GREEN S REPRESENTATION FORMULA IN THE PLANE - Interior Version

Transcription

1 GREEN S REPRESENTATION FORMULA IN THE PLANE - Interior Version Let D be a bounded simply-connected planar region with a piecewise smooth boundary S. Assume u 2 C 1 (D) \ C 2 (D) and u 0 on D. Then Z Q log jp Qj Q [log jp 8 >< >: Qj] ) ds Q 2u(P ); P 2 D (P )u(p ); P 2 S 0; P 2 D e In this, n Q is the inner normal to S at Q, (P ) is the interior boundary angle at P, and D e = R 2 nd.

2 GREEN S REPRESENTATION FORMULA IN THE PLANE - Exterior Version Assume u 2 C 1 (D e ) \ C 2 (D e ); u 0 on D e, and Then sup P 2D e ju(p )j < 1 Z S log jp Qj [log Q = 8 >< >: Qj] ) ds Q 2 [u(1) u(p )] ; P 2 D e 2u(1) [2 (P )] u(p ); P 2 S 2u(1); P 2 D

3 GREEN S REPRESENTATION FORMULA IN R 3 - Interior Version Let D R 3 be a bounded simply-connected region with a piecewise smooth boundary S. Assume u 2 C 1 (D) \ C 2 (D) and u 0 on D. Then Z S Q " jp 1 Qj # = 8 >< >: Q jp Qj ds Q 4u(P ); P 2 D (P )u(p ); P 2 S 0; P 2 D e In this, n Q is the inner normal to S at Q, (P ) is the interior solid angle at P, and D e = R 3 nd.

4 GREEN S REPRESENTATION FORMULA IN R 3 - Exterior Version Assume u 2 C 1 (D e ) \ C 2 (D e ); u 0 on D e, and ju(p )j = O jp j as jp j! 1. Then Z S Q " = jp 8 >< >: 1 ; jru(p )j = O jp j # ) 2 1 ds Q Q jp Qj 4u(P ); P 2 D e [4 (P )] u(p ); P 2 S 0; P 2 D

5 BOUNDARY INTEGRAL EQUATIONS DIRECT FORMULATIONS Solve the exterior planar Neumann problem u(p ) = 0; P 2 P = f(p ); P 2 S and assume S is smooth. For uniqueness, we impose ju(p )j = O jp j as jp j! 1. Then solve the BIE u(p ) + Z Z 1 ; jru(p )j = O jp j Q [log jp Qj] ds Q = f(q) log jp Qj ds Q; P 2 S S This is uniquely solvable. This is a classical BIE of the second kind, and it was used by Fredholm as a means to show existence of a solution to the associated boundary value problem. 2

6 Solve the interior planar Dirichlet problem u(p ) = 0; P 2 D u(p ) = f(p ); P 2 S Do this by solving the BIE Z (Q) log jp Qj ds Q = '(P ); P 2 S S '(P ) = Z S with the Q [log jp Qj] ds Q + (P )f(p ) Q The above BIE of the rst kind has been studied intensively in the past decade; and today it is very well understood for the case that S is a smooth curve.

7 INDIRECT BIE FORMULATIONS IN R 3 Let u i 2 C 1 (D) \ C 2 (D) and u i 0 on D; and let u e 2 C 1 (D e ) \ C 2 (D e ); u e 0 on D e, with ju e (P )j = O jp j 1 ; jru e (P )j = O jp j as jp j! 1.For points P 2 S, introduce [u(p )] = u i (p) u e (P ) " ) i(p e (P P Subtract the exterior version of Green s representation formula from the interior version. This yields Z ( " # " # 1 [u(q)] ds Q Q jp Q jp Qj 8 >< 4u i (P ); P 2 D = (P )u i (P ) + [4 (P )] u e (P ); P 2 S >: 4u e (P ); P 2 D e 2

8 SINGLE LAYER REPRESENTATIONS Given a function u i 2 C 1 (D) \ C 2 (D) and u i 0 on D, choose a function u e 2 C 1 (D e )\C 2 (D e ); u e 0 on D e, with ju e (P )j = O jp j and with 1 ; jru e (P )j = O jp j 2 u e (P ) = u i (P ); P 2 S Then u i has the representation with u i (P ) = Z S (Q) = 1 4 (Q) jp Qj ds Q; P 2 Q # ; Q 2 S This in turn leads to both BIE of the rst kind and second kind, depending on the type of boundary condition being given.

9 DOUBLE LAYER REPRESENTATIONS We can repeat the type of argument given for the single layer to obtain the double layer representation u i (P ) = Z Q " jp 1 Qj Consider the interior Dirichlet problem # ds Q ; u(p ) = 0; P 2 D u(p ) = f(p ); P 2 S This leads to the classic BIE of the second kind K(P ) = Z (2 + K) = f " P 2 D 1 ds Q Q jp Qj + [2 (P )] (P ); P 2 S When the double layer representation is used to solve the interior Neumann problem, one obtains a hypersingular BIE. #

10 EXAMPLES J. Hess (1990) Panel methods in computational uid dynamics, Annual Rev. Fluid Mechanics 22, pp This is a review of potential ow calculations in aerodynamic applications; and Hess has been a major gure in the development of such methods over the past 30 years. J. Newman (1992) Panel methods in marine hydrodynamics, Proc. Conf. Eleventh Australasian Fluid Mechanics Newman and his colleagues at MIT have used panel methods for uid ow calculations, including the study of potential ows around oil-drilling platforms.

11 F. Rizzo, D. Shippy, and M. Rezayat (1985) A boundary integral equation method for radiation and scattering of elastic waves in three dimensions, International Journal for Numerical Methods in Engineering 21, pp Rizzo and his colleagues have applied a combination of BIE methods for studying crack problems in elasticity, developing methods based on the scattering of elastic waves. M. Jaswon and G. Symm (1977) Integral Equation Methods in Potential Theory and Elastostatics, Academic Press. This covers several major problem areas.

