A Singular Function Boundary Integral Method for Elliptic Problems with Singularities hristos Xenophontos, Miltiades Elliotis and Georgios Georgiou Department of Mathematics and Statistics University of yprus P.O. Box 0537 678 Nicosia, yprus January 6, 005 Abstract A singular function boundary integral method for elliptic problems with boundary singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. The main result of this paper is the proof of convergence of the method in particular, we show that the method approximates the generalized stress intensity factors, i.e. asymptotic expansion, at an exponential rate. convergence of the method are also presented. the coecients in the Numerical results illustrating the Key words: Boundary singularities boundary approximation methods Lagrange multipliers stress intensity factors. AMS Subject lassications: 65N5, 65N38, 65N, 65N30. Introduction In many engineering problems (e.g., fracture mechanics applications), governed by elliptic Partial Dierential Equations (PDEs), boundary singularities arise when there is a sudden orresponding author. Email: xenophontos@ucy.ac.cy.
change in the boundary conditions and/or on the boundary itself. Singularities are known to aect adversely the accuracy and the convergence of standard numerical methods, such as nite element, boundary element, nite dierence and spectral methods. Grid renement and high-order discretizations are common strategies aimed at improving the convergence rate and accuracy of the above mentioned standard methods. If, however, the form of the singularity istaken into account and is properly incorporated into the numerical scheme, then a more eective method may be constructed (see, e.g., [7, 6]). (For a recent survey of treatment of singularities in elliptic boundary value problems see [8] and the references therein.) In the case of the two-dimensional Laplace equation, for example, the form of the singularity is visible through the asymptotic expansion for the solution u near the singular point. In polar co-ordinates (r ) centered at the singular point, u is given by [3] u(r )= X j= j r j j () () where j R are the unknown singular coecients, and j R j () are the eigenvalues and eigenfunctions of the problem, respectively, which are uniquely determined by the geometry and the boundary conditions along the boundaries sharing the singular point. The constants j are often called Generalized Stress Intensity Factors (GSIFs) or Flux IntensityFactors. In the case of elasticity problems with singularities, these constants are called Stress IntensityFactors (SIFs), and serve as a measure of the stress at which failure occurs. In most commonly used methods, such as the Finite Element Method (FEM), the SIFs are calculated as a post-solution operation (see, e.g., [4, 3, 4]). If, however, the goal of the computation is the calculation of the SIFs, then methods which calculate these quantities directly may be preferable. The Singular Function Boundary Integral Method (SFBIM), which we will analyze in this article, falls in the latter category. The SFBIM was originally developed by Georgiou et al. [7], and was subsequently rened and expanded by Elliotis et al. [8]{[0]. The method uses the leading terms in the local asymptotic expansion for the solution near the singular point as an approximation, while any Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting problem is posed on the boundary of the domain, hence the dimension of the problem is reduced by one, leading to considerable computational cost reduction. We should also mention here the works of Li et al. [5]{[7], and Arad et al. [], who also developed similar methods. (See also [0] for a review of SIF evaluation and modeling of singularities in boundary integral methods.) The SFBIM has been successfully applied to a number of problems in solid and uid mechanics and excellent numerical results have been obtain thus far [8]{[]. In particular, it was observed that the method: (i) approximates the SIFs at an exponential rate of convergence, (ii) is very ecient, and (iii) compares extremely well with other accurate methods found in the literature. Our main goal in this article is to prove the observed convergence rates of the method.
