Effect of Numerical Integration on Meshless Methods

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2 Effect of Numerical Integration on Mesless Metods Ivo Babuška Uday Banerjee Jon E. Osborn Qingui Zang Abstract In tis paper, we present te effect of numerical integration on mesless metods wit sape functions tat reproduce polynomials of degree k 1. Te mesless metod was used on a second order Neumann problem and we derived an estimate for te energy norm of te error between te exact solution and te approximate solution from te mesless metod under te presence of numerical integration. Tis estimate was obtained under te assumption tat te numerical integration sceme satisfied a form of Green s formula. We also indicated ow to obtain numerical integration scemes satisfying tis property. Keywords: Galerkin metods; mesless metods; quadrature; numerical integration; error estimates 1 Introduction Mesless Metods (MM) were developed in early 90s for numerically solving partial differential equations (PDE). Tis initiative was stimulated by te difficulties in mes generation wen available metods, e.g., te Finite Element Metod (FEM), were used to solve various complex problems in engineering. It was recognized from te very beginning of te development of MM tat numerical integration posed bigger callenge in tis metod tan te FEM, and te issue was discussed in various engineering papers, e.g., [5],[8],[9],[11],[12],[10], Institute for Computational Engineering and Sciences, ACE 6.412, University of Texas at Austin, Austin, TX Tis researc was partially supported by NSF Grant # Department of Matematics, 215 Carnegie, Syracuse University, Syracuse, NY address: banerjee@syr.edu. ttp://banerjee.syr.edu. Tis researc was partially supported by te NSF Grant # DMS Department of Matematics, University of Maryland, College Park, MD address: jeo@mat.umd.edu. WWW ome page URL: ttp:// jeo. Tis researc was supported by NSF Grant # Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzou, , P. R. Cina. address: zqing@mail2.sysu.edu.cn. Tis researc was partially supported by te NSF Grant # DMS

3 [14],[15],[16]. In FEM, te sape functions are piecewise polynomials of degree k 1 and a careful matematical analysis of te effect of numerical integration in FEM was publised 30 years ago in [13]. Te analysis required tat te numerical integration in FEM, wen applied to PDEs wit constant coefficients, must evaluate te stiffness matrix exactly. Tis is easily acieved since te integrands of te elements of te stiffness matrix of FEM are polynomials of degree 2k 2. Te analysis also exploited te fact tat l t order derivatives of te sape functions vanis locally (on eac triangle) for l k + 1. In contrast, te sape functions used in MM are not piecewise polynomials and teir l t order derivatives grow wit l. Moreover, te stiffness matrix cannot be evaluated exactly for PDEs wit constant coefficients. Tus te sape functions used in MM lack te two most important features of te sape functions of FEM. Numerical integration in MM is a bigger callenge primarily because of te lack of tese features. Many interesting ideas on te use of numerical integration in MM were presented in te engineering papers mentioned above, but to te best of our knowledge, a careful matematical analysis of te effect of numerical integration in MM was first reported in [4]. It is sown in tis paper tat te error in te approximate solution, obtained from MM wit standard standard numerical quadrature, does not converge. It is ten sown tat if te stiffness matrix satisfies a condition referred to as te zero row sum condition, te energy norm of te error in te approximate solution is O( + η), were is te standard discretization parameter related to te diameters of te supports of te sape functions and η is te parameter indicating te accuracy of te underlying numerical quadrature. Tus MM, wit numerical integration, does not yield optimal order of convergence unless η = O(). However, te analysis in [4] uses an assumption on te approximation space tat is difficult to verify. We furter note tat te analysis is restricted to MMs wit sape functions tat reproduced polynomials of degree k = 1; it is not clear tat te analysis can be extended to k > 1. In tis paper, we present a matematical analysis of te effect of numerical integration on MM, were te quadrature is required to satisfy certain conditions tat are different from tose required in [4]. We also indicate ow to obtain numerical quadrature scemes satisfying tese conditions. Moreover, in contrast to [4], te analysis presented in tis paper is valid for k 1. We ave sown in tis paper tat te energy norm of te error in te approximate solution obtained from MM wit numerical quadrature (satisfying certain conditions) is O( k 1 (+η)), were η is a parameter related to te accuracy of te numerical quadrature and is te standard discretization parameter. Tus MM does not yield optimal order of convergence for η O(). Certainly if η = O(), we ave te optimal order of convergence. It is important to note tat te parameter η (see (3.10)) associated wit te particular numerical integration used in te FEM, namely te Gauss rule, is O(). We mention tat te numerical integration used in tis paper yields a non-symmetric stiffness matrix. But tis does not pose a serious problem since non-symmetric linear systems could be solved efficiently by iterative metods. 2

