PIECEWISE POLYNOMIAL COLLOCATION FOR BOUNDARY INTEGRAL EQUATIONS

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1 PIECEWISE POLYNOMIAL COLLOCATION FOR BOUNDARY INTEGRAL EQUATIONS KENDALL E ATKINSON AND DAVID CHIEN y Abstract This paper considers the numerical solution of boundary integral equations of the second kind, for Laplace's equation u = 0 on connected regions D in R with boundary S The boundary S is allowed to be smooth or piecewise smooth and we let f K j K N g be a triangulation ofs The numerical method is collocation with approximations which are piecewise quadratic in the parametrization variables, leading to a numerical solution u N : Superconvergence results for u N are given for S a smooth surface and for a special type of renement strategy for the triangulation We show u ; u N is O( 4 log ) at the collocation node points, with the mesh size for f K g Error analyses are given are given for other quantities and an important error analysis is given for the approximation ofs by piecewise quadratic interpolation on each triangular element, with S either smooth or piecewise smooth The convergence result we prove isonlyo( ) but the numerical experiments suggest the result is O( 4 ) for the error at the collocation points, especially for S a smooth surface The numerical integration of the collocation integrals is discussed, and extended numerical examples are given for problems involving both smooth and piecewise smooth surfaces Key words Integral equations, quadrature interpolation, Laplace's equation, numerical integration AMS subject classications 65R0, 5J05, 45L0, 65D05, 65D0 Introduction In this work, we consider the numerical solution of boundary integral equations of the second kind for solving Laplace's equation u = 0on connected regions D in R The collocation method with piecewise polynomial approximations is the numerical method being analyzed Because of the practical need to use easily-computable approximations of the surface, we analyze the eect of using interpolation to approximate the surface of the region We also discuss the eect of numerical integration of the collocation integrals A major consideration in the error analysis of numerical methods for these boundary integral equations is whether the boundary of D, call it S, issmooth or piecewise smooth If S is smooth, then the associated integral operator is compact and there isawealth of results available for the error analysis But if S is only piecewise smooth, then the integral operator is not compact and moreover, the operator can be viewed as involving a Dirac delta function in its denition In this case, other methods of error analysis are required The most widely used techniques originated with Wendland[], in which he adapted and greatly extended a technique introduced in [0] for the theoretical analysis of such integral equations for the planar Dirichlet problem for Laplace's equation We use these ideas of Wendland in our analysis of the collocation method given below inx5 Other approaches for this case are under development for example, see Elschner[0] in which results of Chandler and Graham[] for the planar problem are generalized to Galerkin methods for polyhedral boundaries in R, and see Rathsfeld[7] Two problems for Laplace's equation and their associated boundary integral equations are studied in this paper P The interior Dirichlet problem Let D be a bounded, open, simply connected This researchwas supported in part by NSF grant DMS Department of Mathematics, University ofiowa, Iowa City, Iowa 54 y Department of Mathematics, California State University San Marcos, San Marcos, California 9096

2 K ATKINSON AND D CHIEN region in R, and let its boundary S be piecewise smooth, which is dened more precisely in Section The problem is to nd u C( D) \ C (D) such that u(a) =0 u(p )=f(p ) A D P S We assume u () can be represented as a double layer potential: u(a) = ds Q A Q j A ; Q j S The density function is determined from the integral equation (P )+ () ds Q +[ ; (P )](P )=f(p ) P Q j P ; Q j S For notation, Q denotes the unit normal to S at Q (if it exists), pointing into D The quantity (P ) is the inner solid angle of S at P S and we assume 0 < (P ) < 4: Symbolically, we write the integral equation () as Under suitable assumptions on S, ( + K) = f K : C(S)! C(S) is a bounded linear operator P The exterior Neumann problem Let D and S be as above, and let D e = R nd, the region exterior to D and S The problem is to nd u C( D e )\C (D e ) such that () u(a) =0 A D P = f(p ) P S u(p )=O(j P j ; ) j ru(p ) j= O(j P j ; ) as j P j! It can be shown that such a function u exists (under suitable assumptions on S and f) and that Green's third identity can be applied to u: 4u(A) = f(q) j A ; Q j ds Q ; (4) ds Q Q j A ; Q j To nd u (5) S on S, we solve the integral equation u(p )+ ds Q +[ ; (P )]u(p Q j P ; Q j = S S f(q) j P ; Q j ds Q PS S

3 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE Then (4) gives u on D e Symbolically,we write (5) as ( + K)u = Sf with K as before and S the single layer potential integral operator The integral equations () and (5) are dierent only in their right hand inhomogeneous term With (5), we can study the error in the numerical solution of the integral equation by using problems for which weknow the true solution of () With equation (), we do not know the true solution in general (except when f ) and thus the numerical solution must be checked indirectly by evaluating () numerically and comparing it to a known solution u This turns out to also be of interest, because integral formulas like () are generally known to converge faster than is the density function that solves the integral equation A further discussion is given later In Section, we describe briey the triangulation of the surface S The collocation method and the surface approximation are based on piecewise quadratic isoparametric interpolation, and this is described in Section, together with the numerical integration methods used in evaluating the collocation integrals The collocation method with S smooth is discussed in Section, and numerical examples are given in Section 4 The corresponding results for the collocation method when S is only piecewise smooth are given in Section 5 and Section 6 Some of the methods of this paper follow those of Atkinson[, ] but we also involve the new methods of analysis given in Chien[8], to improve on the error results of the earlier papers Although our analysis is for only quadratic approximation, the method being used will generalize to other degrees of piecewise polynomial approximation The diculty of our argument has led us to specialize to one case and in addition, it is one of the more important cases Preliminaries We describe the triangulation scheme and associated interpolation and quadrature The method being used was discussed in [, ], and we assume a familiarity with those papers, including the notation used in them As discussed in [], we assume the surface S can be written as (6) S = S [ S [[S J where each S i is a closed, \smooth" surface in R The only possible intersection of a pair S i and S j is to be along a common portion of the edges of these two sub-surfaces Assume that for each S j, there is a mapping (7) F j : R { j - onto S j j J where R j is a polygonal domain in the plane and F j C 6 (R j ) In this case, we say S is piecewise smooth Byasmooth surface, we mean that for each point P S, there is a neighborhood on S of P, with the neighborhood having a local six-times continuously dierentiable parametrization in R The surface S of (6) is divided into a triangular mesh (8) f K N j K N g for a sequence N = N N ::: Each S j is to be broken apart into a set of nonoverlapping triangular shaped elements K Nj 's, about which wesay more below In referring to the element K N, the reference to N will be omitted, but understood implicitly Dene the mesh size of (8) by N = max KN diam( K)