12 PLANAR SMOOTH BOUNDARIES SECOND KIND BIE The classical BIE studied in textbooks are usually of this type. Solving the interior Dirichlet problem using a double layer representation for u(p ) leads to a second kind equation ( + K) = f; (t) + Z 2 0 K(t; s)(s) ds = f(t) in which K is smooth and periodic. Everything works well, and Nyström methods with the trapezoidal rule probably work best: n (t) + h nx j=1 K(t; jh) n (jh) = f(t) This is a situation for which all the integral equations literature is well-suited. k n k 1 = O (kk K n k 1 )

13 PLANAR SMOOTH BOUNDARIES FIRST KIND BIE A large literature has developed in the past ten years. Most studied is Z Lx(P ) x(p ) log jp Qj ds Q = f(p ); P 2 S S often called Symm s equation. The curve S is nite, either open or closed. For unique solvability for all f 2 C(S), assume S is not a -contour. Y. Yan and I. Sloan (1988) On integral equations of the rst kind with logarithmic kernels, J. Integral Eqns & Applics, 1, pp For S simple and closed, L : H r 1 1! onto Hr+1 ; r 0 depending on the smoothness of S. It can also be extended to r < 0.

14 DECOMPOSITION OF L Let S have a parameterization r(t); 0 t 2. Then Lx(r(t)) = Z 2 0 [log jr(t) Introduce the equivalent operator K'(t) = Z 2 0 '(s) log jr(t) The equation Lx = f becomes K' = g r(s)j] r 0 (s) x(r(s))ds r(s)j ds with g(t) = f(r(t)); '(s) = r 0 (s) x(r(s)) We can write K' = A' + B'

15 A'(t) = Z 2 0 '(s) log 2e 1 2 sin t 2 s ds B(t; s) = 8 >< >: B'(t) = Z 2 0 B(t; x)'(s) ds log jp e [r(t) r(s)]j t s ; t s 6= 2m 2 sin 2 log p er 0 (t) ; t s = 2m If r 2 Cp m (2), then B(t; s) is in Cp m 1 (2) with respect to both s and t. Consequently, B is a compact operator on most spaces. For example, B is compact from H r to H r+1 for r 0, provided m > 2: Then K' = g is equivalent to ' + A 1 B' = A 1 g This is a second kind equation, with a compact integral operator; and much is known of such equations.

16 For ' 2 L 2 (0; 2), write Then '(s) = 1 p 2 1 X m= 1 2 A'(t) = p 1 6 4b'(0) + 2 b'(m)e ims X jmj>0 b'(m) 3 jmj eims 7 5 The decomposition K = A + B is the major tool used in analyzing many numerical methods for solving K' = g. In essence, A is the case of K with S the unit circle. This shows directly that A : H r 1 1! onto Hr+1 ; r 0 and the de nition of A' extends easily to the case r < 0. Many numerical methods for K' = g are rst studied for A ' = g.

17 NUMERICAL METHODS For a survey of numerical methods for Lx = f, see I. Sloan (1992) Error analysis of boundary integral methods, Acta Numerica 1, pp GALERKIN METHODS. Regard L as a strongly elliptic operator from H 1 2 to H 1 2: (Lx; x) c kxk ; x 2 H 1 2 Then extend nite element theory to this case. For suitable nite element subspaces X n, nd x n solving (Lx n ; v) = (f; v); v 2 X n The left side uses the extension to H 1 2 H 1 2 of the standard inner product on H 0 = L 2. Then kx x n k 12 c inf v2x n kx vk 12 With x n piecewise polynomial of degree r 0; kx x n k 0 ch r+1 kxk r+1 ; x 2 H r+1

18 COLLOCATION METHODS. These divide into two classes, depending on the form of the approximating functions: (a) boundary element methods (b) spectral methods Boundary element methods: Important initial papers are the following. D. Arnold and W. Wendland (1983) On the asymptotic convergence of collocation methods, Numer. Math. 41, pp J. Saranen (1988) The convergence of even degree spline collocation solution for potential problems in the plane, Numer. Math. 53, pp The entire literature of these methods is very well-covered in the excellent survey of Ian Sloan. I. Sloan (1992) Error analysis of boundary integral methods, Acta Numerica 1, pp

19 Spectral methods: The spectral methods de ne X n as the trig polynomials of degree n. For a Galerkin method to solve K' = g, let P n denote the orthogonal projection of L 2 (0; 2) onto X n. De ne ' n 2 X n as the solution of P n K' n = P n g Recalling K = of A, especially A+B, and using the special properties P n A = AP n ; 2 X n we have the equivalent equation ' n + P n A 1 B' n = P n A 1 g A complete error analysis can be based on comparing this with ' + A 1 B' = A 1 g W. McLean (1986) A spectral Galerkin method for boundary integral equations, Math. Comp. 47, pp

20 A discrete Galerkin-collocation method: Let T m (f) = 2 m m 1 X j=0 f 2j m ; f 2 C p (2) the trapezoidal rule over [0; 2], and de ne Introduce B n '(t) = T 2n+1 (B(t; )') (f; g) n = T 2n+1 (f g) a discrete inner product on C p (2), and let Q n denote the discrete orthogonal projection associated with (; ) n ; or equivalently, it is the interpolatory projection onto X n associated with the nodes used in T 2n+1. Then solve Equivalently, Q n [A' n + B n ' n ] = Q n g ' n + Q n A 1 B n ' n = Q n A 1 g

21 We compare with the original equation K' = g in the form ' + A 1 B' = A 1 g For some q > 0:5; this leads to k' ' n k 1 cn q+:5+ ; ' 2 C p (2) \ H q (2) K. Atkinson (1988) A discrete Galerkin method for rst kind integral equations with a logarithmic kernel, Journal of Integral Eqns & Applic. 1, pp W. McLean, S. Prößdorf, and W. Wendland (1993) A fully-discrete trigonometric collocation method, J. Int. Eqns. & Applic., to appear. For S an open smooth planar curve, see K. Atkinson and Ian Sloan (1991) The numerical solution of rst kind logarithmic-kernel integral equations on smooth open arcs, Math. of Computation 56, pp

22 PLANAR SMOOTH BOUNDARIES HYPERSINGULAR BIE Use a double layer potential u(p ) = Z Q " jp 1 Qj # ds Q ; P 2 D to solve the interior Neumann problem for u = 0 on D. We nd by solving H = f Z H(P ) = lim n P r P Z Q " Q 1 jp Qj " # ja 1 ds Q For S = unit circle, let H be denoted by H u. Then H u : e int! jnj e int ; n = 0; 1; ; ::: Qj # ds Q

23 Let D denote the derivative operator, and let C u denote the Cauchy singular integral operator on the unit circle: Then C u '(z) = 1 Z jj=1 '() d H u = idc u = ic u D H u : H r! H r 1 ; r 1 For the general equation H = f, with a parameterization of S proportional to arclength, z H() = 2 L H u() + BD() This forms the basis for developing spectral methods of solution.