The rest of the paper is organized as follows: In Section we present the formulation of the method for a two-dimensional Laplacian problem with a boundary singularity. In Section 3 we present the convergence analysis, and in Section 4 we comment onhow the method can be eciently implemented. Finally, Section 5 includes the results of some numerical computations illustrating the convergence of the method. Throughout the paper the usual notation H k () will be used for spaces containing functions on the domain R with boundary @, having k generalized derivatives in L () the norm and seminorm on H k () will be denoted by kk k and jj k, respectively. Also, the letters and c will be used to denote generic positive constants independent ofany discretization parameters and possibly having dierent values in each occurrence. The Model Problem and its Formulation For simplicity we consider the Laplacian problem stated below and depicted graphically in Figure. Find u such that S where @ = 4 S i. i= u = 0in () @u = 0onS (3) u = 0onS (4) u = f(r )ons 3 (5) @u = g(r )ons 4 (6) Figure : A two-dimensional Laplace equation problem with one boundary singularity. 3
In (){(6), denotes the Laplacian operator, the variables (r ) denote polar coordinates centered at O and the functions f and g are known. It is assumed that f g and the boundary @ are such that there is only one boundary singularity ato. The local solution near O, isgiven by an asymptotic expansion of the form () [3]. Multiplying () by a test function v V (to be specied shortly) and integrating over, we get vu =0 and then, by means of Green's Theorem, we obtain ; rv ru + @ v @u =0: Since @u =0onS and @u = g(r )ons 4,we further have Suppose v is chosen to satisfy rv ru ; S v = 0 in v=0ons v @u ; v @u S4 = vg: (7) one such choice being v r j j () (see eq. ()). Then (7) becomes Now, since u = f on S 3 wehave so adding this to (9), we get or, equivalently, Letting rv ru ; rv ru ; rv ru ; ; @v =0onS (8) v @u S4 = vg: (9) @v (u ; f) =0 v @u ; v @u ; @v S4 (u ; f) = vg @v S4 u = vg ; @v f: = @u @v = (0) 4
the above equation becomes rv ru ; v ; u = S4 vg ; Hence, the variational problem to be solved reads: Find (u ) V V such that B(u v)+b(u v ) =F (v ) 8 (v ) V V () f: where B(u v)= b(u v ) =; F (v ) = S4 rv ru () v ; vg ; u (3) f: (4) The spaces V and V are chosen as follows. Let the trace space of functions in H () be denoted by H = (@) = u H () : uj @ L (@) : (5) With T : H ()! H = (@) denoting the trace operator, the norm of H = (@) is dened as n o k k = @ = inf kuk uh : Tu= : (6) () Then, we dene H ;= (@) as the closure of H 0 (@) L (@) with respect to the norm R @ k'k ;= @ = ' : (7) k k = @ sup H = (@) (See, e.g., [] for more details.) We then dene H () = u H () : uj S =0 (8) and we take V = H () V = H ;= (S 3 ) : (9) The discrete problem corresponding to (), then, reads: Find (u N h ) V N V h H H ;= (S 3 ) such that B(u N v)+b(u N v h )=F (v ) 8 (v ) V N V h (0) with B(u v) b(u v ) and F (v ) given by (){(4), and with V N V h nite dimensional spaces to be chosen shortly. 5
Remark The above formulation will be used in the analysis of the method. For the implementation, the following equivalent formulation will be used, in order to maximize computational eciency: From (9), using Green's formula once more, we get ; From (4) and (8) we obtain uv + @ u @v ; v @u S4 = vg: u @v S4 + u @v ; v @u S4 = vg () which is the equation that we discretize (see Section 4 ahead). Note that all integrals are one-dimensional, hence the computational cost is considerably reduced. Moreover, all integrations are carried out away from the point of singularity, where all integrands are smooth. 3 Error Analysis We begin by dening the nite dimensional spaces V N and V h which will be used in the approximate problem (0). First, with v i r i i () () denoting the singular functions, we dene the nite dimensional space V N = span fv i g N i= : (3) S n Next, let S 3 be divided into quasiuniform sections ; i i= ::: nsuch thats 3 = ; i= i. Let h i = j; i j and set h =max in h i. We assume that for each segment ; i, there exists an invertible mapping F i : I! ; i which maps the interval I =[; ] to ; i and we dene the space V h = h : h j ;i F ; i P p (I) i= ::: n (4) where P p (I) is the set of polynomials of degree p on I =[; ]: In practice, the representation of the boundary S 3 determines the mappings F i i.e.,ifs 3 is represented by a polynomial then an isoparametric mapping may be used, and if S 3 has some (general) parametric representation, then the blending map technique may be used (see h. 6 in []). We have the following theorem. 6