4 4 K ATKINSON AND D CHIEN (9) diam( K )= max p q K j p ; q j : Let denote the unit simplex in the st ; plane = f (s t) j 0 s t s + t g : Let ::: 6 denote the three vertices and three midpoints of the sides of, numbered according to Figure 4 r @ r r 6 Fig The unit simplex - One way of obtaining the triangulation (8) and the mappings from to each K is by means of the parametric representation (7) for the region S j of (6) Triangulations of R j map onto triangulations of S j Since the R j 's are polygonal domains and can be written as a union of triangles, without loss of generality, we assume in this paper that the R j 's are triangles A paraboloid with top is a good example of an S that satises our assumptions but a circular cone is an example of an S for which someofabove assumptions are not valid, because of the discontinuity of the gradient at the vertex Let K b be an element in the triangulation of R j, and let bv, bv, and bv be its vertices Dene (0) m K (s t) =F j (ubv + tbv + sbv ) u =; s ; t (s t) and let K be the image of K b under this mapping Also, if any two elements in this triangulation have a side in common, then their intersection will be an entire side of both triangles Most surfaces S of interest can be decomposed as in (6), with each S j representable as in (7) Also, the surface S could be smooth, and we would often still want to decompose it as in (6) The mapping (0) is used in dening interpolation and numerical integration on K Introduce the node points for K by v j K = m K ( j ) j = ::: 6

5 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 5 bv r % %% A AA % %% A AA bv % %% A r % AA r 4 bv A 5 AA % %% % %% A A AA AA % %% % %% A A AA r r r bv b v6 b v Fig Renement Collectively, the node points of the triangulation f K g will be denoted by with N v f v i j i N v g the number of distinct node points The sequence of triangulations (8) will usually be obtained by successive renements The renement process is based on connecting the midpoints of the sides of a given element b K Given fbv ::: bv 6 g, connect bv 4 bv 5 bv 6 by straight lines, as in Figure, producing four new triangular elements The new elements all are congruent, and they are similar to b K More importantly,any symmetric pair of triangles, as shown in Figure, have the following property: bv r r r bv r r bv v5 4 b Fig A Symmetric pair of triangles b v () bv ; bv = ;(bv ; bv 4 ) and bv ; bv = ;(bv ; bv 5 ) The assumptions on S and the node points that we made in this section are for the use of quadratic interpolation There are other degrees of interpolation that

6 6 K ATKINSON AND D CHIEN can be used, and the assumptions on the smoothness of S and the denition of the nodes will change appropriately But the general process of renement will still remain the same, and we still subdivide K 's in the same way aswe do for the quadratic interpolation To dene interpolation, introduce the six basis functions for quadratic interpolation on Letting u = ; (s + t), dene l (s t) =u(u ; ) l (s t) =t(t ; ) l (s t) =s(s ; ) l 4 (s t) =4tu l 5 (s t) =4st l 6 (s t) =4su: Dene a corresponding set of basis functions f l j K (q) g on K : l j K (m K (s t)) = l j (s t) j 6 K N: Given a function f C(S), dene () P N f(q) = 6X j= f(v j K )l j K (q) q K for K = ::: N This is called the piecewise quadratic isoparametric function interpolating f on the nodes of the mesh f K g for S It is straightforward that P N is a bounded projection operator and kp N k =5= Also, for any f C (S), () kf ;P N fk = O(b N ) where b N is the mesh size of the triangulation f b K N g of R j 's See [] Other kinds of interpolation can be used, such as piecewise cubic isoparametric interpolation In this case, we need ten node points, ::: 0, and ten basis functions for the interpolation on The error analysis is the same, although some what more complicated We also use the same quadratic interpolation scheme to construct an approximate surface e S for S The approximate surface e S is composed of elements e ::: e K, with e K an interpolant of K Write (4) The reference to K (5) em K (s t) = 6X j= m K (s t) = 4 x K (s t) x K (s t) x K (s t) 5 (s t) will be omitted, but understood implicitly Dene m K ( j )l j (s t) = 6 4 P 6 j= x K ( j)l j (s t) P 6 j= x K ( j)l j (s t) P 6 j= x K ( j)l j (s t) 7 5 (s t) Thus, em K (s t) interpolates m K (s t) at f ::: 6 g, and each component is quadratic in (s t) We introduce two major numerical integration schemes that we have used The rst numerical integration method is the -point rule (6) h(s t) d 6 6X j=4 h( j ) :

7 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 7 This method has degree of precision two, integrating exactly all quadratic polynomials Chien[8] shows that the associated composite rule over S is O(b 4 N ) where N b is the mesh size of fb K g The -point rule is mainly for computing integrals if the integrands are continuous In order to get the above results, the integrands are required to be four times continuously dierentiable If the integrands are continuous or smooth on the K, but there is a nearby singularity, we need to use a better numerical integration method The second method is the rule T:5{ from Stroud[, p 4]: 7X (7) h(s t) d j= w j h(r j ) : the weights w j and nodes r j given in the above reference This formula has degree of precision ve Collocation on Smooth Boundaries Our collocation method for solving an integral equation ( + K) = g can be written as (8) ( + P N K) N = P N g = The function g can be the function f of () or Sf of (5) We discuss results for this approximation and then later in the section, we give error results for the eect of using an interpolatory approximation of the smooth surface S An important auxiliary solution for the collocation method is the iterated collocation solution: It satises the equations ^ N = (g ;K N ) (9) (0) ( + KP N )^ N = g P N ^ N = N The questions of stability for (8) and (9) are linked by the identities () ( + KP N ) ; = [I ;K( + P NK) ; P N ] ( + P N K) ; = [I ;P N ( + KP N ) ; K] The solvability of (8) is determined from the standard theory for projection methods for example, see Atkinson[, pp 50-6] With the assumption of (a) compactness for K : C(S)! C(S), and (b) pointwise convergence on C(S) of the projections P N to I, wehave that k(i ;P N )Kk! 0 as n! From this, we have the standard result that if ( + K) ; exists on C(S), then (+P N K) ; exists and is uniformly bounded for all suciently large N,say N N 0 The existence of uniform boundedness of ( + KP N ) ; then follows from ()