4 We address te application of MM on a second order Neumann boundary value problem in tis paper. Te outline of tis paper is as follows: In Section 2, we present te preliminaries, a variational formulation based on Lagrange multipliers and te associated MM. In Section 3, we present a numerical quadrature sceme, togeter wit associated assumptions on te sceme. We present our main results in Section 4, wic are Teorems 4.1 and 4.2. In Section 5, we present a procedure tat indicate ow to obtain quadrature scemes satisfying an assumption given in Section 3. We also present numerical experiments in tis section to illuminate our main results presented in Section 4. Some of tese numerical experiments also indicate te necessity of one of te main assumptions on te quadrature given in Section 3. We provide a few remarks and a brief summary of te paper in Section 6. 2 Preliminaries and Mesless Metod Let Ω R d be a bounded domain wit Lipscitz continuous boundary Γ Ω. We denote te usual Sobolev space by W m,p (Ω) wit te norm and semi-norm, u W m,p (Ω) and u W m,p (Ω) respectively. We will consider only p = 2 and in tis paper; W m,2 (Ω) will be denoted by H m (Ω). Moreover, u L2(Ω), u L (Ω), u L2(Γ), and u L (Γ) will denote te usual norms on L 2 (Ω), L (Ω), L 2 (Γ), and L (Γ) respectively. Exact Problem: We consider te standard Neumann problem u = f in Ω, u = g, n on Γ Ω, (2.1) were n is te outward normal derivative to Γ and f L 2(Ω), g L 2 (Γ) satisfy te compatibility condition f(x)dx + g(s)ds = 0. (2.2) Ω Te associated variational formulation of (2.1) is given by were B(u, v) Ω Find u H 1 (Ω) satisfying, Γ B(u, v) = L(v), v H 1 (Ω), (2.3) u v dx and L(v) Ω f v dx + g v ds. Γ Te compatibility condition (2.2) can be written as L(1) = 0. It is well known tat te problem (2.3) as a unique solution up to a constant. A standard way of specifying a unique solution is to consider a linear functional Φ : L 2 (Ω) R wit 3

5 Φ(1) > 0 and seek te unique solution u satisfying Φ(u) = 0. Te functional Φ(u), for example, could be cosen to be Φ(u) = 1 Ω Ω u dx or Φ(u) = Ω ϕu dx, were ϕ(x) is smoot. Let H Φ = { (v, µ) H 1 (Ω) R : (v, µ) 2 H Φ v 2 H 1 (Ω) + Φ(v) 2 + µ 2 < }. H Φ is a Hilbert space and it is easy to sow tat tere exist positive constants C 1, C 2, suc tat were C 1 (v, µ) 2 H Φ v 2 H 1 (Ω) + µ2 C 2 (v, µ) 2 H Φ, We consider an alternate variational problem given by Find (u, λ) H Φ satisfying, (v, µ) H Φ. (2.4) B Φ (u, λ; v, µ) = L(v), (v, µ) H Φ, (2.5) B Φ (u, λ; v, µ) B(u, v) + λφ(v) + µφ(u) and B(u, v) and L(v) are defined above. Remark 2.1 We note tat te problem (2.5) can equivalently be written as te system B(u, v) + λφ(v) = L(v), v H 1 (Ω) µφ(u) = 0, µ R. Te second equation gives te constraint Φ(u) = 0. Moreover, it is well known tat te first equation is te Euler-Lagrange equation for te constrained extremal problem min J(v), v H 1 (Ω) Φ(v)=0 were J(v) = 1 2B(v, v) L(v). λ is te Lagrange multiplier and te problem (2.5) is known as te variational problem based on Lagrange multiplier. To establis tat te problem (2.5) as a unique solution, we present te following result. Lemma 2.1 (a) Tere is a constant C > 0 suc tat B Φ (u, λ; v, µ) C (u, λ) HΦ (v, µ) HΦ, (u, λ), (v, µ) H Φ. (2.6) (b) Tere exists C > 0 suc tat C < inf (u,λ) H Φ B Φ (u, λ; v, µ) sup. (2.7) (v,µ) H Φ (u, λ) HΦ (v, µ) HΦ 4