Theorem Let (u ) and (u N h ) be the solutions to () and (0), respectively. Suppose there exist positive constants c 0 c and, independent of N and h, suchthatthe following hold: Then B(v v) c 0 kvk and jb(u v)jc kuk kvk 8v V N (5) 9 0 6= v N V N s.t. ku ; u N k + k ; h k ;= with R + independent of N and h. h v N k h k ;= kv N k v kk ;= kvk inf vv N 8 h V h (6) 8v V N : (7) ku ; vk + inf V h k ; k ;= (8) This is similar to the classical result for saddle point problems (cf. [5]), and its proof for our problem appears in the appendix see also Theorem 6. in [5]. Before verifying that (5){(7) hold for the problem under consideration, we will make certain assumptions that will aid in the analysis. First, we note that for any function w which can be expressed in the form given by equation (), we can always write where w N = NX j= w = w N + r N j v j V N r N = We will assume that there exists a (0 ) such that and @r N @r jr N ja N NaN : X j=n + j v j : (9) (A) (A) If r< in (), assumptions (A), (A) hold trivially, since then, by () and the fact that j is harmonic, jr N j X j=n + j j j r j r N+ ; r an 7
with r<a<, and R + independent ofa and N: (The boundedness of the GSIFs was also used above.) Similarly, @r N @r X j=n + d dr j j j r j; = X j=n + j j j r j X j=n + d dr 8 < j j j : d dr r N+ ; r r 0 9 = j; d = d dr Na N : X j=n + j j j 8 < : r 0 j; d If r, one may partition the domain into subdomains in which separate approximations may be obtained, including one near O which isvalid for r <. The solution over the entire domain can then be composed by combining the solutions from each subdomain and properly dealing with their interactions across the interfaces separating them (see, e.g., [9]). Let us now verify that (5){(7) hold for the problem given by equation (0). First, note that B(v v)=jvj so that, by Poincare's inequality, By the auchy-schwartz inequality, B(v v) c 0 kvk 8v H () : (30) B(u v) c kuk kvk 8u v H () (3) so that (30) and (3) give (5). To verify (6), consider the following auxiliary problem (for which a unique solution exists): Find w such that where h Poincare's inequality) 9 = w =0in (3) @w = h on S 3 (33) w =0onS (34) @w =0onS [ S 4 (35) V h in (33). From (3) and (33) we obtain (using Green's formula and h w = with c 0 R + : Also, cf. [], w @w = ww + jrwj = jwj c 0 kwk (36) k h k ;= = @w kwk (37) ;= 8
so that by (36) and (37) with R + independent ofw and h. as given by equation (9). We have and also so that combining (38){(40) we get h w c 0 kwk kwk k hk ;= (38) h w N = Now, let w N V N be such thatw = w N + r N, h w ; h r N (39) h r N k h k ;= kr N k = k h k ;= kr N k (40) h w N kwk k h k ;= ; k h k ;= kr N k : (4) Now, using the reverse triangle inequality, we have kwk = kw N + r N k kw N k ;kr N k which along with (4), gives h w N kw N k ;kr N k k h k ;= ; k h k ;= kr N k kw N k k h k ;= ; ( + ) k h k ;= kr N k : (4) Since by assumption (A), r N converges to 0 exponentially, wehave and for N suciently large we may write 0 < kr Nk kw N k < kr N k kw N k ( + ) (43) where and are the constants from above. ombining (4) and (43) leads to h w N k hk ;= kw N k which in turn gives (6) once we replace w N by v N and = by. ondition (7) follows directly from the denition of the H ;= -norm (see also (40)). The preceding discussion leads to the following theorem. 9