8 8 K ATKINSON AND D CHIEN For the error in N and ^ N, use ; N = ( + P N K) ; ( ;P N ) ; ^ N = ;( + KP N ) ; K( ;P N ) The quantity K( ;P N ) often converges to zero more rapidly than does ;P N Using (0), this will show that N is superconvergent to at the collocation node points We make use of this in the following Theorem Consider the integral equation () and (5) with solution Let S be a smooth surface in R, and assume the unknown function C 4 (S) Then () where b N max in v j (v i ) ; N (v i ) j = O is the mesh size of the triangulation b 4 N logb N n bk N o of the R j 's Proof (a) The major part of the proof is concerned with measuring K(I ; P N )(P ) for all P = v i,anodepoint Later in the proof, we use this to prove () Note we use the exact surface S in this theorem Since the solid angle (P ) = for every P on a smooth surface, the integral equation () can be simplied as (P )+ ds Q = f(p ) P Q j P ; Q j S Using the triangulation scheme in Section, the compact operator K can be written as X K(P ) = (m K (s () j D s m K D t m K j d Q j P ; m K (s t) j For Q = m K (s t), (s t) = Q = D sm K D t m K j D s m K D t m K j with the sign chosen so that Q points into the bounded domain D Without loss of generality, we assume the sign of Q is always positive, and () becomes (4) K(P ) = X K (m K (s t)) (D sm K D t m K ) (P ; m K (s t)) j P ; m K (s t) j ds dt In order to measure the error K(I ;P N )(P ) for P a node point, we need to examine the local error which iscontributed by each K For each K, the integrand of the equation (4) has one singularityat P when P K, and it is smooth over K with P 6 K, although it is increasingly peaked as P and K become closer together We rst compute the error for those K 's which contain P For simplifying notation, we assume P =(0 0 0) and m K (0 0) = (0 0 0) The error in integrating over K equals (5) ((m K (s t)) ;P N (m K (s t))) (D sm K D t m K ) m K (s t) j m K (s t) j ds dt

9 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 9 This integral exists even though j m K (0 0) j= 0 To see this, use the Taylor error formula for the x i K about (s t) =(0 0) Then (5) equals (6) h(s t b 7 ) g(s t b ) d where h and g are polynomials in s and t, and their coecients are of size O(b 7 ) and O(b ), respectively Also, h and g are polynomials of degrees two and three, respectively, which shows the existence of integral (6), and it is O(b 4 ) [When speaking of an order of convergence, say one based on, b the order of convergence is uniform with respect to any absent variable or index] When P 6 K, P ; m K never equals zero for (s t) The kernel function, (s t), (s t) = (D sm K D t m K ) (P ; m K (s t)) j P ; m K (s t) j is smooth Compute the partial derivative s before expanding (s t) about (0 0) s (s t) = [D s(d s m K D t m K )] (P ; m K ) ; (D s m K D t m K ) D s m K j P ; m K j ; [(D sm K D t m K ) (P ; m K )] [D s m K (P ; m K )] j P ; m K j 5 = [D s(d s m K D t m K )] (P ; m K ) j P ; m K j ; [(D sm K D t m K ) (P ; m K )] [D s m K (P ; m K )] j P ; m K j 5 The term (D s m K D t m K )D s m K was dropped because (D s m K D t m K )? D s m K Also j (D s m K D t m K ) (P ; m K ) j = j D s m K D t m K jjp ; m K jjcos j is the angle between the vectors D s m K D t m K and P ; m K,and is a function of s and t (7) j cos jjp ; m K (s t) jconstant 8 (s t) See [6, p 49] Therefore, s is O(b =d K ) where d K =j P ; m K (0 0) j Use a similar calculation, t is also O(b =d K ) We now expand (s t) about (0 0) and have the following formula: (8) (9) where The error of (m K ) ;P N (m K ) is H K (s t) =! (s t) =(0 0) + O(b =d K): (m K ) ;P N (m K )=H K (s t)+o(b 4 ) ) x i (0 0) ; 6X (s + ) x i (0 0)l j (s t) 5 :

10 0 K ATKINSON AND D CHIEN Note that (0 0) and H K (s t) are O(b =d K ) and O( b ), respectively Combining (8) and (9), we have (s t)((m K ) ;P N (m K )) d = b (0 0) + O( ) b = O( 5 b )+O( 6 ) d K d K d K! H K (s t) d for every K which doesnotcontain P We now add all errors contributed by each K Let T 0 be the set of K 's which contain P, and let T be the set of the remaining K 's, which do not contain P Then, (0) K( ; P n )(P )= = X KT 0 X KT = O(b 4 )+ X S (P Q)((Q) ;P N (Q)) ds Q (s t)((m K ) ;P N (m K )) d + (s t)((m K ) ;P N (m K )) d KT (s t)((m K ) ;P N (m K )) d O(b 4 ) is contributed by K 's which are in T 0, and T 0 has at most six elements The error contributed by each K in T is O(b 5 =d K ) Examining the error carefully, we nd that cancellation happens on each symmetric pair of triangles Thus, for the dominant terms in the error (0 0)H i (s t)+(0 0)H j (s t) =0 if i and j are a symmetric pair of triangles This improves the error from O(b 5 =d K ) to O( b 6 =d K ) for each K that is part of a symmetric pair of triangles Let T be the set of these kinds of triangles Let T be the set of triangles that are not in T The error being contributed by triangles in T arises from the term j (0 0) H K (s t) j = (D sm K (0 0) D t m K (0 0)) (P ; m K (0 0)) j P ; m K (0 0) j j H K (s t) j = j (D sm K (0 0) D t m K (0 0)) jj(p ; m K (0 0)) jcos j P ; m K (0 0) j j (D sm K (0 0) D t m K (0 0)) j j P ; m K (0 0) j j H K (s t) j = O( b 5 d K ): j H K (s t) j See (7) Thus, the error analysis has been improved from O(b 5 =d K ) to O( b 5 =d K ) which iscontributed by each triangle in T