24 A. Rathsfeld, R. Kieser, and B. Kleemann (1992) On a full discretization scheme for a hypersingular boundary integral equation over smooth curves, Z. Anal. Anwendungen 11, pp R. Kress (1995) On the numerical solution of a hypersingular integral equation in scattering theory, J. Comp. & Appl. Maths. 61, pp D. Chien and K. Atkinson (1995) The numerical solution of a hypersingular planar boundary integral equation, Rep. on Comp. Maths #74, Dept of Maths, Univ of Iowa. S. Amini and N. Maines (1995) Regularisation of strongly singular integrals in boundary integral equations, Tech. Rep. MCS-95-7, Univ of Salford, UK.

25 PLANAR PIECEWISE SMOOTH BOUNDARIES SECOND KIND BIE This is a topic for which enormous progress has been made, essentially solving the problem. As a particular example, consider solving the interior Dirichlet problem using a double layer potential representation: u(p ) = Z S Then satis es K(P ) Q [log jp Qj] ds Q ; P 2 D Z ( + K) = f Q [log jp Qj] ds Q [ (P )] (P ); P 2 S (P ) denotes the interior angle at P 2 S. It follows that is bounded. K : C(S)! C(S)

26 The behaviour of a solution can be obtained by using (P ) = u(p ) u e (P ) with u e (P ) a solution of an exterior Neumann problem with normal derivative matching that of u(p ) on S. For a harmonic function on a region with a corner, say at the origin with internal angle, with smooth Dirichlet boundary data, the behaviour about the origin is ( O (r ) ; 6= m; a positive integer u(x; y) = O (r log r) ; = m with = =. Letting = (1 ) ; 1 < < 1 (P ) = O jp j ; = jj Therefore, has an algebraic singularity, with 1 2 < < 1

27 In addition to lack of smoothness in, the operator K(P ) = Z Q [log jp [ (P )] (P ) Qj] ds Q is no longer compact on C(S). To gain some insight, look at the special case of a wedge boundary. Consider solving the Dirichlet problem as a double layer potential.

28 Then is equivalent to the system 1 (x) 2 (x) Z 1 0 Z 1 0 ( + K) = f x sin () 2 (y) dy x 2 + 2xy cos () + y 2 = f 1 (x) x sin () 1 (y) dy x 2 + 2xy cos () + y 2 = f 2 (x) for 0 < x 1. The function 1 and 2 are the restrictions of to the bottom branch and upper branch of S, respectively. This system can be uncoupled as (x) 1Z 0 x sin () (y) dy x 2 + 2xy cos () + y 2 = F (x) and each equation can be reduced to a more well-known type of equation.

29 Wiener-Hopf equation. Let x = e t, y = e. Then the integral operator becomes b (t) Z 1 0 sin () b () e t 2 cos () + e t = b F (t) for 0 t < 1. From this perspective, we obtain that the spectrum of K equals [ ; ], clearly showing it is not compact. Mellin convolution equation. Rewrite as (x) 1Z 0 (x) x sin () (y) dy x 2 + 2xy cos () + y 2 = F (x) 1Z 0 x y! (y) dy y = F (x) (u) = u sin () u 2 + 2u cos () + 1 ; 0 u < 1

30 GALERKIN METHODS. Regard L (x) 1Z 0 x sin () (y) dy x 2 + 2xy cos () + y 2; 0 < x 1 as an operator on L 2 (0; 1). Then klk sin < 2 Therefore ( L) = F is uniquely solvable. Let X n be a subspace of piecewise polynomial functions, and let P n be the orthogonal projection of L 2 (0; 1) onto X n. Then ( P n L) n = P n F is uniquely solvable for n 2 X n, the method is stable, and k nk 2 ( P n L) 1 k Pn k 2 Use a graded mesh to obtain an optimal order of convergence.

31 De ne a graded mesh using i q x i = ; i = 0; 1; :::; n; q 1 n Let X n consist of all piecewise polynomial functions of degree r. Recall the order of convergence of the solution about the origin is O(x ) with = (1 + jj) 1. Then choose Then k q > r nk 2 O n (r+1) Choosing q r + 1 will ensure this for all angles. These generalize to results using kk 1, with a suitably larger choice for q. REF: G. Chandler (1984) Galerkin s method for boundary integral equations on polygonal domains, J. Australian Math. Soc., Series B, 26, pp

32 COLLOCATION METHODS. Let X n denote all piecewise polynomials of degree r on our graded mesh, with no continuity restrictions. Introduce 0 0 < 1 < < r 1 and de ne collocation nodes x i;j = x i 1 + j (x i x i 1 ) ; j = 0; :::; r for i = 1; :::; n. [With 0 = 0 and r = 1, there are continuity restrictions on X n.] De ne P n as the piecewise interpolatory projection of C[0; 1] onto X n [actually we use a larger space than C[0; 1] to have a well-de ned theory]. The resulting collocation method ( P n L) n = P n F is more di cult to analyze than Galerkin s method; and having P n x! x for all x 2 C[0; 1] with a suitably graded mesh is no longer su cient for convergence (due to an example of Chandler and Graham).

33 For the numerical method, the approximating subspace X n must be modi ed about the origin, with a corresponding change in P n. On some neighborhood [0; x i ], we must de ne the approximations as piecewise constant, with corresponding collocation nodes the centroids of the subintervals. Assume 0 Z 1 0 i=0 1 ( i ) g()da = 0; deg (g) r 0 Then for i 0 su ciently large, the ( P n L) 1 exist and are uniformly bounded for n N. Moreover, if q > r + r0 + 2 then max j (v i ) n (v i )j O n (r+r0 +2) i This is uniform in if q 2 r + r 0 + 2

34 A crucial role in the analysis is played by the problem of inverting nite section equations. In the Mellin convolution formulation, we need to show the stability over of the approximation (x) 1Z and its relation to the full equation (x) 1Z 0 x y x y!! (y) dy y = F (x); 0 < x 1 (y) dy y = F (x); 0 < x 1 This is also the problem of analyzing the approximation of the Wiener-Hopf equation b (t) 1Z k (t ) b () dy = b F (t); t 0 by the nite section equation 0 Z b (t) k (t ) b () dy = b F (t); t 0 0