6 (c) For any (v, µ) H Φ satisfying (v, µ) HΦ 0, 0 < sup (u,λ) H Φ B Φ (u, λ; v, µ). Proof: (a) Tis follows directly from te Caucy-Scwartz inequality. (b) We sow tat for a given (u, λ) H Φ, we can coose (v, µ) H Φ suc tat B Φ (u, λ; v, µ) C (u, λ) 2 H Φ and (v, µ) HΦ C (u, λ) HΦ. We coose v = u + λ and µ = λ + Φ(u). Ten and B Φ (u, λ; v, µ) = (v, µ) 2 H Φ = v 2 1,Ω + Φ(v)2 + µ 2 B(u, v) + λφ(v) + µφ(u) = B(u, u + λ) + λφ(u + λ) + [ λ + Φ(u) ] Φ(u) = u 2 1,Ω + Φ(u)2 + λ 2 Φ(1) C (u, λ) 2 H Φ, = u + λ 2 1,Ω + [ Φ(u + λ) ] 2 + [ λ + Φ(u) ] 2 = u 2 1,Ω + 2Φ(u) 2 + λ 2 Φ(1) 2 + 2Φ(1)λΦ(u) 2λΦ(u) + λ 2 C [ u 2 1,Ω + Φ(u)2 + λ 2] = C (u, λ) 2 H Φ. Estimate (2.7) follows from tese two inequalities. (c) For a given (v, µ) H Φ, we coose u = v + µ and λ = µ + Φ(v). Using a similar calculation as used in te first part of te proof of (b), we get te desired result. It now follows from Teorem in [1] tat te problem (2.5) as a unique solution. Remark 2.2 We note tat te problem (2.5) as a unique solution (u, λ) for any f L 2 (Ω) and g L 2 (Γ). Let te linear functional Φ(v) be given by Φ(v) ϕv dx, were ϕ(x) is smoot. Ten, if u is smoot, it can be sown Ω tat u is te unique (strong) solution of te Neumann problem u = f u n = g, L(1) ϕ in Ω, (2.8) ϕdx Ω on Γ wit Φ(u) = 0 (see [6]). It can also be sown tat λ = L(1)/Φ(1). If f and g satisfy te compatibility condition (2.2), i.e., L(1) = 0, ten it is clear from (2.8) tat u is te solution of te original Neumann problem (2.1) wit Φ(u) = 0. 5

7 Remark 2.3 Consider te variational problem (2.5), were we assumed tat L(1) = 0. Substituting v = 1 in (2.5), it is easy to see tat λ = 0 and we can also sow tat te problem (2.5) is equivalent to te problem Find u H 1 (Ω) suc tat B(u, v) = L(v) and Φ(u) = 0, v H 1 (Ω). (2.9) Remark 2.4 Te variational formulation (2.5) of te Neumann problem (2.1)- (2.2) is different tan te standard variational formulation used in te literature [7]. We note tat small perturbations in te input data (e.g., caused by te round-off error) or te quadrature error will disturb te compatibility condition (2.2). It is well known tat te compatibility condition is necessary for te existence of te solution of te Neumann problem, and tus te standard variational formulation of te Neumann problem is not well-posed witout a constraint on te perturbation. In contrast, te formulation (2.5) is well-posed witout any constraint on te perturbation of data. We furter note tat tere is obvious freedom in te selection of Φ. Discretization: In order to discretize te variational problem (2.5) by a mesless metod, we consider V H 1 (Ω), a one-parameter family of finite dimensional spaces, given by V = span{φ j : j N }; N is an index set. Te sape functions {φ j (x)} j N are linearly independent. Moreover, φ j s ave compact support and (in a mesless metod) teir construction eiter does not depend, or depends only minimally, on a mes. We let ωj Ω be te interior of te supp φ j. We assume tat ω j is star-saped wit respect to a ball o j ω j and tere exists a constant C > 0 suc tat diam(ω j ) diam(o j ) C, j N. For te definition of star-saped domains wit respect to a ball, we refer to [7]. Often, a sape function φ i (x) is associated wit a particle x i Rd and it is assumed tat te particles are distinct i.e., x i x j if i j. We note tat wen ω i Γ =, ten te associated particle x i ωi Ω. But wen ω i Γ, ten te associated particle x i could be outside Ω. We also divide te set N into two sets, namely, N : ω i Ω} (2.10) N = {i N : ω i Γ } (2.11) We note tat N = N N and N N =. We set N = card N. We now make te following assumptions on te subspace V. 6