11 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE Let d(p ) d = min f d K j P 6 K K = ::: N g : : For simplicity, wetake d K = d d, depending on how far the K is from the point P [A somewhat more complicated argument can be based on a lower bound of a similar type for d K ] Let r = b =d, which is nite for our uniform mesh subdivision scheme and d = O(b N ) Note the indexing ::: N does not indicate distance from P But, there is an arrangement of f K g where the number of triangles at a distance R is proportional to R, with R = d d d The number c i of triangles in T at a distance i d is proportional to i for i = ::: t j Note that for some integer t j, t j d is the longest possible distance from P to triangles in R j Adding the error contributed by each triangle in T,we have X K b K O( 6 )= d K t j X i= b c i O( 6 (i d) ) = O( b 4 ) t j X i= r i i = O( b 4 logb ) : For the triangles in T, the error contributed by each ofthemis O(b 5 K =d K ) The number of triangles of this type at a distance i d is a nite number, and it usually is two or three but the proof is omitted Therefore, where c 0 j () X K O( b 5 K d K )= t j X i= c 0 j O( b 4 i d ) = O( b 4 ) t X j r i i= is either two or three This completes the proof that K(I ;P N )(P ) = O(b 4 logb ) = O(b 4 log b ) uniformly for P a node point in the triangulation f K N g of S [This form of proof is also used in some of the remaining proofs of this paper] (b) To show (), we rst note that the error equation for the iterated collocation solution ^ N is given by () The linear system associated with this is () with ( + KP N )( ; ^ N ) = ;K(I ;P N ) ( + K N )e N = ; N e N i = (v i ) ; ^ N (v i ) = (v i ) ; N (v i ) N i = K(I ;P N )(v i ) i = ::: N v : The matrix of coecients + K N is also the same as that for the linear system associated with the collocation equation (8) As noted earlier following (), ( + KP N ) ; is uniformly bounded for all suciently large N Also, since the iterated collocation equation can be considered as being a Nystrom method, it is a standard derivation that k( + K N ) ; k k( + KP N ) ; k

12 K ATKINSON AND D CHIEN where the matrix norm is the standard row norm Combining these results, (4) k( + K N ) ; k c < N N 0 for some suciently large N 0 and some c>0 Using this result with (), and using () to bound k N k,wehave the desired result () The single layer integral For the exterior problem, we need to evaluate the corresponding single layer integrals on the right hand side of (5) Write (5) S f(q) j P ; Q j ds Q NX K= f(em K (s t)) j P ; em K (s t) j j D s em K (s t) D t em K (s t) j d where P is one of node points Note we are including the use of the approximating surface We can see the integrand in (5) varies from singular to quite smooth To handle this varied behavior, we use two ways to study errors The rst case is for those K 's that contain the point P, and the second case is for the remaining K 's Lemma Let P be anode point in K for some K Then f(m K (s t)) j P ; m K (s t) j j D sm K (s t) D t m K (s t) j d; f N (em K (s t)) j P ; em K (s t) j j D s em K (s t) D t em K (s t) j d = O(b K ) where b K is the diameter of b K Proof There are two cases The rst case is that P is a vertex in some K, and the second case is that P is a midpoint of a side of K Begin with the rst case and, without loss of generality, assume that P = m K (0 0) = em K (0 0) = (p p p ): Before proving the theorem, we show that Compute = = j P ; m K (s t) j d j P ; m K (s t) j d = O( b ; K ): [(p ; x (s t)) +(p ; x (s t)) +(p ; x (s t)) ] (sxs (0 0) + tx t(0 0)) +(sx s(0 0) + tx t(0 0)) = d +(sx s(0 0) + tx t (0 0)) + O(b K )i ;= d

13 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE See (4) for x i 's After integrating the dominant part of the above equation by using polar coordinates in the st-plane about (0 0), we obtain j P ; m K (s t) j d = O( b ; K ): Now, we break the error analysis into three parts f(m K (s t)) j P ; m K (s t) j j D sm K (s t) D t m K (s t) j d; f N (em K (s t)) j P ; em K (s t) j j D s em K (s t) D t em K (s t) j d = E + E + E with (6) (7) (8) E = E = E = f(m K (s t)) ; f N (em K (s t)) j P ; m K (s t) j j D s m K (s t) D t m K (s t) j d f N (em K (s t)) j P ; m K (s t) j (j D sm K (s t) D t m K (s t) j ;jd s em K (s t) D t em K (s t) j ) d j P ; m K (s t) j ; j P ; em K (s t) j j D s em K (s t) D t em K (s t) j f N (em K (s t)) d In equation (6), j E jo(b K ) O(b K ) j P ; m K (s t) j d O( b 5 K ) O(b ; K ) = O( b 4 K ) For the equation (7), we can easily see it has order three: j E jmax j D s m K (s t) D t m K (s t) j;jd s em K (s t) D t em K (s t) j s t f N (em K (s t)) j P ; m K (s t) j d = O( b 4 K ) O(b ; K ) = O( b K ) For E, expand each x i about (0 0), and then integrate it over With a very lengthy calculation, we can show that E is of order three See [8] If P is a midpoint of a side of K,we split into two triangles, and, and we put the singular point atavertex in each of the new triangles see gure 4 for the case with P = m K ( 4 ) We apply an ane change of variables, to move again to an integral over Applying the rst case to these two subtriangles, we again can show the error is of order three Thus, the error contributed by theintegral over K, which contains P,isalways of order three, no matter whether P is a vertex or a midpoint of a side of K

14 4 K ATKINSON AND D CHIEN 6 P r r H H H@ r v v r r Fig 4 Splitting triangles - In the next lemma, we examine the errors from integrating over those triangles K which do not contain P Then, we can combine these two lemmas together and give the global error for the single layer integration Lemma Let P be anode point, and consider all K for which P 6 K Then (9) X K X K f(m K (s t)) j P ; m K (s t) j j D sm K (s t) D t m K (s t) j d; f N (em K (s t)) j P ; em K (s t) j j D s em K (s t) D t em K (s t) j d = O(b K ) where K b is the diameter of K Proof Since P 6 K,we can treat the function = j P ; m K (s t) j as a smooth function All results from Lemma and Theorem (){(7), [8], can be applied with slight changes Let (9) be decomposed as E + + E5 where E = X K E = X K ; X K E = X K f(m K (s t)) [j D s m K D t m K j;jd s em K D t em K j] j P ; m K (s t) j [f(m K (s t)) ; f N (m K (s t))] j D s em K D t em K j j P ; m K (s t) j [f(m K (s t)) ; f N (m K (s t))] j D s m K D t m K j j P ; m K (s t) j [f(m K (s t)) ; f N (m K (s t))] j D s m K D t m K j j P ; m K (s t) j d d d d

