Reproducing polynomial particle methods for boundary integral equations

Transcription

1 DOI /s x ORIGINAL PAPER Reproducing polynomial particle methods for boundary integral equations Hae-Soo Oh Christopher Davis June G. Kim YongHoon Kwon Received: 18 November 2010 / Accepted: 31 January 2011 Springer-Verlag2011 Abstract Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities. Keywords The closed form reproducing polynomial particle (RPP) shape functions Reproducing kernel particle (RKP) shape functions Boundary element method (BEM) H.-S. Oh: Supported by NSF DMS , DMS and Pohang University of Science and Technology, Korea. C. Davis: Pohang Mathematics Institute, Pohang University of Science and Technology, Korea. H.-S. Oh (B) C. Davis Department of Mathematics & Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA hso@uncc.edu J. G. Kim Department of Mathematics, Kangwon National University, Chunchon , Korea Y. Kwon Department of Mathematics, Pohang University of Science and Technology, Pohang , Korea Moving least squares method Reproducing singularity particle shape functions Boundary node method (BNM) 1Introduction Boundary element methods (BEM) that yield numerical solutions of boundary integral equations (BIE) have been widely used to solve important science and engineering problems. Most BEM use a finite element (FE) mesh on the boundaries of physical domains. Since meshless methods [2,5,7,8, 10,12 16,18] havebeenintroducedtoalleviatethedifficulties arising in the conventional finite element method (FEM) [4,29], several meshless BEM have been introduced in which BEM is coupled with meshless methods. Using the moving least squares (MLS) method [14,15] for constructing approximation functions, Mukherjee and Mukherjee [19]proposedaboundarytypemeshlessmethod called the boundary node method (BNM). However, since the MLS nodal approximation functions lack the Kronecker delta property, BNM have difficulties in imposing the boundary conditions. Hence, the computed traction boundary values by BNM are oscillating whenever the true traction boundary data have jumps at some vertices. The system matrices of BNM are non symmetric. Moreover, the convenient tools for error analysis that are available in Galerkin approximation methods can not be used for these approaches. In order to alleviate these difficulties, Li and Zhu [11] introduced a Galerkin boundary node method (GBNM) which is based on an equivalent variational form of a boundary integral formulation for the governing partial differential equation. GBNM uses the MLS approximation functions for the trial and test functions of the variational form and hence the system matrices are symmetric. If the nodal approximation functions do not satisfy the Kronecker delta property, mod-

2 ifying the variational form with use of the penalty methods, the Niethche method, or the Lagrange multiplier method, one can impose the boundary conditions by allowing some variational errors in the Galerkin method [3]. Meshless methods have advantages over conventional FEM. However, these methods also have some difficulties such as inefficiency in handling essential boundary conditions, large matrix condition numbers and so on. Recently, Oh et al. [21 27] introduced various meshless methods that alleviate these difficulties. Specifically, the following have been introduced: 1. Highly regular (piecewise polynomial) reproducing polynomial particle (RPP) shape functions corresponding to uniformly spaced particles [25]. 2. Constructions of various piecewise polynomial partition of unity (PU) functions: convolution PU functions, almost everywhere PU functions, and generalized product PU functions [21,22,24]. 3. Using these partitions of unity, patch-wise RPP shape functions are constructed that correspond to patch-wise uniformly (or non-uniformly) spaced particles [26]. 4. Reproducing singularity particle (RSP) shape functions that can effectively handle elliptic boundary value problems containing singularities [23,27]. It is important to note that almost all particle shape functions mentioned above satisfy the Kronecker delta property. In this paper, with these RPP (RSP) shape functions, we propose three different reproducing polynomial boundary particle methods (RPBPM) to solve BIEs of elliptic problems with or without singularities: 1. Meshfree RPBPM that uses mesh-free RPP shape functions. 2. Patchwise RPBPM in which patchwise RPP shape functions are assigned to each particle on the boundary. 3. Patchwise RSBPM (reproducing singularity boundary particle method) that uses patchwise RSP shape functions for BIE containing singularities. The novelty of this paper is an introduction of effective methods to deal with two- and three-dimensional singularities in the context of BEMs. It is well known that it is very hard to get highly accurate solutions of three-dimensional elliptic problems containing singularities. Thus the dimension reduction by BEM is a big advantage in dealing with singularity problems. In this paper, we use closed form partition of unity functions with flat-top such that the non-flat-tops are only very small areas of their supports. Thus, unlike other meshless methods, the singular approximation functions obtained by the product of the PU functions employed in this paper with global singular functions retain almost the same shapes as the unaltered singular functions. Furthermore, the meshfree particle methods have advantages to bring in any type of singular functions for local approximation functions in BEMs. After introducing definitions and terminologies in Sect. 2, three boundary particle methods are proposed in Sect. 3. In Sect. 4, wedemonstratenumericalexamplesthatshow our methods yield more accurate solutions than BNM and GBNM. In Sect. 5, wealsoshowthatpatchwisersbpm effectively handle BIEs containing singularities. In Sect. 6, three-dimensional extension of the proposed methods is presented. Some concluding remarks are addressed in Sect. 7. Finally, we obtain a highly accurate computed solution for the Motz problem that contains a jump boundary data singularity in Appendix A. 2Preliminary Throughout this paper, α, β Z d are multi indices and x = ( 1 x, 2 x,..., d x), x j = ( 1 x j, 2 x j,..., d x j ) denote points in R d. However, without confusion, we also use the conventional notation for the points in R d or Z d as x = (x 1, x 2,...,x d ) and α = (α 1, α 2,...,α d ). By α β, we mean α 1 β 1,...,α d β d. We also use the following notations: (x x j ) α := ( 1 x 1 x j ) α 1 ( d x d x j ) α d, α := α 1 +α 2 + +α d. Let be a domain in R d.foranynonnegative integer m, C m ( ) denotes the space of all functions φ such that φ together with all their derivatives D α φ of orders α m, are continuous on. The support of φ is defined by supp φ = {x : φ(x) = 0}. In the following, a function φ C m ( ) is said to be a C m - function. A family {U k : k D} of open subsets of R d is said to be apointfiniteopencovering of R d if there is M such that any x lies in at most M of the open sets U k and k U k. For a point finite open covering {U k : k D} of a domain, supposethereisafamily{ψ k : k D} of Lipschitz functions on satisfying the following conditions: 1. For k D,0 ψ k (x) 1, x R d. 2. The support of ψ k is contained in U k, for each k D. 3. k D ψ k(x) = 1foreachx. Then {ψ k : k D} is called a partition of unity (PU) subordinate to the covering {U k : k }. Thecoveringsets{U k } are called patches. By almost everywhere partition of unity,wemean{ψ k : k D} such that the condition 3 of a partition of unity is not