8 A1: (finite overlap) For i N, let S i be te set of indices j suc tat ω i ω j. Tere is a constant κ, independent of i and, suc tat cards i κ. (2.12) A2: Tere are positive constants C, C 1, and C 2, independent of i and, suc tat D α φ i L (Ω) C α, 0 α q for some q 1, (α is a multi-index), (2.13) C 1 d ω i C 2 d and C 1 d 1 ω i Γ C 2 d 1, (2.14) were ω i is te area of ω i in R d and ω i Γ is te lengt of ω i Γ in R d 1. A3: Tere are positive constants C 1 and C 2, independent of and i, suc tat C 1 diam(ω i ) C 2. (2.15) A4: Tere are positive constants C 1 and C 2, independent of and i, suc tat C 1 v 2 L 2(Ω) d vi 2 C 2 v 2 L, 2(Ω) i N (2.16) C 1 v 2 L 2(Γ) d 1 vi 2 C 2 v 2 L, 2(Γ) (2.17) i N C 1 v 2 H 1 (ω i) d 2 j S i (v j v i ) 2 C 2 v 2 H 1 (ω i), i N, (2.18) for all v = i N v i φ i V. A5: Te sape functions reproduce polynomials of degree k, i.e., i N p(x i )φ i (x) = p(x), p Pk (Ω) and x Ω, (2.19) were P k is te space of polynomials of degree k. Remark 2.5 Te assumption A3 implies te first statement of (2.14). However, we stated tem separately since we ave used tese statements in tis paper. We next note tat te inequality (2.16) in A4 implies a strengtened uniform version of linear independence of te sape functions {φ i }. Also note tat (2.16) and (2.17) in assumption A4 imply tat N C d, N C d, and N C (d 1) (2.20) provided te function v = 1 V. Here te notation A B means tat tere are constants C 1, C 2, independent of suc tat C 1 A / B C 2. It is clear from assumption A5 (take p(x) = 1 in (2.19)) tat 1 V and terefore 7

9 {φ i } form a partition of unity. We mention tat te particles {x i } are used in te construction of sape functions satisfying (2.19) (see [18]). We furter note tat it is possible to prove te inequalities (2.16), (2.17), and (2.18) in certain situations for special distributions of te particles {x i }; also see Remark 4.3 in [4]. Tese proofs require te assumption (2.15). Examples of subspaces V satisfying tese assumptions can be found in [2, 17, 20]. We mention tat it is easy to construct smoot sape functions {φ i }, e.g., RKP sape functions wit respect to a smoot weigt function (see [19, 17, 18]). We assume tat φ i Ck+1 (Ω) for all i N. In te rest of te paper, we will write x j, ω j, φ j as x j, ω j, φ j, respectively, for notational simplicity, wit te understanding tat tey depend on. We will use a Lagrange multiplier metod to determine a unique approximate solution for problem (2.5). To tis end, we define a linear functional on V by (v ) = Φ(1) v i, v = v i φ i V. N i N i N is different from Φ, but note tat (1) = Φ(1). Also is a bounded linear functional since (v ) Φ(1) N ( i N 1) 1 2 ( i N v 2 i ) 1 2 C d d 2 d 2 v L2(Ω) = C v L2(Ω), (2.21) were te last inequality is obtained using (2.16) and (2.20). We consider V { (v, µ) V R : (v, µ) 2 V v 2 H 1 (Ω) + (v ) 2 + µ 2 < }. V is a Hilbert space and we can sow tat tere are constants C 1, C 2, independent of, suc tat C 1 (v, µ) 2 V v 2 H 1 + µ 2 C 2 (v, µ) 2 V, (v, µ) V. (2.22) Terefore from (2.4) we see tat te norms (, ) V and (, ) HΦ are equivalent on V and te associated constants are independent of. We note tat te linear functional is not well defined on H 1 (Ω), but is well defined on V. Mesless Metod: A mesless metod to approximate te solution of (2.5) is a Galerkin metod Find (u, λ ) V satisfying B (u, λ ; v, µ) = L(v), (v, µ) V, (2.23) 8