35 The solvability theory for ( + K) = f goes back to Radon in His methods are akin to the present methods, in that the problem is divided into two parts, one on a small wedge shaped section about the corner point and the other on the remaining part of the boundary. This is also the basis for studying the corresponding problems on surfaces in R 3. REF: G. Chandler and I. Graham (1988) Product integrationcollocation methods for non-compact integral operator equations, Math. Comp. 50, pp For Nyström methods, see I. Graham and G. Chandler (1988) High order methods for linear functionals of solutions of second kind integral equations, SIAM J. Numer. Anal. 25, pp For generalizations of all methods and results, see J. Elschner (1990) On spline approximation for a class of non-compact integral equations, Math. Nachr. 146, pp

36 Additional references : M. Costabel and E. Stephan (1985) Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical Models and Methods in Mechanics, Banach Center Publications 15, pp R. Kress(1990) A Nyström method for boundary integral equations in domains with corners, Numer. Math. 58, pp A. Rathsfeld (1992) Iterative solution of linear systems arising from the Nyström method for the double layer potential equation over curves with corners, Math. Methods Appl. Sci. 15, pp Y.-M. Jeon (1993) A Nyström method for boundary integral equations on domains with a piecewise smooth boundary, J. Int. Eqns. & Applic. 5, pp

37 PLANAR PIECEWISE SMOOTH BOUNDARIES FIRST KIND BIE The model rst kind BIE for piecewise smooth planar boundaries is Z Lx(P ) x(p ) log jp Qj ds Q = f(p ); P 2 S S Preliminary results on collocation methods for this equation were obtained in the thesis of Yan Yi; e.g. see Y. Yan (1990) The collocation method for rst kind boundary integral equations on polygonal domains, Math. of Comp. 54, pp The di culty is that the earlier split L = A + B is no longer of much use, at least as de ned earlier.

38 The rst general collocation method was obtained only recently, in J. Elschner and I. Graham (1995) An optimal order collocation method for rst kind boundary integral equations on polygons, Numerische Math. 70, pp The main idea is to use a carefully constructed parameterization of the polygonal boundary, one which will improve the behaviour of the equation in the vicinity of corners of S. For simplicity, consider only a single corner, at the origin. Then we introduce a parameterization : [ ; ]! S with (0) = 0, and we consider here only the corner at the origin. Then Lx = f becomes Kw() w() = Z log j(s) 0 () x(()); ()j w() = g(s) g(s) = f((s))

39 The function is so chosen that all low order derivatives vanish around s = 0. The form of is (s) = O (s q ) for s near to 0, and with q chosen su ciently large. The resulting solution w is smoother around 0; and the operator A can again be used in approximating K, where A'(t) = Z 2 We change Kw = g to h I + A 1 (K 0 It can be shown that '(s) log 2e 2 1 sin t 2 A) i w = A 1 g A 1 (K A) = HD (K A) + E s ds with E compact, D' = ' 0, and H the Hilbert singular integral operator.

40 The operator B = HD (K A) can be shown to be equivalent about the corner at the origin to a Mellin convolution operator plus a compact perturbation. Earlier types of results for such operators can then be brought to bear to analyze the second kind reformulation of Kw = g. For the error in solving for w, it can be shown that if q is chosen su ciently large, and if a magic index i is chosen su ciently large, then a collocation method with splines of degree k and a uniform mesh h will have an error in L 2 of O(h k+1 ). Generalizations using other types of approximating functions are being given, by Elschner, Graham, Jeon, Stephan, and others.

41 SMOOTH BOUNDARIES IN R 3 SECOND KIND BIE The general theory of numerical methods for solving BIE of the second kind with a compact operator on smooth surfaces is basically well-understood; but signi cant practical problems remain, especially concerning numerical integration. Recall the single layer and double layer potential operators: K(P ) = S (P ) = Z S Z Q (Q) jp Qj ds Q; P 2 S " jp 1 Qj # ds Q ; P 2 S For S the smooth boundary of a closed bounded simply connected region, S; K : H r (S)! H r+1 (S)

42 For S = U the unit sphere, S = 2K. Using a representation of s function 2 L 2 (U) in terms of its Laplace series expansion in spherical harmonics, the mapping properties of S are transparent: S : Y m n! 4 2n + 1 Y m n for spherical harmonics Y m n of degree n. An extensive development of the mapping properties of S and K is given in N. Günter (1967) Potential Theory, Ungar Pub., New York. In particular, both K and S are compact on X to X, with X equal to either C(S) or L 2 (S).

43 NUMERICAL METHODS It is straightforward to speak of both Galerkin and collocation methods, especially those of the boundary element type. The convergence theory follows easily from the compactness of K and S. For solving the interior Dirichlet problem as a double layer potential, (2 + K) = f The Galerkin and collocation methods are simply Then (2 + P n K) n = P n f k(i P n ) Kk! 0 implies the existence and uniform boundedness of (2 + P n K) 1 for all n N; and moreover, k n k 2 (2 + P n K) 1 k(i Pn ) k

44 Let T n = BOUNDARY ELEMENT - COLLOCATION (n) k j k = 1; :::; n triangulations of S; and let denote a sequence of h h n = max diam (n) k k Usually we subdivide each element of T n, connecting the midpoints of the sides of (n) k, in order to get the next triangulation T 4n. We usually have a parameterization of each element (n) k, say m k : c(n) k R 2 1 1! onto (n) k Let p(x; y) be a polynomial over c(n) k. Then consider P (m k (x; y)) = p(x; y) as a polynomial over (n) k.

45 For some degree r 0, we de ne X n as the set of all functions that are piecewise polynomial over the triangulation T n of S. In our own work, we often restrict the functions of X n to be continuous; but this is sometimes too restrictive. As a particular example for which I have created a public domain package BIEPACK, we take r = 2 and we base the collocation method on interpolation at the vertices and midpoints of the elements of T n. On the boundary of bounded simply connected region, this leads to n v = 2n + 2 nodes. These are denoted collectively by V n = v (n) i and in terms of the element (n) k to which they belong by n o vk;1 ; :::; v k;6

46 For interpolation over (n) k, introduce Lagrange basis polynomials n o L k;1 ; :::; L k;6 and write P n f (Q) = 6X j=1 f v k;j Lk;6 (Q); Q 2 (n) k We often also approximate the surface S by interpolation, of the same or greater degree. For a piecewise polynomial surface approximation of degree q 1, interpolate the parameterization function m k (x; y) by a polynomial of degree q over c(n) k. In BIEPACK, we use quadratic interpolation. The panel method uses piecewise constant interpolation of functions combined with piecewise planar approximations of the surface. It is widely used in the aircraft industry.