3 satisfied only at finitely many points (2D) or lines (3D) on a part of the boundary. A weight function (or window function) is a non-negative continuous function with compact support and is denoted by w(x). Consider the following conical window function: For x R, { (1 x w(x) = 2 ) l, x 1, (1) 0, x > 1, where l is an integer. Then w(x) is a C l 1 -function. In R d, the weight function w(x) can be constructed from a onedimensional weight function as w(x) = d i=1 w(x i ), where x = (x 1,...,x d ). In this paper, we use the normalized window function defined by ( x ) wδ l (x) = Aw, (2) δ where A =[(2l + 1)!]/[2 2l+1 (l!) 2 δ] [8] istheconstantthat makes R wl δ (x)dx = Meshfree particle shape functions Adopting those terminologies and notations of [2,12 14,16], we have the following: For j = ( j 1, j 2,..., j d ) Z d, and the mesh size 0 < h 1, let x h j = ( j 1h,..., j d h) = hj. Then the points x h j are called uniformly distributed particles. Let φ be a continuous function with compact support that contains the origin 0. Then the particle shape functions associated to the uniformly distributed particles is defined by φ h j (x) = φ ( x jh h ) ( x1 j 1 h = φ h,..., x d j d h h ), for j Z d and 0 < h 1. Then these particle shape functions are translation invariant in the sense that x h i+ j = xh i + x h j, φh j (x ih) = φh i+ j (x). In this paper, we assume that the mesh-free particle shape functions are translation invariant on the uniformly distributed particles,unlessstatedotherwise.moreover,usingpartition of unity and patches(background mesh), the particle shape functions associated with non-uniformly distributed particles are constructed in Sect Without loss of generality, in the constructions of meshfree particle shape functions, we assume that h = 1and hence φ h j (x) = φ(x j). Let be an index set and denotes a bounded domain in R d.let{x j : j } be a set of uniformly (or non-uniformly) distributed points in R d, that are called particles. Definition 2.1 ([2,8,10,12]) Let k be a nonnegative integer. Then the functions φ j (x) corresponding to the particles x j, j are called the reproducing kernel particle (RKP) shape functions with the reproducing property of order k (or simply, of reproducing order k ) if and only if it satisfies the following condition: for x R d, (x j ) α φ j (x) = x α, and for 0 α k, α =α 1 + +α d. j By applying a similar argument to [2,8], one can show that (3)isequivalentto (x x j ) β φ j (x) = δ β 0, for 0 β k and x Rd. j This characterization of meshfree shape functions has no direct relation with the window functions. Using (4), Oh et al. [25] constructedpiecewisepolynomialmeshfreeparticle shape functions that have the polynomial reproducing property. These meshfree shape functions constructed without using window function were called reproducing polynomial particle (RPP) shape functions. One of the salient features of RPP shape functions is that they satisfy the Kronecker delta property [22,24 27]. We refer to [25]forvariousclosedformRPPshapefunc- tions with high polynomial reproducing order and high order of regularity. For example, 1-dimensional C 0 -RPP shape functions, φ j (x), j Z, of reproducing order 3 can be constructed as follows: Suppose the particles x j are the integer points and φ j (x) = φ(x j) for j Z. Then, from (4), we have a system of functional equations, (x + 1) x (x 1) (x 2) (x + 1) 2 x 2 (x 1) 2 (x 2) 2 (x + 1) 3 x 3 (x 1) 3 (x 2) 3 1 = 0 0 for x (0, 1). 0 ˆφ(x + 1) ˆφ(x) ˆφ(x 1) ˆφ(x 2) Now using the solutions of this system, we construct the basic RPP shape function φ(x), fromwhichotherparticleshape functions are obtained by translations, in the following way. φ(x) ( 2, 1) = ˆφ(x + 1) shifted by + 1, φ(x) ( 1,0) = ˆφ(x + 1) shifted by 1, φ(x) (0,1) = ˆφ(x), φ(x) (1,2) = ˆφ(x 2) shifted by 2, φ(x) = 0 forx / [ 2, 2]. Connecting these pieces together at 2, 1, 0, 1, 2, we have the following piecewise polynomial C 0 -RPP shape function (3) (4)

4 defined by φ ([ 2,2];0;3) (x) (x + 1)(x + 2)(x + 3) x [ 2, 1] = (x 1)(x + 1)(x + 2) x [ 1, 0] 1 2 (x 2)(x 1)(x + 1) x [0, 1] 1 6 (x 3)(x 2)(x 1) x [1, 2] 0 x [ 2, 2], where the subscripts [ 2, 2], 0, 3, respectively, stand for the support, the regularity, and the order of reproducing polynomial property of φ ([ 2,2];0;3). By sacrificing polynomial reproducing order by one from φ ([ 2,2];0;3) (x), we construct C 1 -RPP shape function φ ([ 2,2];1;2) (x) with reproducing order 2 in the following: 1 2 (x + 1)(x + 2)2 x [ 2, 1] 2 1 (x + 1)(3x2 + 2x 2) x [ 1, 0] φ ([ 2,2];1;2) (x) = 1 2 (x 1)(3x2 2x 2) x [0, 1] 2 1 (x 2)2 (x 1) x [1, 2] 0 x [ 2, 2] We refer to [25] fortheproof.letusnotethatthehigher dimensional piecewise polynomial RPP shape functions constructed by solving the system (4) are actually the same as the tensor product of 1-dimensional RPP shape functions. 2.2 One-dimensional partition of unity functions with flat-top In this section, we briefly review one dimensional partition of unity with flat-top. For details of this construction, we refer to [24], in which we showed that PU functions with flat-top lead to a small matrix condition number. Throughout this paper, we reserve the small real number δ,usually,0.01 δ 0.1, for the width of non flat-top part of the PU functions with flat-top. For any positive integer n, C n 1 -piecewisepolynomial basic PU functions were constructed as follows: For integers n 1, we define a piecewise polynomial function by φ (pp) g n (x) = (5) φg L n (x) := (1 + x) n g n (x) if x [ 1, 0] φg R n (x) := (1 x) n g n ( x) if x [0, 1] 0 if x 1, (7) where g n (x) = a (n) 0 + a (n) 1 ( x) + a(n) 2 ( x)2 + +a (n) n 1 ( x) n 1 whose coefficients are inductively constructed by the following recursion formula: 1, if k = 0 a (n) k k = j=0 a(n 1) j, if 0 < k n 2, (8) ), if k = n 1. 2(a (n) n 2 (6) For example, from the recurrence relation (8), we have g 1 (x) = 1; g 2 (x) = 1 2x; g 3 (x) = 1 3x + 6x 2. Then, φ g (pp) n has the following properties whose proofs can be found in [24]. φ g (pp) n (x) + φ g (pp) n (x 1) = 1forallx [0, 1]. Hence, {φ g (pp) n (x j) j Z} is a partition of unity on R. φ g (pp) n (x) is a C n 1 -function. The gradient of the scaled basic PU function is bounded as follows: d [ dx φ (pp) g n ( x )] C 2δ δ Note that the constant C is 0.9 forn 3. Using the basic PU function φ g (pp) n,weconstructac n 1 - PU function with flat-top whose support is [a δ, b + δ] with (a + δ) <b δ as follows: ( ) φ L x (a+δ) g n 2δ if x [a δ, a + δ] ψ (δ,n 1) 1 [a,b] (x)= ( ) if x [a + δ, b δ] φg R x (b δ) n 2δ if x [b δ, b + δ] 0 if x / [a δ, b + δ]. (10) (9) Here, in order to make a PU function have a flat-top, we assume δ (b a)/3. Actually, it was shown in [24] that ψ (δ,n 1) [a,b] (x) is the convolution, χ [a,b] (x) wδ n 1 (x), of the characteristic function χ [a,b] and the scaled window function wδ n 1,definedby(2). Suppose a < b < c, and let ϕ :[0, 1] [b δ, b + δ] be a linear transformation defined by ϕ(ξ) = 2δξ + (b δ). Then for x [b δ, b + δ], using (10), we have ψ (δ,n 1) [a,b] (x) + ψ (δ,n 1) (x) = φg R n ([ϕ(ξ) (b δ)]/(2δ)) [b,c] +φ L g n ([ϕ(ξ) (b + δ)]/(2δ)) = φ R g n (ξ) + φ L g n (ξ 1) = 1 which show that the functions defined by (10)make PUfunctions with flat-top subordinate to the patches [a = b 0, b 1 ], [b 1, b 2 ], [b 2, b 3 ],...,[b n 1, b n = c] of =[a, c]. 2.3 Boundary integral equation Let u be the solution of Laplace s equation 2 u = 0 in R 2. and G(x, P) the fundamental solution of 2 G(x, P) + δ(x P) = 0,