15 E 4 = X K ; X K E 5 = X K PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 5 fn (m K (s t)) j P ; m K (s t) j ; fn (m K (s t)) j P ; m K (s t) j ; fn (m K (s t)) j P ; m K (s t) j ; The integrand of E in Theorem, we get that E For E, = X K f N (m K (s t)) j D s em K D t em K j ds dt j P ; em K (s t) j f N (m K (s t)) j P ; em K (s t) j f N (m K (s t)) j P ; em K (s t) j is O(b 6 K =d K )+O( b 5 K =d K is of order three j D s m K D t m K j ds dt j D s m K D t m K j d ) Using the calculation we had f(m K (s t)) ; f N (m K (s t)) fjd s m K D t m K j;jd s em K D t em K jg d j P ; m K (s t) j O(b ) O(b 4 ) j P ; m K (s t) j d = O( b 7 ) for every K where P 6 K Adding errors from each triangle, we have that E is O(b 5 ), as we discussed in computing E For E,wehave the error from f(m K (s t)) ; f N (m K (s t)) j P ; m K (s t) j d K j D s m K D t m K j d is O(b 6 =d K ) for every triangle Again, following the argument in Theorem, E is O(b 4 ) Analyzing E 4,wehave (40) and j P ; m K (s t) j ; j P ; em K (s t) j j P ; m K (s t) j ; j P ; em K (s t) j b K 7 = O( ) d K j P ; m K (s t) j ; j P ; em K (s t) j for each K After adding up errors, E 4 = O(b 5 lnb ) b K = O( ) d : K f N (m K (s t)) j D s m K D t m K j d; f N (m K (s t)) j D s em K D t em K j)dsdt For E 5, each triangle give us an error of O(b 5 K =d K ) When adding errors together, cancellation happens at every symmetric pair of triangles and errors become O(b 6 K =d K ) Thus, as we discussed in computing E E 5 is O(b ) After going through E {E 5, the global error for the single layer integral, in which P ; m K (s t) is nonzero for every K, is O(b ) This result is uniform as P ranges over the node points of the triangulation

16 6 K ATKINSON AND D CHIEN Combining the above lemma, we get the following result, which gives the total error for evaluating the single layer integral at any node point We use this later to assess the eect on N of using an approximation to the single layer Theorem 4 Let S be a piecewise smooth surface, and let P be anode point on S Assume the unknown function f C 4 (S i ) \ C(S) i = ::: J Then S f(q) j P ; Q j ds Q ; NX K= f(em K (s t)) j P ; em K (s t) j j D s em K (s t) D t em K (s t) j d = O(b ) : Proof Combine Lemma and Lemma Using the approximate surface When using the approximate surface es N, the linear system for () for the Dirichlet problem becomes e N (v i )+[ ; (v i )] e N (v i )+ NX 6X K= j= e N (v j K ) (4) = f(v i ) i = ::::N v : l j K (s t) (D s em K (s t) D t em K (s t)) (v i ; em K (s t)) j v i ; em K (s t) j For a smooth surface S, wewould expect to use N (P )=(P)=, thus simplifying the above system However, for the piecewise smooth surfaces considered in Section 5, we need to consider an approximation to (P ) and from the numerical examples in Section 4, it is also useful to consider approximations of (P ) for S a smooth surface Using the (4) (P ) = ds Q P Q j P ; Q j we dene (4) N (P ) = NX 6X K= j= Later, in Theorem 5 of Section 5, we showthat (44) S l j K (s t) (D s em K D t em K ) (P ; em K (s t)) j P ; em K (s t) j max j (v i ) ; N (v i ) j = O(b N ) in v Empirically for a smooth surface S, in Section 4, it appears the approximation error is actually O(b N ), although we have not been able to prove this The linear system (4) is denoted here by d d (45) ( + e KN )e N = g N with e N i e N (v i ) g N i f N (v i ) i = ::: N v : When solving the integral equation (5) for the exterior Neumann problem, we also approximate the right-hand side, now a single layer integral, using (5) In the above

17 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 7 frame work, and consistent with earlier notation, we write (46) eg N i = NX 6X K= j= f(em K (s t)) j D s em K D t em K j j v i ; em K (s t) j for i =,:::, N v For convergence when using the approximate surface SN e,wehave the following theorem In Section 4, we give experimental results which suggest that the below convergence results can be improved Theorem 5 Consider the integral equations () and (5) with solution Let S be a smooth surface inr, and assume the unknown function C 4 (S) Then d (47) max j (v i ) ; e N (v i ) j = O(b N ) in v Proof We use a perturbation analysis, based on regarding the system (45) as a perturbation of the corresponding system (48) (I + K N ) N = g N for the projection method analyzed in Theorem which used the exact surface S From earlier in (4), ( + K N ) ; is uniformly bounded for all suciently large N The present analysis uses the result (49) kk N ; e KN k = O(b N ) with the matrix row norm The proof of this is essentially the same as that for (44), and thus we defer the proof of (60) to Theorem 5 Using (49), and the invertibility of + K N with the uniform boundness of ( + K N ) ;, for all suciently large N, wehave by standard arguments that the same is true for the inverse of + KN e : (50) k( + e KN ) ; k c < N N 0 for some N 0 and some c>0 By straightforward manipulation of (45) and (48), we have (5) N ; e N = ( + e KN ) ; h ekn ; K N i g N +( + e KN ) ; [g N ; eg N ] The rst term on the right sideis O(b N ), from (49) The second term is either zero or O(b N ), from Theorem 4 When consider with Theorem, this shows the result (47) 4 Numerical Examples: Smooth Surface Case The collocation method of x, with the use of the quadratic isoparametric interpolation of the surface S, was implemented with a package of programs which work for a wide variety of smooth and piecewise smooth surfaces This package was rst described in [, ] and it has since been updated and improved in several ways [Eventually, the package will be made available publicly, with an accompanying user's manual] There are two crucial aspects of the practical implementation that were not discussed in x: the calculation of the collocation integrals and the solution of the large linear systems that often arise from the discretization The iterative solution of such