47 The collocation system. Consider the linear system associated with We write and solve 2 n + n (Q) = nx v (n) i 6X k=1 j=1 (2 + P n K) n = P n f 6X j=1 n vk;j Lk;6 (Q); Z n vk;j L k;j Q = f v (n) i Q 2 (n) k v(n) i 1 ; i = 1; :::; n v The most time-consuming part of this method is the setup of the coe cient matrix, which means the accurate and e cient numerical integration of the integrals over the elements. It is often an order of magnitude more costly than is the solution of the linear system. Q ds Q

48 ASSORTED REMARKS With piecewise quadratic interpolation de ning the quadrature, and with the re nement of T n to T 4n based on subdividing each element into 4 new elements based on connecting the midpoints of the sides, we have max j(v v i ) n (v i )j = O h 4 log h i whereas k n k 1 = O h 3 This appears to generalize to collocation with X n piecewise polynomial of even degree. When piecewise polynomial collocation of degree r, and approximating the surface S with a polynomial of degree q, the best we have been able to prove is k n k 1 = O h minfr+1;qg although empirical results are better than this. For r = q = 2, we obtain something consistent with max v i j(v i ) n (v i )j = O h 4 log h or O h 4

49 REFERENCES W. Wendland (1985) Asymptotic accuracy and convergence for point collocation methods, in Topics in Boundary Element Research, Vol. II : Time-Dependent and Vibration Problems, Springer-Verlag, Berlin, pp K. Atkinson and D. Chien (1995) Piecewise polynomial collocation for boundary integral equations, SIAM J. Scienti c Computing 16, pp K. Atkinson (1993) User s Guide to a Boundary Element Package for Solving Integral Equations on Piecewise Smooth Surfaces, Reports on Computational Mathematics #43, Dept of Mathematics, University of Iowa, Iowa City.

50 SPECTRAL METHODS - GALERKIN For the equation (2 + K) = f K(P ) = Z Q " jp 1 Qj # ds Q ; P 2 S change the integration region to the unit sphere U. Let S be given by a smooth mapping with a smooth inverse M 2(P ) + 2b(P ) + Z Z M : U 1! 1 onto S 1. Then our BIE S K(P; Q)(Q) ds Q = f(p ); S ck(p; Q)b(Q) ds Q = b f(p ); b(p ) (M(P )); b f(p ) f(m(p )) P 2 S P 2 U ck(p; Q) = K(M(P ); M(Q))J M (Q); Q 2 U with Jacobian J M (Q).

51 Denote the new integral equation by 2 + b K b = b f Let X N denote all spherical polynomials of degree N, which has dimension d N (N + 1) 2. The standard orthogonal basis is the set of spherical harmonics fy m n (; ) j 0 n N; n m ng which we assume here are of norm 1. Also denote these by f k j 1 k d N g Let P N denote the orthogonal projection of L 2 (U) onto X N. The Galerkin method is simply 2 + PN b K b N = P N b f Write the numerical solution as b N (P ) = d NX k=1 k k

52 It is obtained by solving 2 i + d NX k=1 k b K k ; i = (f; i ) ; i = 1; :::; d N The coe cients b K k ; i are double surface integrals; and as such, they must be evaluated carefully to have a method of acceptable cost. For the speed of convergence, say on C(U), k N k P N K 1 I P N 1 For 2 C [`;] (U), a Hölder space of `-times di erentiable functions with index, and for S su ciently smooth, we have k N k 1 O N (`+ 0:5)

53 A Discrete Galerkin method. For integration on U, introduce I m (f) = m 2mX mx i=1 j=1 w j f(cos i sin j ; sin i sin j ; cos j ) with i = i=m, and n o n o cos j ; wm;j the Gauss- Legendre nodes and weights of order m on [ 1; 1]. This formula I m (f) is exactly I(f) for all spherical polynomials of degree 2m 1. Introduce a discrete inner product on C(U) using (f; g) N = I N+1 (f g) On X N, (f; g) N (f; g), the true inner product for L 2 (U). Let Q N denote the associated discrete orthogonal projection operator. A semi-discrete Galerkin method can be obtained with 2 + QN b K b N = Q N b f but the theory for this and for more complete discretizations is not adequate to handle the case that K is a double layer potential operator.

54 REFERENCES K. Atkinson (1982) The numerical solution of Laplace s equation in three dimensions, SIAM J. Num. Anal. 19, pp K. Atkinson (1985) Algorithm 629: An integral equation program for Laplace s equation in three dimensions, ACM Trans. on Math. Soft. 11, pp M. Ganesh, I. Graham, and J. Sivaloganathan (1993) A pseudospectral 3D boundary integral method applied to a nonlinear model problem from nite elasticity, SIAM J. Numerical Analysis 31, pp

55 PIECEWISE SMOOTH BOUNDARIES IN R 3 : SECOND KIND BIE The most studied of such equations is (2 + K) = f; K(P ) = Z " 1 ds Q Q jp Qj + [2 (P )] (P ); P 2 S which can arise from either the interior Dirichlet problem or the exterior Neumann problem. The rst important paper in the study of the numerical analysis of this equation is # W. Wendland (1968) Die Behandlung von Randwertaufgaben im R 3 mit Hilfe von Einfach und Doppelschichtpotentialen, Numerische Math. 11, pp Detailed analyses are given on the mapping properties of single and double layer operators, including that K is a bounded map of C(S) to C(S).

56 The properties Wendland assumed for the surface are still those assumed for most error analyses. In essence, the surface is assumed to be piecewise smooth, S = S 1 [ [ S J with each S j the smooth image of a planar polygonal region. In all cases, the solid angle is assumed to satisfy 0 < (P ) < 4 Also, each face S j is assumed to have a piecewise smooth boundary with no cusps. The region is triangulated as in the case of a smooth surface S, S = n[ k=1 with the triangulation T n = edges and corners of S. (n) k (n) k respecting the

57 We let X n be a space of piecewise polynomial functions of some degree r, as before. These may be restricted to be continuous, as in most of my BIEPACK. Let P n denote the interpolatory projection of C(S) onto X n. The collocation method for solving (2 + K) = f is equivalent to (2 + P n K) n = P n f The methods of Wendland (1968) are based on a pair of assumptions: (a) at each point P 2 S, the tangent cone is convex or its complement is convex; (b) sup j2 P 2S (P )j sup kp n k < 2 n1 With piecewise constant interpolation and piecewise linear interpolation (at the vertices of an element), this last assumption is always true. A convergence theory can be based on the geometric series theorem for (2 + P n K) 1. This work assumes well-behaved solution functions, so that uniform meshes work well.