5 where x = (x, y), P = (ξ, η), and δ(x P) = the Delta function centered at P. Then,usingGreen stheorem,we have the following boundary integral equation (BIE) u(ξ, η) + u = Ɣ Ɣ G(x, P) dɣ u G(x, P)dƔ for (ξ, η) (11) However, using the properties of the Delta function, we can write the integral equation for any point P R 2 as follows: G(x, P) u c(p)u(p) + u dɣ = G(x, P)dƔ, (12) where Ɣ G(x, P) = 1 log r, 2π r = (ξ x) 2 + (η y) 2 (13) 0 if P is outside 1 if P C(P) = 1 if P Ɣ and Ɣ is smooth at P (14) 2 α if P Ɣ and Ɣ is not smooth at P 2π Here α is the internal angle of at P (refer to [9]forproof). In boundary integral equation (12), the unknowns are u( P) and u(x) or If P, G(x, P) u(p) + u dɣ = Ɣ Ɣ Ɣ u (x) for x Ɣ. u G(x, P)dƔ, (15) has no singular integral. However, if P and Ɣ is smooth at P, 1 2 u(p) + G(x, P) u u dɣ = G(x, P)dƔ, (16) Ɣ Ɣ has a singular integral that diverges in sense of improper integral. Thus, we use the Cauchy principal value (PV) in order for this singular integral to have a finite definite value. Using (16), u Ɣ and u Ɣ can be determined. Together with these boundary values, one can use (15)toget the value of u at all points in (the solution of Laplace s equation). In other words, firstly, we solve for the boundary data (u and its normal derivative along the boundary Ɣ). Secondly, the volume data can be found as a post processing. Numerical methods to compute u Ɣ and u Ɣ are called the BEMs which obviously, have advantages over the FEM whenever the fundamental solution could be found in a simple form. However, the fundamental solutions are not known except for some special differential equations, whereas the FEM does not have such restrictions. In this paper, we are concerned with boundary particle methods using meshfree particle shape functions corresponding to particles. For brevity, we present the proposed methods to the two-dimensional Laplace equation on polygonal domains. Three-dimensional extension is presented in Sect Boundary element methods Two main approximations, and hence the two main sources of error in the calculation are the interpolation of the boundary and the interpolation of the boundary functions. Since we are concerned with Laplace s equation on polygonal domains, we only develop numerical methods that efficiently approximate the boundary functions. Suppose is a polygonal domain with edges, Ɣ 1,...,Ɣ N. That is, Ɣ = N Ɣ j. j=1 Then Eq. (12) becomes c(p)u(p) + N u j=1ɣ j G(x, P) dɣ(x) N u = G(x, P) dɣ(x), (17) j=1ɣ j where G(x, P) represents a fundamental solution at the point P. Let P α, α = 1,...,M, be chosen boundary particles and α, α = 1,...,M, be approximation functions corresponding to these boundary particles such that Ɣ N supp j and α (P β ) = δα β. (18) j=1 For each element Ɣ j,supposep j1,...,p jn( j) are those nodes among P α that lie on Ɣ j.thenu and u are approximated by n( j) u Ɣ j jk (x)u jk (19) k=1 q Ɣ j = u n( j) Ɣ j jk (x)q jk. (20) k=1 where u jk and q jk are values of u and q at node P jk on element Ɣ j. From Eq. (18), we have u(p α ) = N n( j) jk (P α )u jk = j=1 k=1 N n( j) δ α j k u jk j=1 k=1