18 8 K ATKINSON AND D CHIEN linear systems by two-grid methods is discussed in Atkinson[6] and thus we restrict our attention here to the numerical integration of the collocation integrals For the numerical integration, we have currently settled on the following schema, after much experimentation with other approaches We nd that the numerical integration of the collocation integrals is by far the most time-consuming part in solving the boundary integral equation One must have integrals that are suciently accurate, to match the accuracy of the \pure" collocation solution N But it is very wasteful of computing time to calculate these integrals with more accuracy than is needed The collocation integrals in the matrix of coecients of (4) are given by (5) (v i em k (s t))l j (s t) j D s em K D t em k j d In this, i = ::: N v, j = ::: 6, and K = ::: N and (P Q) denotes the kernel function for the double layer integral operator For the exterior Neumann problem, we also need to evaluate the corresponding single layer integrals (5) f(em K (s t)) j v i ; em K (s t) j j D s em K D t em K j d Recall from x that em K : ;! K e is a one-to-one and onto parametrization of the triangle approximating K We consider two cases in evaluating (5), depending on whether v i is inside or outside of K If v i K, then (v i Q) is singular We useachange of variable based on [9] This was introduced in [, p 40], where we noted that it removed all singular behavior in both the double layer integrals (5) and the corresponding single layer integrals Subsequently, wediscovered that the change of variables is equivalent to that introduced in [] Others who have since made use of this transformation include [] and [] The latter paper carries out a detailed analysis of the method and an extension of the transformation to other singular integration problems arising in solving boundary integral equations Assuming the collocation node v i = m K (0 0), introduce the change of variables s =(; y)x t = yx 0 x y With this, the new integrands in (5) and (5) will be well-behaved For K a surface with C m dierentiability, m, the transformed integrand for (5) will be C m; times dierentiable and if the density f(q) is m-times dierentiable on K, the transformed integrand for (5) will be C m; times dierentiable We then evaluate the transformed integral using a product Gaussian quadrature formula, with N g nodes in both the x and y coordinates (thus using Ng integration nodes) If v i = m K (0 ) or m K ( 0), then we use an ane transformations to convert back to the case just discussed If v i = m K (q j ) with j =4,5,or6,thenwe divide into two parts and treat the integral over each part as described above As an example, suppose v i = m K (0 :5) See Figure 4 for the appropriate subdivision of, for which we use an ane transformation to map each subtriangle onto in such away that the singular point occurs at (0,0) The above change of variables is used to remove the singularity in the integration over each triangle For cases of N = 5 faces, we have found N g =0 tobevery sucient to preserve the accuracy of the collocation solution and smaller values of

19 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 9 N g are sucient for smaller values of N Note that the number of integrals (5) with v i K, for some i and K, is of order N v, whereas the total number of integrals to be computed is of order Nv Thus when considering operation counts, the singular integrals are the less important oftheintegrals (5) to be considered For v i = K, the integrand in (5) is analytic but it is increasingly peaked as the distance between v i and K decreases A method to evaluate integrals such as (5) and (5) over is based on (7), the quadrature rule T:5- of [] Let an integer parameter N d 0 be given If v i = K and dist(v i K ) N where N is the mesh size of f K g as dened in (9), then integrate (5) using (7) with N d levels of subdivision of [thus dividing into 4 Nd subtriangles, with (7) applied to the integral over each of the corresponding subintegrals] If v i = K and N < dist(v i K ) N then integrate (5) using (7) with maxfn d ; 0g levels of subdivision of If v i = K and N < dist(v i K ) N then integrate (5) using (7) with maxfn d ; 0g levels of subdivision of Continue with this in the obvious way We have found that as N is increased to 4N, then raising N d to N d + is sucient tointegrate (5) and (5) with the needed accuracy For all of our examples, for both smooth and piecewise smooth surfaces, the largest value of N d that we have needed to use has been N d =Wehave used larger values of N d in our experiments, to check the accuracy when using the lower values of N d When v i = K, other methods have been tried for evaluating (5) and (5) for example, a method with automatic error control was described in [] and [4] But the method described here has proven to be the most ecient Nonetheless, the integrations of (5) and (5) are still the most expensive parts of our computation, far exceeding the cost of solving the linear system () for the discretized boundary integral equation 4 The Surfaces Two smooth surfaces were used in our experiments Surface # (denoted by S#) was the ellipsoid x y z + + = a b In Tables {4 given below for this ellipsoid, (a b c)=( :5 ) The ellipsoid is convex and symmetric For that reason, we also devised and used a surface which is not symmetric and which is slightly non-convex Surface # (S#) is dened by with (x y z) =( )(A B C) + + = ( ) =; [( ; :) +( ; :) ; ( ; :) ]= c

20 0 K ATKINSON AND D CHIEN solid curve: =0, dash curve: = =4, dot curve: = = Fig 5 Cross sections of \squash" surface and A, B, C>0, 5 The case we use here is = 0 and (A B C) =( ) Figure 5 gives the cross-sections of S# when intersecting S with vertical planes containing the z-axis, intersecting at angles of = 0, =4, = with respect to the positive x-axis Experiments were done with other choices of and (A B C), corresponding to surfaces with a more pronounced lack of symmetry and convexity But in order to obtain error results with some regularity in asymptotic behavior, we chose the parameters given above, giving the surface illustrated in Figure 5 4 The Solid Angle At allpoints P S, the solid angle (P )= In Table,wegive the approximate values of the solid angle for S# as computed using N (P ) in (4) The points P at which these are given are v =(0 0 ) v =( 0 0) v =(0 :5 0) v 7 =( p p 4:5 0) v 8 =( p p :5 0) v 9 =(0 p :5 p 4:5) The subscripts refer to the indexing of node points used in our triangulation package The empirical rate of convergence is approximately O(b N ) The integration parameters used were N g = 0 and N d = The columns E, E, E, and E4 denote the errors for N = 8,, 8, 5 respectively [Note that for a given N, the number of nodes on S is N v =(N +):]