58 GALERKIN METHODS When S is a polyhedral Lipschitz boundary, J. Elschner has given a general theory for Galerkin boundary element methods. This includes using a graded mesh to compensate for singular behaviour in the solution in the vicinity of edges and corners of S. J. Elschner (1992) The double layer potential operator over polyhedral domains I: Solvability in weighted Sobolev spaces, Applic. Analysis 45, pp [Includes regularity results.] J. Elschner (1992) The double layer potential operator over polyhedral domains II: Spline Galerkin methods. Math. Methods Appl. Sci. 15, pp The author shows that local to each corner of S, the operator K can be regarded as an operator valued Mellin convolution operator; and using related results, he is able to study the solvability of (2 + K) = f and analyze a Galerkin method for its solution.

59 Convergence of the numerical method: The author requires that the in nite cone generated by each corner of the surface S satisfy a nite section assumption. This amounts to something akin to the invertibility of the - nite section equation in the planar case. For cases in which either the interior or exterior cone is convex, this assumption is satis ed. With this assumption, the author constructs a mesh which is graded towards each edge of S, and it is doubly-graded towards the vertices of S. The subspace X n is de ned as the set of piecewise polynomial functions of some degree r 0; and there is no continuity restriction on the approximants in X n. With a properly graded mesh, Elschner is able to prove an optimal order of convergence: k n k L 2 (S) = O m (r+1) ; n = O m 2 The grading is of the same algebraic type as used for planar problems.

60 COLLOCATION METHODS When S is a polyhedral Lipschitz boundary, A. Rathsfeld has given a general theory for collocation boundary element methods for solving (2 + K) = f. This includes using a graded mesh to compensate for singular behaviour in the solution in the vicinity of edges and corners of S. A. Rathsfeld (1992) The invertibility of the double layer potential operator in the space of continuous functions de ned on a polyhedron: The panel method, Appl. Analysis 45, pp The author makes a nite section assumption, which is known to be true in either of two cases (assuming the surface is Lipschitz) (a) Either the interior or the exterior solid angle is convex at each point of the surface; or

61 (b) the surface S is made up of faces that are parallel to one of the three coordinate planes in R 3. Rathsfeld de nes a graded mesh, with n = O(m 2 ) elements, using lines parallel to the edges of S, at distances proportional to the mesh i q ; i = 0; 1; 2; :::; m; q 1 m For some r 0, we de ne X n as the functions of degree r in the parameterization variables on each face of S. As in the planar case, this is too general to allow a successful error analysis for collocation methods. In the vicinity of edges and corners of S, special modi cations of the approximating space X n are needed; and the collocation nodes are chosen with some care as well. With this, Rathsfeld proves convergence and stability of the collocation scheme, with k n k 1 = O m (r+1)

62 GENERAL FRAMEWORK With both the work of Elschner and Rathsfeld, the authors study the double layer integral operator K(P ) = Z " 1 ds Q Q jp Qj + [2 (P )] (P ); P 2 S by localizing it to each corner of the polyhedral surface S. Denote by the in nite cone determined by a generic corner P 0, and consider K W de ned over. Let intersect a unit sphere centered at P 0, and call this intersection. Parameterize by!(s), 0 s L. Write a typical point on as P = r!(s). For g 2 C 0 ( ), Wg(r!(s)) = Z 1 0 r B with B(t) an operator on C() to C(): # g (!()) (!(s)) d (B(t)h) (!(s)) = Z L 0 h(!()) tn!()!(s) jt!(s)!()j d; t 0

63 The equation (2 + W) = f is a Mellin convolution equation using the operator-valued Mellin convolution kernel B. The authors use a known literature for such operators to obtain convergence and stability results for the numerical solution of boundary element methods for solving (2 + W) = f. This is also used by them in obtaining a solvability theory for the original integral equation (2 + K) = f, including regularity results for the solution. An alternative approach. Using results on the solvability of the associated boundary value problems for u = 0, combined with the Green s representation formulas, one can obtain results on solvability of some BIE, including regularity results on their solutions. For such an approach, see T. von Petersdor and E. Stephan (1990) Regularity of mixed boundary value problems in R 3 and boundary element methods on graded meshes, Math. Methods in the Appl. Sci. 12, pp

64 SMOOTH BOUNDARIES IN R 3 : FIRST KIND BIE There is a well-developed theory for those BIE which can be regarded as strongly elliptic pseudo-di erential equations on suitable Sobolev spaces. The most studied of such equations is S (P ) Z S (Q) jp Qj ds Q = f(p ); P 2 S In this, we usually assume S is the boundary of a bounded simply-connected D R 3, and we assume S is as smooth as needed. The basic paper in the study of this equation is J. Nedelec (1976) Curved nite element methods for the solution of singular integral equations on surfaces in R 3, Computer Methods in Appl. Mechanics and Engin. 8, pp

65 Nedelec shows that S : H 2 1 1! 1 onto H 1 2 and (S'; ') c k'k ; ' 2 H 1 2 with (; ) denoting the extension of the usual inner product on L 2 (S) to H 2 1 H 1 2. Then he applies the standard theory of nite element methods to obtain a convergent nite element Galerkin method. Using a triangulation T n = (n) k of S, let X n denote all functions which are piecewise polynomial of degree r in the parameterization variables. The Galerkin method is de ned by Then k By standard arguments, (S n ; ') = (f; '); ' 2 X n nk 12 c inf '2X n k 'k 12 k nk 0 ch r+1 k k r+1 ; 2 H r+1

66 Approximating the surface. Nedelec also considers the approximation of the boundary by interpolation, of some degree k 1. Denoting the resulting solution by b n, the convergence results become b n 1 c h h r+1 k k 0 r+1 + h k i k k 0 n for 2 H r+1 (S). In this, n is a special map from the interpolating surface to the original surface S.

67 These results generalize to more general strongly elliptic pseudo-di erential operator equations. For example, see W. Wendland (1987) Strongly elliptic boundary integral equations, in The State of the Art in Numerical Analysis, ed. by A. Iserles and J. Powell, Clarendon Press, 1987, pp W. Wendland (1990) Boundary element methods for elliptic problems, in Mathematical Theory of Finite and Boundary Element Methods, by A. Schatz, V. Thomée, and W. Wendland, Birkhäuser, Boston, pp For Galerkin methods based on using spherical polynomials as the approximants, see Y. Chen and K. Atkinson (1994) Solving a single layer integral equation on surfaces in R 3, SIAM J. Num. Anal., Reports on Computational Mathematics #51, Dept of Mathematics, University of Iowa.