6 Substituting Eqs. (19) and(20) intoeq.(12) wehave c(p α )u(p α ) + = Let a α j k = N n( j) q jk j=1 k=1 PV Ɣ j b α j k = α j k (x) N n( j) j=1 k=1 u jk PV Ɣ j jk (x) G(x, P α ) dɣ(x) Ɣ j G(x, P α ) jk (x)dɣ(x) (21) G(x, P α ) dɣ(x) and Ɣ j G(x, P α ) α j k (x)dɣ(x) (22) Then (21) canbewrittenasfollows: c(p α )u(p α ) + N n( j) u jk a α j k = j=1 k=1 N n( j) q jk b α j k (23) j=1 k=1 that holds for each particle P α. Depending on the given boundary conditions on Ɣ, at each α, eitheru α or q α is unknown. Equation (23) hasmunknowns and considering Eq. (23) for each node P α, α = 1,...,M, wehavethem-equations for the M-unknowns. Remark 2.1 Suppose S is the line segment on some part of Ɣ connecting P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) and let P = (ξ, η). Theintegralsforthecoefficientsa jk, b jk are of the form h(x) I n (P) = ds(x), or x P n S I log (P) = h(x) log x P ds(x), where P Ɣ S When P S these integrals become singular and special techniques must be used to evaluate these integrals. For polygonal domains in R 2 when P S, a jk = 0because G(x, P) S = G(x, P) n = 0. (24) so the only singular integral to consider comes from b jk which is of the form I log (P). To evaluate I log (P), split the segment S into S 1 connecting P 1 and P and S 2 connecting P and P 2.Then I log (P) = h(x) log x P ds(x) + h(x) log x P ds(x) S 1 Consider the first integral, the second may be evaluated in asimilarmanner.set x(t) = (P P 1 )t + P 1, where t [0, 1] S 2 Then h(x) log x P ds(x) S 1 = 1 0 h((p P 1 )t + P 1 ) log (P P 1 )(t 1) P P 1 dt = log P P 1 P P 1 + P P h((p P 1 )t + P 1 )dt h((p P 1 )(1 t) + P 1 ) log(t)dt The first integral may be computed by standard Gaussian Quadrature rules whereas the second integral should be computed using Logarithmic Gaussian Quadrature rules. Without taking special care in treatment of the singular integrals, the high accuracy of the numerical solution will be lost. 3Reproducingpolynomialboundaryparticlemethods (RPBPM) In this section, we introduce three RPBPMs that yield numerical solutions of BIEs. We also demonstrate that these proposed methods are superior to the BNM [19] whenthetrue solution is a polynomial. 3.1 Meshfree RPBPM that use uniformly spaced meshfree RPP shape functions Suppose R 2 is a convex polygonal domain as shown in Fig. 1 in which particles are edge-wise uniformly spaced. That is, the width between two adjacent particles on different edges are allowed to be different. 1. (Construct edge-wise equally spaced particles on each edge of )Plantuniformlyspacedparticlesoneachedge so that no particles are vertices of the polygonal domain. Determinethenumberofparticlessothatthewidth h between two adjacent particles becomes small enough for the required accuracy to be obtained. 2. (Select RPP shape function on the reference domain) Pick abasicrppshapefunctionφ ([ K,K ];0;2K 1) (x) whose support is [ K, K ] among those listed in [25]. For each edge Ɣ j, set up a parametrization ϕ j from an interval [ K, K ] into an extended edge ˆƔ j so that the integer points in [ K, K ] are mapped onto equally spaced nodes on ˆƔ j,asshowninfig (Construct approximation functions corresponding to particles on edges) Construct particle shape functions corresponding to each particle P jk along ˆƔ j by a translation

7 .. Γ Γ 4 3. Γ Γ Γ1. ϕ Fig. 1 Diagram of meshfree particles planted on edges of polygonal domain and basic RPP shape functions φ ([ 2,2];0;3) (x) with RPP order 3 of φ ([ K,K ];0;2K 1) ϕ j.forexample,supposeh is the width of uniformly spaced particles in Fig. 2. Then, for each k = 1, 2,...,12, the RPP shape function corresponding to the particle x k is ( ) x xk φ xk (x) = φ ([ K,K ];0;2K 1). h Then, in order to keep the reproducing order for all points in Ɣ j, 2K particles should be outside of the edge Ɣ j.for those nodes shown in Fig. 2, noboundaryconditionsare available, and hence the boundary data on the external particles can be extrapolated by least squares method and the property of reproducing polynomials as described below. 4. (Extrapolating boundary data for those particles outside of the boundary): Without loss of generality, we may assume that K = 2, and P ik = P i1 + h(k 1) for each k = 1,...,N i.thenthefourparticlesp i1, P i2, P i(ni 1), P ini go outside the ith edge as shown in Fig. 2. Supposethe Dirichlet boundary condition is imposed along the edge Ɣ i. Then for the approximation u Ɣi N i k=1 u ikψ ik (x), the amplitudes u ik are known for k = 3, 4,...,N i 2. We want to extrapolate the boundary conditions u i1, u i2, u i(ni 1), u ini corresponding to the four external (active) particles by the least squares method. For example, u i2 minimizes E(u i2 ) defined by, E(u i2 ) = x(p i4 ) x(p i3 ) [ 5 2 u ik φ ik (x) u(x)] dx k=2 x1 x2 x3 x4 x5 x11 x12 x13 Inside Particles Fig. 2 Diagram of external nodes whose boundary data should be extrapolated by using information on the boundary data on the inside nodes In other words, x(p i4 ) x(p 5 / i4 ) u i2 = u(x) u ik φ ik (x) φ i2 dx φ i2 (x)dx x(p i3 ) k=3 x(p i3 ) (25) where u i2 = 0ifthedenominatorvanishes,andx(P i3 ) is the coordinate of particle P i3 in a parametrization of edge Ɣ i. Similarly, the boundary condition u i1 corresponding to the external node P i1 minimizes E(u i1 ) = x(p i3 ) x(p i2 ) [ 4 2 u ik φ ik (x) u(x, y)] dx k=1 5. (Numerical Integration): By (24), the normal derivative of the fundamental solution G(x, P) of Laplace s equation is zero whenever P and x are on the same line of the boundary. Hence, there is no double layer potential. However, for the highly accurate computed solutions as shown in Sect. 4, itisessentialtouselogarithmicquadrature for Ɣ u G(x, P)dƔ after it is parameterized as described in remark 2.1. Remark 3.1 In Sect. 3.1, inorder touseparticleshape functions with high order reproducing polynomial property defined by (5), we placed edge-wise uniformly spaced particles along the boundary. This uniformly spaced particle method uses no mesh at all. However, it has a domain restriction and some active particles go outside the domain as shown in Fig. 1. These difficulties can be fixed by introducing background meshes as shown in Figs. 3 and 4. Using partition of unity shape functions with flat-top, one can construct the particle shape functions corresponding to arbitrary spaced particles in Sects. 3.2 and 3.3. ThesupportsofPUfunctions are the background meshes and the corresponding particle shape functions also have the Kronecker delta property and the reproducing polynomial property. 3.2 Patchwise RPBPM that use RPP shape functions corresponding to patchwise arbitrary spaced particles In order to avoid extrapolating the boundary values corresponding to those nodes that are outside of the boundary, we