Theorem 3 Let (u ) and (u N h ) be the solutions to () and (0), respectively. If H k (S 3 ) for some k, then there exists a positive constant, independent of N h and a (0 ), such that where m = minfk p +g: ku ; u N k + k ; h k ;= Proof. From Theorem we have ku ; u N k + k ; h k ;= inf vv N n pna N + h m p ;k o (44) ku ; vk + inf V h Now, inf ku ; vk ku ; w N k = kr N k vv N with w N r N given by (9). Using (A) and (A) we get inf vv N k ; k ;= : ku ; vk kr N k 0 + jr N j na N + p o Na N p Na N (45) with R + independent ofn. Next, let I be the p th ;order interpolant of on the partition f; i g n i= of S 3.Wehave (cf. [3]) k ; I k 0 h m p ;k kk k (46) where m =minfk p +g and >0 a constant independent ofh and p. Now, inf V h k ; k ;= k ; I k ;= k ; I k 0 (47) so that, since H k (S 3 ), (45){(47) give the desired result. Remark 4 The above theorem shows that if the number of singular functions N is increased then u N converges to u at an exponential rate. The theorem also shows that the convergence of h to can occur in one of three ways: (i) by keeping p xed andreducing h, (ii) by keeping h xed and increasing p, or (iii) by doing both. These three \options" loosely correspond to the three versions of the FEM, namely the h, p and hp versions (cf. [3]). Remark 5 Based on the above theorem, one may obtain the \optimal" matching between N and h, i.e. the relationship between the number of singular functions and the number of Lagrange multipliers used in the method, by choosing them in such a way so that the error in (44) is balanced. For example, in the case when p is kept xed and h! 0, we take h p+ p Na N. This leads to the following approximate expression for N: N (p +) ln h ln a : (48) 0
The approximation of the GSIFs is given by the following orollary. orollary 6 Let and u = u N = X j= NX j= j r j j () N j r j j () satisfy () and (0), respectively, with j N j denoting the true and approximate GSIFs. Then, there exists a positive constant, independent of N and a (0 ), such that j ; N j Proof. This is a direct consequence of (A) and the fact that (See also eq. (45).) 4 Implementation j ; N j a N : (49) ku ; un k 0 : We now give a brief description of the implementation in order to emphasize the properties of the method. As described in Remark, the equation which we discretize is We seek such that u @v S4 + u @v ; v @u S4 = vg: u N = NX j= N j v j (50) @v i u N S4 + @v i u N ; @u N v i S4 = v i g (5) with v i given by (). The normal derivative ofu N above is replaced by the Lagrange multiplier function (cf. equations (0) and (4)) h = MX k= k k V h (S 3 ) (5)
where k R (to be determined), and V h (S 3 )=span f k g M. The boundary condition k= on S 3 is weakly enforced by requiring that i (f ; u N )=0 i= ::: M: (53) ombining (5){(53), we obtain the (N + M) (N + M) linear system K K ;!;! " ;!G;! # K T = (54) 0 F where ;! = N ::: N N T ;! =[ ::: M ] T and [K ] j k = [K ] j i = ; h ;!G i h ;!F i j i = = S4 @v j v k S4 + v k @v j j= ::: N k = ::: N v j i j= ::: N i= ::: M v j g j= ::: N f i i= ::: M: It is easily shown that the coecient matrix in (54) is symmetric. This matrix, however, is singular if N < M. Hence, we shouldchoose N larger than M, but not too large since for excessively large values of N the linear system (54) becomes ill-conditioned and the results obtained are unreliable. The relationship between N and h (hence M) described in Remark 4 should be our guide in deciding how large to choose these values { see also Section 5 ahead. As a nal remark in this section, we should point out that all integrals involved in the determination of the coecient matrix (and right hand side) in (54) are along the sides of the domain that do not contain the singularity. Moreover, they are one-dimensional and can be approximated by standard techniques, such as Gaussian quadrature. 5 Numerical Results We consider the model problem depicted in Figure, originally studied in [4]. The local asymptotic solution may be written as u = X j= j v j