68 NUMERICAL INTEGRATION We consider the numerical evaluation of the integrals in the collocation solution of the double layer equation (2 + K) = f: K(P ) = The collocation system is: 2 n v (n) + nx i 6X k=1 j=1 for i = 1; :::; n v. Z " 1 ds Q Q jp Qj + [2 (P )] (P ); P 2 S Z n vk;j L k;j Q + 2 (v (n) i ) n v (n) i 2 # v(n) i = f Q v (n) i ds Q

69 In this, the integral L k;j Q k v(n) i 1 Q ds Q must be evaluated numerically in all but the simplest cases. The functions L k;j (Q) are Lagrange basis functions for quadratic interpolation over k. [In the case of piecewise constant interpolation over polyhedral surfaces, these integrals can be done analytically.] These are the most time consuming aspect of most boundary element codes. Case (a). v (n) i 2 k. Then the integrand is singular. This appears to be di cult, but there is a simple x, called the Du y transformation.

70 Consider the integral I = Z g(s; t) d with the unit simplex in the plane: = f(s; t) j 0 s; t; s + t 1g Further assume the function has a point singularity at the origin. Introduce the change of variables s = (1 y) x; t = yx; 0 x; y 1 I = Z 1 Z xg ((1 y) x; yx) dx dy When the original integral arises from a collocation integral of either single or double layer type, then the new integrand contains no singularity, including in its derivatives. Then simply apply Gaussian quadrature to both integrals, usually with only a few nodes (e.g. 5) in each direction. See C. Schwab and W. Wendland (1992) On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62, pp

71 Case (b). v (n) i Z =2 k. Now the L k;j Q k v(n) i 1 Q ds Q are nonsingular. But they vary from almost singular, when v (n) i is near k ; to having very well-behaved integrands, when v (n) i is distant from k. [Note that of the O(n 2 ) integrals in the collocation matrix, only O(n) are of singular type.] We are using the following scheme in order to more e ciently evaluate the nonsingular integrals. Introduce a parameter Maxlev to indicate a number of level of subdivisions of k for a composite quadrature scheme. If dist (v i ; k ) h, use levels of subdivision of k and apply a given quadrature scheme to each of the resulting 4 subelements.

72 If h < dist (v i ; k ) 2h, use 1 levels of subdivision of k and apply a given quadrature scheme to each of the resulting 4 1 subelements. Continue this until no subdivision of k takes place, so that the quadrature scheme is applied directly to k. As n increases to 4n to 16n, it seems best to increase to + 1. We give an example of solving (2 + K) = f for an exterior Neumann problem on an ellipsoid. Note the errors with varying, and then consider the costs. n = 0 = 1 = 2 = E E E E E E E E E E E E E E E E 5 Table 1: Max errors in n

73 n = 0 = 1 = 2 = 3 Iteration < 0: Table 2: Timings for graded mesh quadrature (sec) In the nal table, we use levels of subdivision over all elements. n = 0 = 1 = 2 = Table 3: Timings for uniform mesh quadrature (sec) For an improved scheme that is theoretically more e - cient, see C. Schwab (1994) Variable order composite quadrature of singular and nearly singular integrals, Computing 53, pp

74 ITERATION Denote the collocation system 2 n v (n) + nx i 6X k=1 j=1 i = 1; :::; n v, by Z n vk;j L k;j Q + 2 (v (n) i ) n (2 + K n ) v n = y n v (n) i Two-grid iteration methods are examined in v(n) i = f Q v (n) i ds Q K. Atkinson (1994) Two-grid iteration methods for linear integral equations of the second kind on piecewise smooth surfaces in R 3, SIAM J. Scienti c Computing 15, pp

75 Multigrid methods are examined in W. Hackbusch (1994) Integral Equations: Theory and Numerical Treatment, Birkhäuser Verlag, Basel. A general examination and comparison of these and other iteration methods is given in A. Rathsfeld (1995) Nyström s method and iterative solvers for the solution of the double layer potential equation over polyhedral boundaries, SIAM J. Num. Anal. 32, pp For problems over smooth surfaces S, there is no di culty in applying any of these methods; and their error analysis is relatively straightforward. For piecewise smooth surfaces, one must precondition the system. To do this, I solve exactly the portion of the linear system corresponding to the nodes on the surface that are at an edge, a corner, or are nearby to such. Examples are given in both the Atkinson and Rathfeld articles.

76 ITERATION FOR SECOND KIND BIE Second kind BIE such as (2 + K) = f lend themselves well to two-grid and multigrid iteration methods. For a collocation method (2 + P n K) n = P n f denote the associated linear system by (2 + K n ) v n = y n and let m < n be an index for which the linear system is solvable. 1. Given v n (0), compute 2. Compute K n r n. 3. r n = y n (2 + K n ) v (0) n c n = P m;n (2 + K n ) 1 R n;m K n r n 4. De ne v (1) n = v (0) n [r n c n ]

77 For smooth boundaries in R 3, with boundary elements re ned by subdividing elements into four new elements, we can show v n v (k) n thus proving convergence. 1 = O k(p n P m ) Kk k With piecewise smooth boundaries S, the operator K is usually not compact and the convergence fails, both in theory and practice. To treat it, we must divide the linear system based on a subdivision of the boundary into a portion containing small neighborhoods of each corner and the remaining portion of S. Let S 1 contain all node points close to a corner of S, and let S 2 denote the remainder of S. Partion each vector v based on S 1 and S 2. Rewrite the linear system as K(1;1) n K n (1;2) K n (2;1) 2 + K n (2;2) v(1) n v n (2) 3 5 = 2 4 y(1) n y n (2) Solve exactly linear systems associated with 2 +K (1;1) n. Then we can apply a convergent two grid method to the remaining part of the linear system. 3 5

78 H. Schippers (1985) Multi-grid methods for boundary integral equations, Numer. Math. 46, pp F. Hebeker (1988) On the numerical treatment of viscous ows against bodies with corners and edges by boundary element and multi-grid methods, Num. Math. 52, pp K. Atkinson and I. Graham (1990) Iterative variants of the Nyström method for second kind boundary integral equations, SIAM J. Sci. and Stat. Comp. 13, pp [Planar problems with corners] S. Vavasis (1992) Preconditioning for boundary integral equations, SIAM J. Mat. Anal. 13, pp A. Edelman (1993) Large dense numerical linear algebra in 1993: The parallel computing in uence, University of California - Berkeley. T. von Petersdor and E. Stephan (1990) On the convergence of the multigrid method for a hypersingular integral equation of the rst kind, Numer. Math. 57, pp

79 FAST MATRIX-VECTOR MULTIPLICATION METHODS Consider a discretization of the single layer operator: S v (n) i nx 6X k=1 j=1 v (n) k;j Z k L k;j (Q) v(n) i Q ds Q for i = 1; :::; n v, using a piecewise quadratic interpolation of. Denote the vector of values v (n) i by n, and let A n denote the matrix multiplication inherent in the above approximation. Then all iteration methods for solving a linear system involving A, say Av = y, will involve repeated multiplications Aw for various vectors w. To evaluate A explicitly requires O n 2 operations or more, and to evaluate Aw requires 2n 2 operations. Reducing the cost of computing Aw is of little consequence if we must still evaluate A explicitly.