10 were B (u, λ ; v, µ) B(u, v) + λ (v) + µ(u ). We will give a rationale for using (in place of Φ) in 2.23 later in Remark 3.1. Let z C(Ω). We define I z V, called te V -interpolant of z, as I z i N z(x i )φ i (x). (2.24) Strictly speaking, I z is a quasi-interpolant of z, since I z(x j ) z(x j ). We furter note tat te function z as to be suitably extended outside Ω in order to define I z, since some of te particles x i may be outside Ω. Te extension of z to define I z as been discussed in [3]. In te following result, we present te interpolation error estimate. Lemma 2.2 Let z W k+1,p (Ω) C(Ω), p = 2,. Ten tere exists a constant C, independent of z and suc tat z I z W s,p (Ω) C k+1 s z W k+1,p (Ω). (2.25) Te proof of tis teorem can be found in [17, 2]. Remark 2.6 It is important to note tat te proof of Lemma 2.2 needs an additional assumption. We assume tat for eac i N, tere is a ball B i of diameter ρ suc tat j S i ω j B i, were ρ 1 is independent of i. We furter note tat te proof also depends on te assumption A5, wic in turn imposes certain restrictions on te distribution of te particles {x i }, wic we do not elaborate ere. To address te existence and uniqueness of te solution of te problem (2.23), we state te following result: Lemma 2.3 (a) Tere is a constant C > 0, independent of, suc tat B (w, ν; v, µ) C (w, ν) V (v, µ) V, (w, ν), (v, µ) V. (b) Tere exists C > 0, independent of, suc tat C < inf (w,ν) V sup (v,µ) V B (w, ν; v, µ) (w, ν) V (v, µ) V. (c) For any (v, µ) V satisfying (v, µ) V 0, 0 < sup B (w, ν; v, µ). (w,ν) V 9

11 Te proof of tis result depends on te fact tat V contains constants (because of assumption A5) and follows te same arguments used in te proof of Lemma 2.1. Remark 2.7 Since V contains constants, by considering v = 1 in te variational problem (2.23) it is clear tat λ = 0 and we can also sow tat te problem (2.23) is equivalent to Find u V suc tat B(u, v ) = L(v ) and (u ) = 0, v V. (2.26) We note tat te constraint (u ) = 0 gives a non-singular stiffness matrix (wic will oterwise be singular). Tis feature, possibly wit a different coice of, is always incorporated in a standard FEM code. Now it is immediate from (2.9) and (2.26), B(u u, v ) = 0, v V and terefore, using Lemma 2.3, u u H1 (Ω) inf v V u v H1 (Ω) C k u H k+1 (Ω). (2.27) Tus u converges to u only in te energy norm, i.e., te H 1 -seminorm. Moreover using (2.25), we also obtain u I u H 1 (Ω) u u H 1 (Ω) + u I u H 1 (Ω) C k u H k+1 (Ω), (2.28) wic will be used later in te paper. 3 Numerical Integration in Mesless Metod To motivate te quadrature in te mesless metod, we first look at te problem (2.23) in detail. We write u, te solution of (2.23), as u = j N c j φ j. Ten te problem (2.23) can be written as γ ij c j + λ (φ i ) + µ Φ(1) c j = l i, for i N, N j N j N were γ ij = B(φ j, φ i ) = = φ j φ i dx Ω φ j φ i dx = φ j φ i dx ω j ω i ω i 10