where Figure : Geometry and boundary conditions for the model problem. v j = r (4j;3)=3 sin 3 (4j ; 3) : The numerical results presented here correspond to the following choices of all relevant parameters: The Lagrange multiplier function h used to impose the Dirichlet condition along S 3 is expanded in terms of quadratic basis functions i (see equation (5)), and boundary S 3 is divided into n quadratic elements of equal size. For the integration, boundary S 4 is also subdivided into n intervals of equal size. All integrals involved are calculated numerically by subdividing each interval above into 0 subintervals and using a 5-point Gauss-Legendre quadrature over each one. In computing the coecient matrix in (54), its symmetry is taken into account. To determine the relationship between the number of singular functions N, and the number of Lagrange multipliers M, we proceed as follows: Since for h we are using p = and h ==n, wehave M =n +: For the moment, we xn = 8 (say), which amounts to M = 7, and solve the linear system (54) for various values of N>M(e.g., N =9 3 :::). We concentrate only on the calculation of the rst GSIF N and record our results in Table. From the table we see that, for this choice for M, thevalue of a N is converged up to 4 signicant digits once N =35. Moreover, from (48) we have N (p +) ln h ln a ln a p + N =(p +)ln(=n) ln a ln p M ; ) a ln; p M; =(p +) ln a p M ; p+ N hence, using M =7andN = 35, we nd that a 0:89. With a known, we maynow compute the rest of the GSIFs and/or any other quantities of interest by choosing N and 3 )
N a N 9 :7979988335 :7980768903 3 :79844468 5 :79804003968 7 :798040076 9 :79804005877 3 :79804005983 33 :79804005939 35 :79804005939 Table : Approximate GSIF a N with M = 7. M via equation (48). For example, for M =4(i.e. h ==0), we nd that N 60. Table below shows the converged approximate values for the rst ve GSIFs obtained using the SFBIM (with M =4andN =60),aswell as the hp-version of the FEM, which is considered to be the state-of-the-art for problems with singularities [] the values from reference [4] are also included for comparison. These results suggest that j a N j (SFBIM) a N j (hp-fem) a N j (Ref. [4]) :79804005939 :79800 :80 0:69933866505 0:6993387 0:699 3 ;0:0304097399348 ;0:03049 ;0:030 4 0:0034796658 0:0034755 0:0035 5 0:0009557099 0:00096 0:0009 Table : Approximate GSIFs a N j, j = ::: 5. the SFBIM can be an attractive (and often preferable) method for problems in which the GSIFs are the main goal of the computation. Figure 3 shows the convergence of the leading GSIFs witn N { in particular, the gure shows a semilog plot of the percentage relative error in a N j j = ::: 5. Since each error curve is essentially straight, this illustrates the exponential convergence of the method, as predicted by orollary 6. References [] M. Arad,. Yosibash, G. Ben-Dor and A. Yakhot, \omparing the ux intensity factors by a boundary element method for elliptic equations with singularities," ommun. Numer. Meth. Eng., Vol. 4, pp. 657{670 (998). 4
0 0 3 0 4 GSIFs using the SFBIM j = j = j = 3 j = 4 j = 5 0 5 Percentage relative error 0 6 0 7 0 8 0 9 0 0 0 0 5 0 5 0 5 30 35 N Figure 3: onvergence of a N j j = ::: 5 using the SFBIM. [] I. Babuska, \The nite element method with Lagrangian multipliers", Numer. Math., Vol. 0, pp. 79{9 (973). [3] I. Babuska and M. Suri, \The p and h-p versions of the nite element method, basic principles and properties", SIAM Review, Vol. 36, No. 4, pp. 578{63 (994). [4] S.. Brenner, \Multigrid methods for the computation of singular solutions and stress intensity factors I: orner singularities", Math. omp. Vol. 68, pp. 559{583 (999). [5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag (99). [6] P. G. iarlet, The Finite Element Method for Elliptic Problems, lassics in Applied Mathematics 40, SIAM (00). [7] L. G. Olson, G.. Georgiou and W. W. Schultz, \An ecient nite element method for treating singularities in Laplace's equation", J. omp. Physics Vol. 96, pp. 39{ 40 (99). [8] M. Elliotis, G. Georgiou and. Xenophontos, \The solution of Laplacian problems over L-shaped domains with a singular function boundary integral method", 5