8 x0 ψ ψ ψ ψ x11 x12 x21 x22 x31 x x1 x2 x3 x4 2δ 2δ 2δ xj1 = xj δ xj2 = xj + δ Fig. 3 Schematic diagram of partition of unity functions ψ i, i = 1, 2, 3, 4 Q41 Q42 x 8 x0 Γ 4 x9 Q33 x 7 x 1 Q32 Γ 3 Ω Γ 1 x6 x 2 Q31 Q11 Q12 Q13 Fig. 4 Schematic diagram of particles for patches, Q11, Q12, Q13 of Ɣ 1 ;patchesq21, Q22, of Ɣ 2 ;patches,q31, Q32, Q33 of Ɣ 3 ; patches Q41, Q42, of Ɣ 4 construct patchwise RPP shape functions that correspond to patchwise uniformly (or non-uniformly) spaced particles. (1) Suppose = n k=1 Ɣ k is union of the edges of a polygonal domain. (2) Suppose Ɣ k =[x 0, x 1 ] [x 1, x 2 ] [x 2, x 3 ] [x 3, x 4 ] as shown in Fig. 3 and x 0, x 4 are vertex nodes of the polygonal domain. For example, Ɣ 1 = Q 11 Q 12 Q 13 ; Ɣ 2 = Q 21 Q 22 ; Ɣ 3 = Q 31 Q 32 Q 33 ; Ɣ 4 = Q 41 Q 42, in Fig. 4. Let us choose a small number δ with 0.01 δ < min{0.1,(x j x j 1 )/3, j = 1, 2, 3, 4}. If χ [x j 1,x j ] is the characteristic function of an interval [x j 1, x j ] defined by χ [x j 1,x j ](x) = then for x Ɣ k, { 1 : x [x j 1, x j ] 0 : otherwise χ [x0 δ,x 1 ](x) + χ [x1,x 2 ](x) + χ [x2,x 3 ](x) +χ [x3,x 4 +δ](x) = 1 and hence the convolutions of these functions with window function w l δ (x), ψ 1 = χ [x0 δ,x 1 ] w l δ, ψ 2 = χ [x1,x 2 ] w l δ, ψ 3 = χ [x2,x 3 ] w l δ, Γ 2 ψ 4 = χ [x3,x 4 +δ] w l δ x 4 x5 x 3 Q22 Q21 shown in Fig. 3, become a partition of unity subordinate to the patches [x 0 2δ, x 1 + δ], [x 1 δ, x 2 + δ], [x 2 δ, x 3 + δ], [x 3 δ, x 4 + 2δ]. Specifically, with (10), these PU functions with flat-top can be written as (3) Let ψ 1 (x) = ψ (δ,n 1) [x 1 δ,x 1 ] (x), ψ 2(x) = ψ (δ,n 1) [x 1,x 2 ] (x), ψ 3 (x) = ψ (δ,n 1) [x 2,x 3 ] (x), ψ 4(x) = ψ (δ,n 1) [x 3,x 4 +δ] (x). {x 11, x 12,...,x 1K1 }, {x 21, x 22,...,x 2K2 }, {x 31, x 32,...,x 3K3 }, and {x 41, x 42,...,x 4K4 }, be distinct particles planted (not necessary uniformly spaced) intheintervals [x 0 + δ, x 1 + δ], [x 1 δ, x 2 + δ], [x 2 δ, x 3 + δ], and [x 3 δ, x 4 δ], respectively. We assume that all (K 1 + K 2 + K 3 + K 4 )- particles are distinct. Remark 3.2 Recall that the Chebyshev polynomial T n (x) of degree n 1hasn simple roots in [ 1, 1] at x k = cos((2k 1)π/2n), for each k = 1, 2,...,n. It is known that these zeros are an optimal choice for nodes for the Lagrange interpolating polynomial. Thus, we suggest to use the shifted x k = cos((2k 1)π/2n) into appropriate intervals for particles x k, j, j = 1,...,K k, k = 1, 2, 3, 4. However, instead of an optimal choice of nodes, we use easy choices of nodes for Lagrange polynomials in numerical examples of this paper. (4) For j = 1,...,K 1,letL 1 j (x) be the j-th Lagrange interpolating polynomial of degree (K 1 1) corresponding to particles x 11, x 12,...,x 1K1.Thenthesefunctions have the reproducing polynomial property of order (K 1 1). Similarly, we construct the Lagrange interpolating polynomials L 2 j (x), for j = 1,...,K 2 ; L 3 j (x), for j = 1,...,K 3 ; L 4 j (x), for j = 1,...,K 4. Using 4 i=1 ψ i (x) = 1, we prove the following theorem.

9 Theorem 3.1 On [x 0, x 4 ], ψ 1 (x)l 1 j (x), j = 1,...,K 1 ; ψ 2 (x)l 2 j (x), j = 1,...,K 2 ; ψ 3 (x)l 3 j (x), j = 1,...,K 3 ; ψ 4 (x)l 4 j (x), j = 1,...,K 4 ; are C l 1 piecewise polynomial RPP shape functions with reproducing order k, where k = min{k 1 1, K 2 1, K 3 1, K 4 1} Then, on the boundary Ɣ j, u and u are approximated by 4 K k u Ɣi ψ k (x)l kl (x)u ikl (26) k=1 l=1 q Ɣi = u Ɣ i 4 K k ψ k (x)l kl (x)q ikl. (27) k=1 l=1 (5) Similarly, for each i = 1,...,n, weconstructhighly smooth patchwise RPP shape functions corresponding to particles planted on patches of edge Ɣ i as shown in Fig Patchwise reproducing singularity boundary particle method (RSBPM) that use patchwise reproducing singularity particle (RSP) shape functions as well as RPP shape functions Suppose an isotropic non-convex polygonal domain has a corner point at which the internal angle is π/α,whereα is a real number with 0 < α < 1. Then an elliptic equation on contains a singularity of type r α f (θ). The number α is called the intensity of a singularity. To deal with elliptic problems containing singularities, Oh et al. [27] constructed reproducing singularity particle (RSP) shape functions that reproduce polynomials as well as singular functions that resemble the singularities. In this section, we are concerned with BEM for elliptic equations containing corner singularities or jump boundary data singularity (Motz problem). Since we consider problems on polygonal domains, we may assume that f (θ) is constant, and hence the singularity along edges is of type r α with 0 < α < 1. Let us note that α = 1/2 forthecrackandthejump boundary data (Motz Problem) singularities. (1) Let T : [0, b] [0, b α ] be a singular mapping defined by ξ = T (x) = x α. (28) (2) Let 0 < ξ 0 < ξ 1 < < ξ 2N b α be particles in [0, b α ].IfL j (ξ) be the j-th Lagrange interpolating polynomial associated with above 2N + 1nodes,thenL j (ξ), j = 0, 1,..., 2N, have the polynomial reproducing property with reproducing order 2N.Thatis, 2N j=0 ξ k j L j(ξ) = ξ k, for 0 k 2N. (29) Using those arguments in [27] one can easily prove the following: Lemma 3.1 For j = 0, 1,...,2N, let j (x) = L j(t (x)) and x j = T 1 (ξ j ).Then 1. j (x i) = δ j i (Kronecker delta property). 2. If α = 1/2 then meshfree particle shape functions: j, j = 0, 1,..., 2N, generate the polynomials 1, x, x 2,...,x N and the singular functions x 1/2+k, k = 0, 1, 2,...,(N 1). (3) We now modify the basic PU function (10)asfollows: 1 if x (0, b δ] ψ 0 (x) = φ g R n ( x (b δ) 2δ ) if x [b δ, b + δ] (30) 0 if x / (0, b + δ], which is a one-dimensional version of almost everywhere partition of unity [22]. A schematic diagram for ψ 0 (x) is shown in Fig. 5. Notethatψ 0 (x) is undefined at (0, 0), the singularity point. Γ Γ 1 4 (0,0) (1,0) Γ 5 ψ ψ 0 1 x00 x05 x06 (0,0) (1,0) x10 x11 x15 x16 Fig. 5 Diagram of the L-shaped domain with corner singularity. Schematic diagram of two PU functions ψ 0, ψ 1 and 14 singular particles corresponding to RSP shape functions that are the images of seven particles in each reference patch by the patch mapping x = ξ 3 :[0, 1] Ɣ 1. x 10 = 0.125, x 11 = 0.166, x 12 = 0.216, x 13 = 0.343, x 14 = 0.512, x 15 = 0.729, x 16 = 0.97 Γ 3 Γ 6 Γ 2