12 and l i = L(φ i ) = = fφ i dx + gφ i ds Ω Γ fφ i dx + ω i gφ i ds f i + g i. Γ ω i (3.1) Te integrals γ ij, f i, and g i are computed using quadrature. We define γ ij = ω i φ j φ i dx (3.2) and l i = ω i fφ i dx + Γ ω i gφ i ds f i + g i, (3.3) were ffl represents te numerically computed integral. We note tat te matrix {γij } is not symmetric, since γ ij = ω i φ j φ i dx ω j φ i φ j dx = γ ji. We next note tat for v = j N v j φ j and w = i N w i φ i in V, we ave B(v, w) = γ ij v j w i i,j N and L(v) = f i v i + g i v i. i N i N So we naturally define and B (v, w) = γij v jw i i,j N L (v) = fi v i + gi v i. (3.4) i N i N Under tis definition, te form B (, ) is bilinear on V V and L ( ) is linear on V. Since {γ ij } is not symmetric, it is clear tat B (v, w) is also not symmetric. Moreover, it can be easily sown tat j N γ ij = 0, i N, (3.5) i.e., te row-sum of te matrix {γij } is 0, wic implies tat B (1, w) = 0, w V. (3.6) 11

13 But, in general, te column-sum of {γij } is not 0 and tere is v V suc tat B (v, 1) 0. We also observe tat L (1) = fi + gi 0. (3.7) i N i N It is important to note tat several oter approaces for performing numerical integration ave been proposed in te literature to approximate γ ij. We mention some of tem below: (i) Te domain Ω is partitioned into small and appropriately selected subdomains suc tat eac ω i and ω i ω j are also unions of tese subdomains [16]. Numerical integration is ten performed on eac of tese subdomains. Te coice of subdomains in tis approac allows te numerical integration of only smoot functions. Te computed stiffness matrix {γij } is also symmetric, but partitioning process could be expensive wen ω i s are not simplices. (ii) In a procedure proposed in [11, 12, 10], te domain Ω is partitioned into subdomains suc tat eac ω i and ω i ω j are also unions of tese subdomains. Moreover, tese subdomains contain exactly one particle x j. Te main feature of tis procedure is tat te numerical integration is performed only on te boundary of te subdomains. Also te matrix {γij } is symmetric. Te effectiveness of tis approac for te lowest order, i.e., wen k = 1 in (2.19), was sown in [11, 12]; te iger order metods, i.e., k 2 in (2.19), were addressed in [10]. (iii) Te element γ ij of te te stiffness matrix could be approximated by performing numerical integration on ω i ω j. Tis approac was used in [14, 15], were te domains ω i s were considered as speres. Consequently, te domains ω i ω j were lens-saped and special quadrature formulae were developed to numerically integrate over suc lens-saped domains. Te matrix {γij }, obtained from tis procedure, is symmetric, but te main drawback of is tat te matrix {γij } does not satisfy te zero row sum condition (3.5). (iv) Numerical integration on ω i ω j, togeter wit a simple correction, to approximate γ ij was suggested in [4]. Tis correction ensured te zero row sum condition (3.5) for te matrix {γij }; te matrix was also symmetric. Te matematical analysis presented in tis paper required a certain assumption on te discretization, wic is not easy to ceck. Also te analysis is valid for sape functions satisfying k = 1 in (2.19); it could not be generalized for k 2. We note tat rigorous matematical error analysis is not available for te procedures described in (i) (iii). In tis paper, we considered numerical integration on ω i (not on ω i ω j ) of te form (3.2) (3.3) to approximate γ ij and, unlike [4], our error analysis is valid for all k 1 in (2.19). Te mesless metod wit quadrature to approximate te solution u of te problem (2.5) is given by Find (u, λ ) V satisfying B (u, λ ; v, µ) = L (v), (v, µ) V, (3.8) 12

