.0.5 0 .5

.13

.33

.65

.21

.19

.5

.56

//5

.9999461.2098511.5610702.235215.968910.765039.999947

2.435367

U --U4 U --/-/5

5.4E-5 5.4E-52.3E-3 1.3E-45.1E-3 7.1E-49.3E-4 2.1E-32.7E-6 2.7E-6

-2.0E-3 -1.7E-45.3E-5 5.3E-59.0E-3 3.8E-3

Increasing the accuracy of the integration of UN(A) by changing to 132 does notdecrease the error in the calculated Us. Note again that fairly reasonable accuracy hasbeen obtained by solving a linear system of order 36, which is quite small for athree-dimensional problem.

Example 4. We use another ellipsoidal surface,2 2

x + + =1,

which is more ill-behaved in the sense of being thin and narrow in shape. We solve (1.1)for two cases,

b/(1) [(X 5)2 "-(y --4) + (Z 3)2]-1/2

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NUMERICAL SOLUTION OF LAPLACE’S EQUATION 273

and u (2) given by (6.1). Table 4 contains the results of solving for/17, with the Galerkincoefficients computed using 516. The coefficients (f, hi) and UN were computed with 132.

TABLE 4

0 0 0 -5.6E-6.1 .1 .11 -5.7E-6.25 .25 .28 -8.6E-6.5 .5 .57 -6.7E-5

0 1.0 .20 1.9E-60 2.0 .40 6.7E-6

0.1.25.5

00

1.41426981.44902841.50444041.60711321.49069311.5429666

1.0035 -3.5E-31.2086 1.4E-31.5665 -4.7E-32.2958 -5.8E-21.0031 -3.1E-31.0025 -2.5E-3

The function /1(2) varies greatly and rapidly over some sections of this region; andthat leads to the need for a higher degree N in order to accurately represent the doublelayer density p2) associated with u 2). The function u 1) is a better behaved function.

Example 5. Our final surface S is defined by

(x, y, z) R (sin 0 cos b, 2 sin 0 sin b, cos 0),

R /cos (20)+ /1.1 -sin2 (20).

Loosely speaking, it is a "peanut-shaped" region, based on the "ovals of Cassini". Thecross-section in the x, z-plane is shown in Fig. 1.

FIG.

We will solve for the solutions u1 and U(2) used with Example 4. The Galerkin

coefficients were calculated with 16, and the cofficients (f, hi) and the solution uN werecomputed with I32. Table 5 contains the results for N 7. The results are acceptable inboth cases; but they are much better for u () than for u (2), just as was true in the previousexample.

Concluding remarks. The method of this paper results in linear systems that arequite small for solving the three-dimensional Laplace equation. It is also a fairly rapidmethod, especially after the Galerkin coefficients have been calculated and stored ondisk. The method is not suitable for regions which do not have a smooth boundary, and

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274 . E. ATI,ZNSON

x y z

0 0 00 0 .40 0 .80 0 1.2.03 .06 .04.07 .14 .10.14 .28 .20

.28

.56

.84

.11

.25

.49

TABLE 5

(1) (1) U(71)b/7 b/

.14142 -4.8E-7

.14473 -3.1E-5

.14771 -1.5E-5

.15041 -6.3E-5

.14288 9.9E-7

.14491 8.1E-6

.14857 1.4E-5

u 2)

1.0001 -5.3E-51.0009 -9.4E-4.9950 5.0E-3.9852 1.5E-2

1.0597 9.2E-51.1386 7.5E-41.2692 6.7E-3

these probably form the large majority of potential theory problems. But for thatsmaller class of problems which do have smooth boundaries, the present method shouldbe superior to the use of a finite element method for solving the integral equations, andto most direct methods of solution of the Laplace equation. In the sequel [4] to thispaper, we consider the extension of the present work to the exterior Dirichlet problemand to the interior and exterior Neumann problems.

As an addendum at the time of this paper’s acceptance, we have developed a bettermethod of calculation for the Galerkin coefficients. This work will appear in [5].

Acknowledgments. I wish to thank Vladimir Oliker, University of Iowa, forhelpful discussions, and I thank Dick Askey, University of Wisconsin, for references tothe literature on the convergence of the Laplace series.

REFERENCES

1. M. ABRAMOWITZ AND I. STEGUN, eds., Handbook of Mathematical Functions, National Bureau ofStandards, Washington, DC, 1964.

2. T. APOSTOL, Mathematical Analysis, Addison-Wesley, Reading, MA, 1957.3. K. ATKINSON, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the

Second Kind, Society for Industrial and Applied Mathematics, Philadelphia, 1976.4. , The numerical solution of Laplace’s equation in three dimensions--II, Numerical Treatment of

Integral Equations, J. Albrecht and L. Collatz eds., Birkhauser Verlag, Basel, 1980, pp. 1-23.5. ,Numerical integration on the sphere, J. Australian Math. Soc., Series B, to appear.6. C. BAKER, The Numerical Treatment of Integral Equations, Oxford Press, Cambridge, 1977.7. P. GARABEDIAN, Partial Differential Equations, John Wiley, New York, 1977.8. T. GRONWALL, On the degree of convergence of Laplace’s series, Trans. Amer. Math. Soc., 15 (1914),

pp. 1-30.9. N. M. GONTER, Potential Theory, F. Ungar, New York, 1967.

10. M. JASWON AND G. SYMM, Integral Equation Methods in Potential Theory and Elastostatics, AcademicPress, New York, 1977.

11. O. D. KELLO66, Foundations of Potential Theory, Dover, New York, 1953 (reprint of 1929 book).12. T. M. MACROBERT, Spherical Harmonics, 3rd ed., Pergamon Press, 1967.13. W. PO6ORZELSKI, IntegralEquations and TheirApplications, Vol. I, Pergamon Press, New York, 1966,14. D. RA6OZIN, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer.

Math. Soc., 162 (1971), pp. 157-170.15. ., Uniform convergence of spherical harmonic expansions, Math. Ann., 195 (1972), pp. 87-94.16. A. H. STROUD, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, NJ,

1971.17. A.H. STROUD AND D. SECREST, Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ,

1966.18. A. TAYLOR, Introduction to Functional Analysis, John Wiley, New York, 1958.19. R. WAIT, Use offinite elements in multidimensional problems in practice, Numerical Solution of Integral

Equations, L. Delves and J. Walsh, eds., Clarendon Press, Oxford, 1974, pp. 300-311.

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