10 Now RSP shape functions to deal with singularity of type r α are defined as follows: j (x) = ψ 0 (x) j (x), j = 0, 1,...,2N. (31) 4Examplesonconvexpolygonaldomains An application of a meshless method to BEM, the boundary node method, in which MLSs method was applied to construct nodal approximation functions, was introduced in ([11,19], and reference cited there). However, the meshless approximation functions constructed by MLS do not satisfy the Kronecker delta property and hence meshless BNM failed to capture the traction (Neumann) boundary values when it has jump discontinuities. However, in this section, we demonstrate that our methods exactly capture the traction boundary values even when traction functions have big jumps at the corners whenever the true solutions are polynomials. Moreover, our methods also correctly approximate the true solution and its derivatives at the interior points near boundary In the following examples, we consider Laplace equations whose true solutions are polynomials. Example 4.1 Consider Laplace s equation u = 0on = [0, 4] [0, 4] whose true solutions are polynomials and boundary conditions are as follows: (I: Dirichlet problem) The true solution is Im(z 2 )/2, z = x+iy.displacement(dirichlet)boundaryconditionsare prescribed from the true solution along the entire boundary. Then the exact Dirichlet boundary values are continuous along the entire boundary, whereas the exact traction boundary values have jump discontinuities, [ u ]=4, at the corner points (4, 0) and (0, 4). (II: Mixed problem 1) The exact solution is Re(z 2 ). Mixed boundary conditions are prescribed from the true solution on the boundary as follows: u = 0onthelinesx = 0 and y = 0 u = Re(z 2 ) on the lines x = 4 and y = 4 The traction boundary values q have the large jump discontinuities, [ u ]=16, at the corner points (4, 0) and (0, 4). Examples (I) and (II), respectively, are examples 5 and 6 of [19]. (III: Mixed problem 2) The true solution is Re(z 4 ).The boundary conditions are similar to (II). Then, the traction boundary values q have even larger jump discontinuities, ]=256, at the corner points (4, 0) and (0, 4). [ u From our numerical tests of the above three examples, we observe the following: 1. Meshfree RPP shape functions and patchwise RPP shape functions with reproducing order k, respectively, are able to interpolate polynomials of degree k exactly. Hence, our methods can get the exact boundary values as shown in Table 1 (for Dirichlet problem), Tables 2 and 3 (for mixed problem 1), and Table 4 (for mixed problem 2) even when the boundary data have a large jump. 2. However, figures 8 and 9 of [19]showthatthecomputed traction boundary values are quite different from the true traction boundary values (the maximum difference 0.5 for problem I and the maximum difference 2forproblem II ) when MLSs approximation functions are used at boundary nodes. 3. If degree of freedom (DOF) in meshfree RPP decreases, then the space between uniformly spaced particles increases. Thus, the boundary extrapolations (25) for particles outside of the boundary could be less accurate. 4. Since our methods are able to calculate the boundary values exactly, via post processing, our methods can accurately calculate the solutions and their derivatives inside and near the boundary of the domain as seen in Table Examples 5 and 6 of [19]statethattheirmeshlessBNMis unable to handle those problems whose traction boundary data have jumps at corners. Moreover, it is worse when the traction data have lager jump discontinuities. However, Table 4 shows that our method yields the exact boundary values even when the traction jumps are as big as [q] =256 at the corner points. 6. Since in the case (III) of Example 4.1, u Ɣ is a polynomial of degree 4, the RPP shape functions of reproducing order 2cannotexactlyinterpolateneitheru nor q as shown in Table 4. However,the RPP shape functions of reproducing order 4 generate x k, k = 0, 1, 2, 3, 4, and hence they can exactly capture the true boundary values as shown in Table 4. Thus far, we have shown that our methods are able to get the exact boundary values when the true solutions are harmonic polynomials. Table 1 Absolute error in maximum norm of the computed traction boundary values ( q) of the case (I) of Example 4.1 Methods RPP order DOF q q,ɣ Meshfree RPP (Sect. 3.1) E 12 Patchwise RPP (Sect. 3.2) E 13 MLS (Fig. 8 of [19]) 64 Larger than 0.5 The true traction boundary values (q) have jump discontinuities [ u ]= 4, at the corner points (4, 0) and (0, 4)