14 were B (w, ν; v, µ) B (w, v) + ν(v) + µ(w) (3.9) and B (, ) and L ( ) are as defined in (3.4). We refer to u as te quadrature approximation to u. It is clear from (3.7) tat te compatibility condition is not satisfied. Remark 3.1 We note tat te main reason for using ( ) in (2.23) and (3.8) instead of Φ( ), is tat te numerical approximation of te linear functional Φ( ) is not required in (3.8). We assume tat te numerical quadrature satisfies te conditions described below: QA1. Tere exist positive constants η and τ, small enoug and independent of i and, suc tat dx dx η ωi L (ω i) (3.10) ω i ω i and ω i Γ ϑ ds ϑ ds τ ωi Γ ϑ L ( ω i Γ), (3.11) ω i Γ for a class of functions W m1, (ω i ) and ϑ W m2, ( ω i Γ) satisfying and D α L (ω i) C [ diam(ω i ) ] α L (ω i), α m 1 (3.12) D α ϑ L ( ω i Γ) C [ diam(ω i ) ] α ϑ L ( ω i Γ), α m 2 (3.13) were C > 0 is independent of i and m 1, m 2 1 may depend on te numerical quadrature as well as on q in assumption A2. QA2. (a) Tere is a constant C > 0, independent of, suc tat B (w, ν; v, µ) C (w, ν) V (v, µ) V, (w, ν), (v, µ) V. (3.14) (b) Tere exists C > 0, independent of, suc tat C < inf (w,ν) V sup (v,µ) V B (w, ν; v, µ). (3.15) (w, ν) V (v, µ) V (c) For any (v, µ) V satisfying (v, µ) V 0, 0 < sup B(w, ν; v, µ). (3.16) (w,ν) V 13

15 QA3. For eac i N, let G i : C2 (ω i ) R be a linear functional given by G i (v) = ω i v φ i dx + ω i v φ i dx v n φ i ds, (3.17) ω i Γ were n is te outward normal to ω i Γ. We assume tat G i (p) = 0, p P k and i N, (3.18) were P k is te space of polynomials of degree k. We first note tat te assumption QA2, i.e., (3.14), (3.15), and (3.16) ensures tat te problem (3.8) as a unique solution. In te following lemma, we sow tat (3.14), (3.15), and (3.16) old under a somewat restricted condition on te parameter η. Lemma 3.1 Suppose tere is a positive constant C suc tat te quadrature satisfies (3.10) wit η C. Ten B (w, ν; v, µ) is bounded, and for C small enoug, B (w, ν; v, µ) satisfies te inf-sup conditions, i.e., (3.15) and (3.16) are satisfied. Proof: Let w = i N w i φ i, v = i N v i φ i V. We first estimate B(w, v) B (w, v). Recalling te definition of η in (3.10) and using (3.6), (2.18), we ave for i N, B(w, φi ) B (w, φ i ) = w φ i dx w φ i dx ω i ω i = (w w i ) φ i dx (w w i ) φ i dx ω i ω i w j w i φ j φ i dx φ j φ i dx j S ω i ω i i η ω i φ j φ i L (ω i) ( j S i w j w i 2 ) 1 2 κ C 2 η d 2 d 2 2 w H 1 (ω i) κ C η d 2 1 w H1 (ω i), (3.19) were we used (2.13) wit α = 1. Terefore, squaring bot sides of te above inequality and summing over i N, we get [ B(w, φi ) B (w, φ i ) ] 2 C η 2 d 2 w 2 H 1 (ω i) i N i N C η 2 d 2 w 2 H 1 (Ω). 14

16 Tus, recalling tat v = i N v i φ i and using (2.16), we get B(w, v) B (w, v) = [ B(w, φi ) B (w, φ i ) ] i N v i ( i N v 2 i ) 1/2 [ [ B(w, φi ) B (w, φ i ) ] ] 2 1/2 i N = C η d 2 1 w H 1 (Ω) ( i N v 2 i ) 1/2 C η 1 w H 1 (Ω) v L2(Ω). (3.20) We now prove te boundedness of B, i.e., (3.14). From te definition (3.9) of B (w, ν; v, µ), we get B (w, ν; v, µ) = B (w, ν; v, µ) [ B(w, v) B (w, v) ]. (3.21) Terefore using (3.20) and part (a) of Lemma 2.3, we get B (w, ν; v, µ) B (w, ν; v, µ) + B(w, v) B (w, v) C 1 (w, ν) V (v, µ) V + C 2 η 1 w H 1 (Ω) v L2(Ω) C(1 + η 1 ) (w, ν) V (v, µ) V. Tus by taking η C (C does not ave to be small enoug) in te above inequality, we get (3.14). We now prove te inf-sup condition (3.15). For a given (w, ν) V, we coose v = w+ν V and µ = ν +(w). It can be sown following te proof of (2.7) tat Terefore form (3.21), B (w, ν; v, µ) C 1 (w, ν) 2 V and (v, µ) V C (w, ν) V. (3.22) B (w, ν; v, µ) C 1 (w, ν) 2 V B(w, v) B (w, v). (3.23) Since v = w + ν, v could be written as v = i N v i φ i, were v i = w i + ν. Terefore, from (2.16) and (2.20) v 2 L 2(Ω) C d vi 2 = C d (w i + ν) 2 i N i N C d (wi 2 + ν2 ) i N = C d i N w 2 i + C d i N ν 2 C [ w 2 L 2(Ω) + ν2]. (3.24) 15