21 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE Table Solid angle approximations on S# at selected v i i E E E E=E E=E E=E4 5E; 0E; 54E; E; 09E; 8E; E; 55E; 96E; E; 5E; 0E; E; 00E; 56E; E; 9E; 409E; Similar results for the approximate solid angle are true for S# 4 Solution of the exterior Neumann problem The problem () was solved with the normal derivative f so chosen that the true solution is known The two cases used here are u (P )= r u (P )= r ex=r cos(z=r ) with P =(x y z) and r =j P j In this case, = u andweuse u and u N in our discussion Tables and contain the maximum error at the node points for solving boundary integral equation (5) for S# and S#, respectively The integration parameter N g = 0 and for N d,we used 0,,, for the cases N = 8,, 8, and 5 respectively, for both S# and S# The results in Table for S# are consistent with an asymptotic rate for the error of O(b 4 N ) or O(b 4 N logb N ), in agreement with the theoretical result in Theorem for the collocation method with the exact surface In the case of S# in Table, the asymptotic pattern for the maximum error appears to be O(b N ) and to check in more detail whether the error is truly O(b N ), Table 5 gives the errors at a representative sampling of the 8 nodes used in the coarsest triangulation of S (for N = 8), along with the ratios by which these errors decrease The columns E, E, E, and E4 denote the errors for the parameter N = 8,, 8, and 5, respectively When looking at the individual errors, there is a pattern of an O(b 4 N ) rate of convergence at a large numberofthepoints and we conjecture that with larger values of N, an asymptotic error of O(b 4 N ) would emerge for the maximum error Table Maximum errors on ellipsoid ku ; u N k Ratio ku ; u N k Ratio 8 9E; 9E; 44E; 4 85E; E; E; E;6 59 6E;5 56 Table Maximum errors on surface S# ku ; u N k Ratio ku ; u N k Ratio 8 76E; 549E; 540E; 4 485E; 8 870E;4 6 5E; 6 5 E; E;4 68 Since these are smooth surfaces, why not use the true value of (v i )=, rather than incorporating the approximation (4) into the discretization of (5)? Table 5 gives the values of the maximum error at the node points fv i g with u = u on S#, with (v i )= at all node points Note that now the error is O(b N ), which is worse than the convergence rate of O(b 4 N log b N ) predicted by Theorem for the solution u N

22 K ATKINSON AND D CHIEN The use of the approximation (4) is forcing a favorable cancellation to occur in forming the discretized linear system (4) Another way of looking at what is happening is the following The matrix of coecients (4) is forced to have 4 as an eigenvalue, with the eigenvector being the vector with all components equalto This makes the discretized system exactly like the original integral equation (5), in which the function u(p ) is an eigenfunction of the left side of (5), with the eigenvalue being 4 Table 4 Errors at representative v i on S#, for u = u N E E E4 E=E E=E E=E4 ;56E; ;477E;4 ;4E; ;45E; ;8E;4 ;78E; ;54E; ;8E;4 ;4E; ;09E; ;805E;5 0E; ;6 8 ;458E; ;80E;4 ;6E; ;540E; ;46E;4 ;5E; ;6E; ;5E;4 84E; ;5 5 ;5E; ;79E;5 98E;6 8 9 ;86 8 ;5E; ;5E;4 ;04E; It is clearly preferable to use the approximate solid angle rather than the exact one The cost of using the approximation (4) is minimal, since all quantities used have been calculated in setting up the linear system (4) Table 5 Errors for u = u on the ellipsoid S# with use of the exact solid angle = N ku ; u N k Ratio 8 975E; 5E; E; E; The interior Dirichlet problem We solvetheintegral equation (), for the interior Dirichlet problem, with the same procedures as described above for the exterior Neumann problem To complete the solution process, we must then calculate numerically the integral () Letting ~ N denote the approximate density function thus obtained, we must evaluate u N (A) = ~ N (54) ds Q A Q j A ; Q j S From [8], the rate of convergence will be O(b 4 N ) when the quadrature is based on standard symmetric numerical integration rules over the unit simplex with a suf- ciently high degree of precision, eg the rules (6) and (7) Expand the integral in (54) as nx (55) k= ~ N Q j A ; Q j The triangulation f K g being used here need not be the same as the one used in obtaining ~ N butthetwo triangulations should be compatible in sense that one is a renement of the other For those triangles K which are close to the eld point A, theintegration should be done with more accuracy than for those triangles which are relatively far from the eld point ds Q

23 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE It has been our experience that the density function ~ N can be relatively inaccurate, and quite acceptable accuracy in the solution u N (A) can still be obtained The accuracy in the solution u N is dependent much more on the accuracy of the numerical integration of (55) than on having high accuracy in ~ N This should not be especially surprising, as it is well known that integration is a \smoothing" operation, and the eect of errors in the integrand, including ~ N, are reduced Extended examples to illustrate this are given in the technical report [4], and we omit them here for reasons of space 5 Collocation on Piecewise Smooth Boundaries As in x, we rstana- lyze the collocation method ( + P N K) N = P N Sf for (5) by assuming the exact representation of the surface is used in all integrations and following that, we analyze the eect of using a quadratic interpolatory representation of the surface For polyhedral boundaries, there is no need to approximate the boundary, and these are the cases analyzed in [0] and [7] As in [], we use a stability analysis based on Wendland[] and then as in x, we analyze the discretization error for the iterated collocation solution: ^ N = (g ;K N) In [], a piecewise constant collocation method is dened and analyzed The proofs given there generalize easily to our collocation method based on quadratic isoparametric interpolation In Wendland's paper, he makes several assumptions about the piecewise smooth surface S, in addition to those described in x Assumption V of his paper states that at all points of S, either the inner or the outer tangent cone must be convex and assumption V4 states that all edges of S must be piecewise continuous and must not contain any cusps Within this setting, it is straightforward to prove the following Theorem 5 Let S satisfy the assumptions given above and earlier in x and let S also satisfy the assumptions V and V4 of [], as discussed above preceding the theorem Moreover, assume (56) 5 sup j ; (P ) j< PS Let P N denote the interpolatory projection of (), based on quadratic isoparametric interpolation over the triangulation f K j K = ::: Ng Then for all suciently large N, say N N 0, and for some c<, (57) k( + P N K) ; kc N N 0 Moreover, this implies that (58) k( + KP N ) ; kc N N 0 For the error, (59) k ; N k O(b N ) Proof We refer to the derivation in [] Essentially, the problem of analyzing ( + P N K) N = P N g is divided into two parts Begin by decomposing the surface