11 Table 2 Absolute errors of computed traction boundary data ( q) in maximum norm and computed Dirichlet data (ũ) of the case (II) of Example 4.1 Methods RPP order DOF u ũ,ɣ q q,ɣ Meshfree RPP (Sect. 3.1) E E 13 Patchwise RPP (Sect. 3.2) E E 14 MLS (Fig. 9 of [19]) Larger than 2.0 The true traction boundary values have jump discontinuities [ u ]=16, at the corner points (4, 0) and (0, 4) Table 3 Absolute errors of the computed solutions of the case (II) of Example 4.1 at interior points along the line y = 4 x connecting corner points (0, 4) and (4, 0) where q has jump discontinuities [ u ]=16 (x, y) u ũ u x ũ x u xx ũ xx (0.01, 3.99) 2.309E E E 8 (0.10, 3.90) 1.066E E E 8 (0.20, 3.80) 1.243E E E 9 (0.30, 3.70) 0.000E E E 9 (0.70, 3.30) 3.553E E E 9 (1.00, 3.00) 3.553E E E 8 (2.00, 2,00) 1.178E E E 9 (2.50, 1.50) 3.553E E E 9 (3.00, 1.00) 1.776E E E 8 (3.20, 0.80) 3.553E E E 9 (3.30, 0.70) 8.882E E E 8 (3.70, 0.30) 3.553E E E 8 (3.80, 0.20) 3.553E E E 8 (3.90, 0.10) 1.776E E E 8 (3.99, 0.01) 2.487E E E 9 u and ũ, respectively,representthetrueandthecomputedsolutions at (x, 4 x). u x and u xx are the first and the second derivative of u, respectively points of the domain. The computed solutions, its first and second derivatives are evaluated in Table 5. From Table 5, one can see that our method also gives good results at points near the boundary. 3. Since RPP shape functions for our method satisfy the Kronecker delta property, our method gives highly accurate computed solutions when the polynomial reproducing order is high (see, Table 5). 4. It is known that p-fem yields better results than h-fem at the same number of DOF [29] when the true solution is smooth. Similarly, the combination of RPP shape functions of high reproducing order and a small number of patches (the counterpart of p-fem) yields better results than the combination of RPP shape function of lower reproducing order and a large number of patches (the counterpart of h-fem) in general. Remark 4.1 In the Examples of this section, the normal derivatives (q) have no jumps inside any edges. In the case where q has a jump at a point inside an edge, that point should be treated as a vertex as we will do in the next section to deal with BIEs containing singularities. In the following example, we test the performance of patchwise RPBPM when the true solution is not a polynomial, but a smooth function (example 1 of [11]). Example 4.2 Consider Laplace s equation u = 0ona square =[ 1, 1] [ 1, 1] with essential boundary condition prescribed from the exact solution u(x, y) = sin(x 2 y 2 ) exp(2xy) on all boundary. From this example, we observe the following: 1. Even though the boundary values obtained by patchwise RPP shape functions are the same as the true boundary values at particles, the interpolation function by the RPP shape functions can not be the same as the true boundary values. Therefore, unlike the previous examples, our method can not exactly get the true boundary values. 2. However, the results in Table 5 show that our method is also yield very good approximation at various inside 5PatchwiseRSBPMforellipticproblemscontaining singularities 5.1 Laplace s equation with corner singularities In this section, we consider BEM for the Laplace equation with a corner singularity. An elliptic problem on the L-shaped domain (Fig. 5) has asingularityoftyper 2/3 f (θ) at the re-entrant corner point (0, 0) Example 5.1 (Laplace s equation on the L-shaped domain) Consider u = 0on =[ 1, 1] [0, 1] [ 1, 0] [0, 1] with displacement boundary condition prescribed from the exact solution r 2/3 sin(2θ/3) along all boundary. The exact solution has a corner singularity at (0, 0) of intensity 2/3. We construct RSP shape functions to deal with the corner singularity of type r 2/3

12 Table 4 Absolute error of the computed boundary values of the case (III) of Example 4.1 in maximum norm Methods RPP order DOF u ũ,ɣ q q,ɣ Patchwise RPP (Sect. 3.2) E E+00 Patchwise RPP (Sect. 3.2) E E 14 The true traction q has large jump discontinuities [ u ]=256, at the corner points (4, 0) and (0, 4) Table 5 Computed solutions (ũ), the true solution (u), relative errors in percentage (( (u ũ)/u 100), of Example 4.2 at various points inside of the domain (x, y) Type k = 4(DOF= 40) k = 6(DOF= 56) k = 8(DOF= 72) True solution (0, 0.2) u % 0.002% % (0, 0.2) u,x % % % (0, 0.2) u,xx % % % (0.4, 0.8) u % % % (0.4, 0.8) u,y % % % (0.4, 0.8) u,yy % 0.209% 0.114% (0.99, 0) u % 0.013% % (0.999, 0) u % 0.015% 0.001% (0.9999, 0) u % 0.048% 0.035% DOF degree of freedom (the total number of particles). k stands for the reproducing order of the RPP shape functions 1. Label the six edges of as follows: Ɣ 1, Ɣ 2, Ɣ 3, Ɣ 4, Ɣ 5, Ɣ 6, respectively, that are the line segments on y = 0, x = 1, y = 1, x = 1, y = 1, x = 0, and define PU functions on the edge Ɣ 1 as follows: 1 ( ) if x (0, b δ] ψ 0 (x) = φg R x (b δ) n 2δ if x [b δ, b + δ] 0 if x / (0, b + δ], ( ) φ L x (b+δ) g n 2δ if x [b δ, b + δ] ψ 1 (x) = 1 if x [b + δ, 1] 0 if x / [b δ, 1], (32) (33) Then ψ 0 (x) + ψ 1 (x) = 1, for all x Ɣ Suppose we choose N = 3forEq.(29)andδ = Let L 0 j (ξ), j = 0, 1,...,6, be the Lagrange interpolating polynomials associated to the nodes: 0< ξ 00 < ξ 01 < ξ 02 < ξ 03 <ξ 04 <ξ 05 < ξ 06 = (b+ δ) 1/3. Let L 1 j (ξ), j = 0, 1,...,6, be the Lagrange interpolating polynomials associated to the nodes: (b δ) 1/3 = ξ 10 < ξ 11 < ξ 12 < ξ 13 < ξ 14 < ξ 15 < ξ 16 = 0.99 < 1 For each k = 0, 1, and each j = 0, 1,...,6, define RSP shape functions to deal with the corner singularity of intensity 2/3 by kj (x) = ψ k (x) L kj (x 1/3 ). Then, for an approximation of boundary functions on Ɣ 1, we use u Ɣ1 = 1 k=0 j=0 6 u kj kj (x) (34)