ommunications in Numerical Methods in Engineering, Vol. 8, No. 3, pp. 3{ (00) [9] M. Elliotis, G. Georgiou and. Xenophontos, \Solving Laplacian problems with boundary singularities: A comparison of a singular function boundary integral method with the p/hp version of the nite element method", accepted for publication in Appl. Math. omp., 004. [0] M. Elliotis, G. Georgiou and. Xenophontos, \Solution of the planar Newtonian stick-slip problem with a singular function boundary integral method", submitted, 004. [] M. Elliotis, G. Georgiou and. Xenophontos, \The singular function boundary integral method for a two-dimensional fracture problem", submitted, 004. [] G. Georgiou, A. Boudouvis, A. Poullikkas, \omparison of two methods for the computation of singular solutions in elliptic problems", J. omput. Appl. Math., Vol. 79, pp. 77{90 (997). [3] P. Grisvard, Singularities in Boundary Value Problems, Springer-Verlag, Berlin, 99. [4] T. Igarashi and I. Honma, \A boundary element method for potential elds with corner singularities", Appl. Math. Modelling, Vol. 0, pp. 847{85 (996). [5].. Li, ombined Methods for Elliptic Equations with Singularities, Interfaces and Innities, Kluwer Academic Plublishers, Boston, Amsterdam (998). [6].. Li, \Penalty combinations of Ritz-Galerkin and nite-dierence methods for singularity problems", J. omput. Appl. Math., Vol. 8, pp. {7 (997). [7].. Li, Y. L. han, G. Georgiou and. Xenophontos, \Special boundary approximation methods for Laplace's equation with singularities", accepted for publication in omp. Math. Appl., 004. [8].. Li and T. T. Lu, \\Singularities and treatments of elliptic boundary value problems," Math. omput. Model., Vol. 3, pp. 97{45 (000). [9].. Li, R. Mathon and P. Sermer, \Boundary methods for solving elliptic problems with singularities and interfaces", SIAM J. Num. Anal. Vol. 4, No. 3, pp. 487{498 (987). [0] N. K. Mukhopadhyay, S. K. Maiti and A. Kakodkar, \A review of SIF evaluation and modelling of singularities in BEM," omp. Mech., Vol. 5, pp. 358{375 (000). 6
[] J. Pitkaranta, \Boundary subspaces for the nite element method with Lagrange multipliers", Numer. Math. Vol. 33, pp. 73{89 (973). [] B. A. Szabo and I. Babuska, Finite Element Analysis, Wiley, 99. [3] B. A. Szabo and. Yosibash,\Numerical analysis of singularities in two dimensions. Part : omputation of the eigenpairs", Int. J. Numer. Methods Eng. Vol. 38, pp. 055{08 (995). [4] B. A. Szabo and. Yosibash,\Numerical analysis of singularities in two dimensions. Part : omputation of generalized ux/stress intensity factors", Int. J. Numer. Methods Eng. Vol. 39, pp. 409{434 (996). A Proof of Theorem Here, for completeness, we give a proof of Theorem. First, we note that since (u ) also satisfy (0), we have 8 (v ) V N V h B(u ; u N v)=;b(u ; u N v ; h )= v( ; h )+ Since u = f on S 3 and R (u N ; f) =08 V h,wehave u N = and thus the last integral in (55) is zero. Hence, Letting w = u N ; v v V N,we get B(u ; u N v)= (u ; u N ): (55) u 8 V h (56) v( ; h ): (57) B(u N ; v w) = B(u N ; u w)+b(u ; v w) =B(u ; v w) ; = B(u ; v w) ; By means of (56), the above equation becomes B(u N ; v w) =B(u ; v w) ; w( ; ) ; 7 w( ; ) ; w( ; h ) w( ; h ) ( h ; )(u ; v): (58)
Now, from (5), (7) and (58) we get c 0 ku N ; uk = c 0 kwk ku ; vk kwk + kwk k ; k ;= + ku ; vk k h ; k ;= nku ; vk + k ; k ;= kwk + ku ; vk k h ; k ;= o : This is an inequality of order : where x = kwk c 0 x bx + d b d>0 b= d = ku ; vk k h ; k ;= Therefore, we obtain the bound ku ; vk + k ; k ;= " k h ; k ;= + " ku ; vk ">0: x b + p b +4c 0 d c 0 or, equivalently, ku ; vk + k h ; k ;= + " ku ; vk kwk + "k h ; k ;= : (59) Next, using (6) with h = h ; we nd that there exists 0 6= v N V N such that k h ; k ;= R h ; )v N ( kv N k : (60) Also, it follows from (57) that ( h ; )v N = ( h ; )v N + ( ; )v N = B(u ; u N v N )+ ( ; )v N c ku ; u N k kv N k + kv N k k ; k ;= (6) so that by (60) and (6), o k h ; k ;= nku ; u N k + k ; k ;= o nku ; vk + kv ; u N k + k ; k ;= : 8
Since kv ; u N k = kwk,wehave from (59) k h ; k ;= n ku ; vk + k ; k ;= + " ; ku ; vk o + " k h ; k ;= with R + independent ofv and ": Letting " = weget o k h ; k ;= nku ; vk + k ; k ;= : (6) Using the triangle inequality, we immediately get k ; h k ;= k ; k ;= + k h ; k ;= o nku ; vk + k ; k ;= : (63) Similarly, using (59) and (63), we get ku ; u N k ku ; vk + kv ; u N k = ku ; vk + kwk o nku ; vk + k ; k ;= which gives the desired result. 9