80 In the past ten years, several approaches have been used to speed the evaluation of Aw, from the usual 2n 2 operations to something like O (n log n) operations. Fast multipole method. L. Greengard, V. Rokhlin Clustering methods. Hackbusch Wavelet compression methods. Petersdorf, Schwab; Dahmen, Prößdorf, Schneider, and others; Michelli, Xu All methods produce a vector u =Aw + at a smaller cost than O n 2, with of acceptably small size.

81 FAST MATRIX-VECTOR MULTIPLICATION METHODS Consider a discretization of the single layer operator: S v (n) i nx 6X k=1 j=1 v (n) k;j Z k L k;j (Q) v(n) i Q ds Q for i = 1; :::; n v, using a piecewise quadratic interpolation of. Denote the vector of values v (n) i by n, and let A n denote the matrix multiplication inherent in the above approximation. Then all iteration methods for solving a linear system involving A, say Av = y, will involve repeated multiplications Aw for various vectors w. To evaluate A explicitly requires O n 2 operations or more, and to evaluate Aw requires 2n 2 operations. Reducing the cost of computing Aw is of little consequence if we must still evaluate A explicitly.

82 In the past ten years, several approaches have been used to speed the evaluation of Aw, from the usual 2n 2 operations to something like O (n log n) operations. Fast multipole method. L. Greengard, V. Rokhlin Clustering methods. Hackbusch Wavelet compression methods. Petersdorf, Schwab; Dahmen, Prößdorf, Schneider, and others; Michelli, Xu All methods produce a vector u =Aw + at a smaller cost than O n 2, with of acceptably small size.

83 FAST MULTIPOLE METHODS The problem posed is to calculate the potential at points 1 ; :::; p as determined by charges of strengths q 1 ; :::; q k at k points Q 1 ; :::; Q k : ( i ) = kx j=1 q j ; i = 1; :::; p i Q j Often f i g = fq i g, with the proviso of omitting the term in the sum for j = i. The size of k can be quite large, say k = 10; 000. The methods used depend on the use of spherical harmonics expansions: Let = (r; ; ) and Q = (; ; ). Then for < r, 1 X 1 j Qj = P n (cos ) n=0 n with the angle between and Q, and with P n (u) the Legendre polynomial of degree n. r To give a avor to the method s techniques, we give a theorem from the development of the method.

84 Theorem. Suppose that k charges of strengths q 1 ; :::; q k are located at the points Q i = ( i ; i ; i ), i = 1; :::; k, and assume that i < a, a given number. Then for any = (r; ; ) with r > a, the potential () is given by () = 1X nx n=0 m= n M m n r n+1y m n (; ) For any ` 1, () M m n = `X nx n=0 m= n kx i=1 M m n m q i n i Y n ( i ; i ) r n+1y m n (; ) A r a a `+1 r A = kx i=1 jq i j The functions fy m n (; )g are spherical harmonics of degree n and order m. The formula for () is called the multipole expansion.

85 The Fast Multipole Method is based on creating graded subdivisions of space such as to make it possible to calculate an estimate of () at p points in much fewer than O(pk) operations. The best reference is the original document: L. Greengard (1988) The Rapid Evaluation of Potential Fields in Particle Systems, The MIT Press, Cambridge, Massachusetts, Chap. 3. For papers discussing its implementation, see K. Nabors, F. Korsmeyer, F. Leighton, and J. White (1994) Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional rst-kind integral equations of potential theory, SIAM J. Sci. Comp. 15, pp

86 CLUSTERING METHODS We assume the creation of a sequence of triangulations of S, each a re nement of the preceding one. Call these T 0, T 1, T 2,...,T I, with T 0 = f k j 1 k ng the nest mesh. For simplicity here, also assume S is polyhedral. Restricting ourselves to single layer potentials, we want to evaluate S n (v i ) nx 6X k=1 j=1 Z L k;j (Q) n vk;j jv i Qj ds Q k for i = 1; :::; n v. As the distance from v i is increased, the integrand becomes smoother.

87 For each element in some T l, with P =2, we expand the simple layer in a Taylor series about the centroid Q of : jp 1 Qj X l2i m l (P ; Q ) l (Q) The degree of the Taylor series is m, and I m denotes an index set for the expansion. The accuracy of this expansion will depend on the distance of P from Q and the size of. We call a cluster of elements from the nest level T 0. Note that l (Q) is independent of P and Q, which is critically important. The functions l (Q) are to be simple monomials in the components of Q, and thus they are simple to integrate over elements k of T 0.

88 Let n (Q) denote the piecewise quadratic, n (Q) = 6X j=1 n vk;j Lk;j (Q); Q 2 k ; 1 k n Let the cluster consist of a set of elements from T 0, say 1 ; :::; p. Then Z n (Q) ds Q jp Qj px j=1 l (P ; Q ) X l2i m With preprocessing, we will have the values of Z j n (Q) l (Q) ds Q Z j n (Q) l (Q) ds Q in O(n) operations as we allow both l and j to vary. Because both n (Q) and l (Q) are polynomials in Q, these integrals are straightforward to evaluate.

89 For each point P = v i, we do an initial preprocessing to produce the needed clustering of elements of T 0, to have S = [ 1 [ [ c ] [ h i1 [ [ iq i Then we have Z S n (Q) ds Q jp Qj = cx Z j=1 n (Q) ds Q + jp Qj j qx Z j=1 ij n (Q) ds Q jp Qj With a careful attention to detail and with preprocessing where possible, we obtain a way to evaluate an approximation of [S n (v i )] in O n log 3 n operations. The main reference is: W. Hackbusch and Z. Nowak (1989) On the fast matrix multiplication in the boundary element method by panel clustering, Num. Math. 54, pp