13 1 6 q Ɣ1 = q kj kj (x)/(x 1/3 ). (35) k=0 j=0 ( 1, 1) q = 0 Γ 4 q = 0 Γ 3 (1, 1) Note that L 0 j (x 1/3 ), j = 0, 1,...,6, generates q = 0 Γ 5 Γ 2 u = 500 1, x 1/3, x 2/3, x, x 4/3, x 5/3, x 2. and L 0 j (x 1/3 )/(x 1/3 ), j = 0, 1,...,6, generates Γ 6 ( 1, 0) (0, 0) u = 0 Γ 1 q = 0 (1, 0) x 1/3, 1, x 1/3, x 2/3, x, x 4/3, x 5/3. These two sets of functions are able to approximate r 2/3 sin(2θ/3) and its derivative, respectively. Similarly, RSP shape functions are constructed to approximate the boundary functions along Ɣ 6. Let us note that the vertices (0, 0), (1, 0), (0, 1) should not be particles for RSP shape functions. 3. We use RPP shape functions for an approximation of boundary functions on the edges Ɣ k, k = 2, 3, 4, 5. Similarly, vertices (1, 1), ( 1, 1), ( 1, 1), (0, 1) should not be particles for RPP shape functions on Ɣ 2, Ɣ 3, Ɣ 4, Ɣ The error of the computed traction q and the error of the computed solution ũ in maximum norm on entire domain are shown in Table 6. Table6 shows that the solutions obtained by using mapping techniques (RSBPM) are far better than those obtained by RPBPM that does not use mapping techniques. 5.2 The Motz problem that contains a jump boundary data singularity Consider the Motz problem, the bench marking problem that contains a jump boundary data singularity of type O(r 1/2 ) at (0, 0).For numerical solutions of this problem, many computational techniques are suggested in the literature. Especially, we refer to [1,17] thatiscloselyrelatedtothispaper. Example 5.2 (Motz problem) Consider u = 0 on a rectangular domain =[ 1, 1] [0, 1] with mixed boundary Table 6 Errors in maximum norm of Laplace s equation on the L-shaped domain with Dirichlet boundary conditions prescribed by the true solution u = r 2/3 sin(2θ/3) k DOF u ũ,ɣ q q,ɣ u ũ, 4 60 RSBPM 3.766E E E 5 RPBPM 4.031E E E RSBPM 1.663E E E 6 RPBPM 1.530E E E 3 Fig. 6 The rectangular domain for the Motz problem conditions prescribed as follows: u = 0onƔ 6, u = 500 on Ɣ 2 u 5 = 0onƔ 1 Ɣ i, i=3 where Ɣ 1 =[0, 1] {0}, Ɣ 2 ={1} [0, 1], Ɣ 3 =[0, 1] {1}, Ɣ 4 =[ 1, 0] {1}, Ɣ 5 ={ 1} [0, 1], Ɣ 6 =[ 1, 0] {0}, as shown in Fig. 6. Then,theasymptoticexpansionof the true solution in a polar coordinates centered at (0, 0) is of the form ( u(r, θ) = A k r 2k 1 2 cos (2k 1) θ ). (36) 2 k=1 For each k 50, the coefficients A k are computed in appendix by using the least squares method. We construct the RSP shape functions to approximate singular boundary functions on Ɣ 1 and Ɣ 6 and the RPP shape functions to approximate smooth boundary functions on Ɣ 2, Ɣ 3, Ɣ 4 and Ɣ For example, we choose N = 3forEq.(29), δ = 0.05 and b = Let L 0 j (ξ), j = 0, 1,...,6, be the Lagrange interpolating polynomials associated to the nodes: 0 < ξ 00 < ξ 01 < ξ 02 < ξ 03 < ξ 04 < ξ 05 < ξ 06 = (b + δ) 1/2. where ξ 06 ξ 05 >(2δ) 1/2. Let L 1 j (ξ), j = 0, 1,...,6, be the Lagrange interpolating polynomials associated to the nodes: (b δ) 1/2 = ξ 10 < ξ 11 < ξ 12 < ξ 13 < ξ 14 < ξ 15 < ξ 16 = 0.99 < 1, where ξ 11 ξ 10 >(2δ) 1/2. Suppose ψ 0 and ψ 1 are PU functions defined by (32)and (33), respectively. For each k = 0, 1, and each

14 j = 0, 1,...,6, we define singular approximation functions by kj (x) = ψ k (x) L kj (x 1/2 ). Then, for an approximation of boundary functions on Ɣ 1, we use u Ɣ1 = q Ɣ1 = 1 k=0 j=0 1 k=0 j=0 6 u kj kj (x) (37) 6 [ q kj kj (x)/ x ]. (38) 2. Then, L 0 j ( x), j = 0, 1,...,6andL 1 j ( x), j = 0, 1,...,6areRSPshapefunctionscorrespondingtonodes (ξ 0 j ) 2, j = 0, 1,...,6and(ξ 1 j ) 2, j = 0, 1,...,6, respectively, which reproduce 1, x, x 2, x 3, x 1/2, x 3/2, x 5/2, (39) over the interval (0, 1]. Moreover, for each k = 0, 1, the set of functions {L kj ( x)/ x : j = 0, 1,...,6} generate x 1/2, x 1/2, x 3/2, x 5/2, 1, x, x 2. (40) u Ɣ1 O(x 1/2 ) and q Ɣ1 O(x 1/2 ). Thus, comparing (39) with(40), one can see that the particle shape functions, [ kj (x)/ x ], k = 0, 1; j = 0, 1,...,6, can be used for the approximations of both u Ɣ1 and q Ɣ1 instead of (37) and(38). 3. Noting that for x Ɣ 6 = ( 1, 0), 0 < x and hence RSP shape functions for particles on Ɣ 6 are constructed as follows: For each k = 0, 1, and each j = 0, 1,...,6, let kj (x) = ψ k ( x) L kj (( x) 1/2 ). Then, for an approximation of boundary functions on Ɣ 6, we use u Ɣ6 = q Ɣ6 = 1 k=0 j=0 1 k=0 j=0 6 ũ kj kj (x) (41) 6 [ q kj ψ k ( x) L kj ( ( x))/ ] ( x). (42) The vertices (0, 0), ( 1, 0), (0, 1) should not be particles for neither RSP nor RPP shape functions. 4. We use RPP shape functions for an approximation of boundary functions on the edges Ɣ k, k = 2, 3, 4, 5. Similarly, vertices (1, 0), (1, 1), ( 1, 1), ( 1, 0) should not be particles for RPP shape functions on Ɣ 2, Ɣ 3, Ɣ 4, Ɣ 5. That is, for particles on the edges that do not contain singularity, we assign RPP shape functions constructed in Sect We observe the following: (i) The true solution of motz problem is unknown. However, for the calculation of errors in Tables 7 and 8, we assume that the computed true solution (u) is the sum of first 50 terms in Eq. (36) withcoefficientsin Table 12 and the true normal derivative (q) is Eq. (48) with coefficients in Table 12. (ii) Table 7 shows the maximum errors of q and u along the boundary. Table 8 shows the maximum errors of the computed solution (ũ) in the whole domain. Tables 7 and 8 show that RSBPM effectively handles BIEs with ajumpboundarydatasingularity.forcomparingpurposes, parts of Table A.II of [17] arereproducedin Table 8. Themaximumerror u ũ, of the computed solutions obtained by RSBPM are compared with those results (Table A.II of [17]) obtained by the p-fem coupled with method of auxiliary mapping (MAM), developed by Oh and Babuška [20]. Moreover, from Table A.1 of [17], we observe the following results: u ũ, = 3.52 when DOF= 525, 825 using ELLPACK (h-fem code with piecewise linear basis) Table 7 Errors in maximum norm of q and u along the boundary of the Motz problem for k = 4, 6, 8, the order of reproducing polynomial k DOF u ũ,ɣ q q,ɣ 4 80 RSBPM 2.585E E RSBPM 1.396E E RSBPM 6.979E E 4 DOF stands for the total number of particles Table 8 Errors in maximum norm on the inside of the domain of the Motz problem for k = 4, 6, 8, the order of reproducing polynomial RSBPM RPP order k p-fem with MAM DOF u ũ, p-degree DOF u ũ, E E E E E E 6 DOF stands for the total number of particles. RSBPM is compared with the results by p-fem with mapping technique