24 4 K ATKINSON AND D CHIEN S into two subdomains based on distance to an edge or vertex of S Let T denote the union of all edges and vertices of the surface S For a given >0, let S = fp S j dist(p T) g and let S be the closure of S ; S Consider spaces C(S i ), i =, and dene integral operators K ij : C(S i )! C(S j ) by (K ij )(P )= ds Q +[ ; (P )](P ) PS i C(S j ) S Q j P ; Q j The nal term [ ; (P )](P ) needs to be included only when i = j = For (i j) 6= ( ), the operators K ij are compact Dene XC(S i ) C(S j ) Then the original boundary integral equation (5) for the exterior Neumann problem, ( + K) = Sf, and the collocation equation for its solution, ( + P N K) N = P N Sf, can be reformulated, respectively, as (60) + K K = K + K + PN K P N K P N K + P N K N N = g g (PN g) (P N g) We assume that the interpolation operator P N is so dened that P N j S i depends on at only the node points within S i Then we can dene P i : C(S i )! C(S i ) by (6) P in = P N j S i C(S i ) i = Using the methods of [], it is straightforward to show that if is chosen suciently small, then +K : C(S ) { - onto C(S ) and moreover, for all suciently large N, (6) k( + P N K ) ; k c< Using this, operate on (60) and (6) to obtain I ( + K ) ; K K I + K I ( + P N K ) ; P N K P NK I + P NK ( + PN K = ) ; (P N g) (P N g) We write these equations in the simpler forms = ( + K ) ; g N N g (6) (I + H)~ = r (I + H N )~ N = r N respectively, with ~ =[ ] T,~ N =[ N N ] T The operator H : X! X is compact and the family fh N g is a pointwise convergent and collectively compact family, converging pointwise to H With the

25 PIECEWISE POLYNOMIAL COLLOCATION FOR BIE 5 known invertibilityof +K on C(S), we can obtain the invertibilityof I +H Using the theory of collectively compact operator approximations, we have the existence and uniform boundedness of (I+H N ) ; for all suciently large N and this leads directly to the result (57) asserted in the theorem The result (58) follows from the identity () given earlier For convergence of the collocation solutions f N g, the standard result (64) k ; N k k( + P N K) ; kk ;P N k implies k ; N k = O(b N ) from the bound () for interpolation error The condition (56) and the other assumptions of [] on the solid angle are quite restrictive and it is clear from the numerical examples that they are not necessary in practice Other somewhat less restrictive assumptions on S are given in [4], [5], [7]{[9] but for our proof of stability, we still require (56) Our results on rates of convergence assume only the stability results (57) and (58), not on how they are obtained Other tools for proving stability are given in [0] and [], and it may be possible to adapt them to our use of piecewise polynomial isoparametric interpolation Again, they consider only polyhedral surfaces, and thus do not need to approximate the surface We cannotshow superconvergence of ^ N at the node points (which was shown in Theorem for S a smooth surface) For S only piecewise smooth, K is no longer a smoothing operator, and that appears to prevent superconvergence 5 Using the approximate surface In practice, we solve the linear system (4), which uses the approximate surface SN e We also approximate the solid angle (P ) by the quantity N (P ) dened in (4) Theorem 5 Let S be a piecewise smooth surface, and let P be anode point on S Then (P ) ; N (P )=O(b N ) : Proof We rst compute the error contributed by K which contains P Without loss of generality, assume P = m K (0 0) Let We break error over K (65) and (66) E = E = P = (p p p ) = m K (0 0) = em K (0 0) : (D s m K D t m K ) (D s em K D t em K ) into two parts: P ; m K (s t) j P ; m K (s t) j ; (D s em K D t em K ) P ; em K (s t) j P ; m K (s t) j ; (D s em K D t em K ) P ; em K (s t) j P ; m K (s t) j ds dt P ; em K (s t) j P ; em K (s t) j d :

26 6 K ATKINSON AND D CHIEN We now manipulate the rst part of the integrand of (65) (D s m K (s t) D t m K (s t)) P ; m K (s t) j P ; m K (s t) j (67) = (x s x t ; x s x t x s x t ; x s x t x s x t ; x s x t ) (p ; x p ; x p ; x ) [(p ; x ) +(p ; x ) +(p ; x ) ] = Using the Taylor error formula for the x i about (s t) =(0 0), the numerator of equation (67) becomes (x s x t ; x s x t x s x t ; x s x t x s x t ; x s x t ) (p ; x p ; x p ; x ) =(x t x s ; x t x s)(s x ss +stx st + t x tt)+(x t x s ; x t x s)(s x ss +stx st + t x tt) +(x t x s ; x t x s )(s x ss +stx st + t x tt )+O( b 5 K ) : Computing the corresponding part of the second term of (65) with the same formula as we had above, (68) (D s em K D t em K ) (P ; em K (s t)) Thus, =(x t x s ; x t x s)(s x ss +stx st + t x tt)+(x t x s ; x t x s)(s x ss +stx st + t x tt) +(x t x s ; x t x s)(s x ss +stx st + t x tt)+o(b 5 K ) (D s m K D t m K ) (P ; m K (s t)) ; (D s em K D t em K ) (P ; em K (s t)) = O(b 5 K ): Expanding each x i about (0 0), the denominator of (67) is Then, = O(b (69) K ) [(p ; x ) +(p ; x ) +(p ; x ) ] ;= = O(j b K j ; ) (D s m K D t m K ) (P ; m K (s t)) ; (D s em K D t em K ) (P ; em K (s t)) j P ; m K (s t) j Note there are at most six triangles containing the node point P, and the total error contributed from the K 's which contain P is O(b K ) To analyze E,we need to know the error from the following: j P ; m K (s t) j ; j P ; em K (s t) j j P ; m K (s t) j ; + j P ; em K (s t) j j P ; m K (s t) j (j P ; m K (s t) j)(j P ; em K (s t) j) + j P ; em K (s t) j O(b K ) O(b ; K ) = O( b ; K ) d