17 Using te above in (3.20), we get B(w, v) B (w, v) C η 1 [ w H1 (Ω) w L2(Ω) + ν ] Terefore from (3.23), we ave C η 1[ w 2 H 1 (Ω) + w 2 L + ν 2] 2(Ω) C 2 η 1[ w 2 H 1 (Ω) + (w) 2 + ν 2] = C 2 η 1 (w, ν) 2 V. (3.25) B(w, ν; v, µ) [ C 1 C 2 η 1] (w, ν) 2 V. Finally, considering η 1 small enoug suc tat [ C 1 C 2 η 1] C > 0, we get B(w, ν; v, µ) C (w, ν) 2 V. We ave already seen from (3.22) tat (v, µ) V C (w, ν) V. Tus we proved te inf-sup condition (3.15). Te proof of (3.16) is similar to Lemma 2.1(c) and we do not provide it ere. Remark 3.2 We note tat Lemma 3.1 was proved under a restrictive condition on η, namely, we required tat η C, wit C sufficiently small. Computations suggest tat te condition η = O() is not necessary for te existence of a unique solution of te problem (3.8) (η sufficiently small, independent of, is sufficient). We will furter comment on te dependence of η on later in tis paper. Remark 3.3 We will indicate in tis remark tat it is possible to coose a quadrature rule tat yields a small η in (3.10) in te assumption QA1. We consider te set ω i = {ξ R d : ξ = x/diam(ω i ), were x ω i }. Clearly, diam(ω i ) = 1. For W m1, (ω i ) satisfying (3.12), we define (ξ) (x) = (ξ diam(ω i )). Ten it is easy to sow tat L (ω i) = L (ω i) and D α L (ω i) C L (ω i), α m 1. (3.26) We now consider a n i -point quadrature rule on ω i suc tat (ξ)dξ (ξ)dξ η D α L (ω i), α = m 1, (3.27) ω i ω i were m 1 depends on te quadrature rule and η is inversely proportional to n i. For example, we may consider n i -point composite trapezoidal rule on ω i. It is well known tat for n i -point trapezoidal rule, (3.27) is true wit m 1 = 2 and 16

18 η = n 2 i /12. We may also consider an n i -point Gaussian quadrature rule, in wic case we ave η = O(n m1 ). Now from (3.26) and (3.27), we ave ω i (ξ)dξ i ω i (ξ)dξ η L (ω i), α = m 1, (3.28) were η = Cη is inversely proportional to n i. Tus we can coose a quadrature rule (i.e., number of quadrature points n i ) suc tat te associated η is small. Finally we get (3.10) by employing a standard scaling argument to te inequality (3.28). Using similar arguments, we can sow tat we can coose a quadrature rule wit a small τ in (3.13). We furter note tat te functions and ϑ (in (3.10) and (3.11) respectively) tat we numerically integrate in tis paper satisfy te conditions (3.12) and (3.13). Remark 3.4 For eac i N, we define te linear functional G i : H 2 (ω i ) R as follows: G i (v) = v φ i dx + v φ i dx v nφ i ds. (3.29) ω i ω i ω i Γ It follows directly from Greens formula tat G i (p) = 0, p P k. (3.30) In fact, (3.30) is true for any smoot function p. Te linear functional G i, defined in (3.17), is obtained by using numerical integration on eac integral in G i. In general, (3.30) is not true if G i is replaced by G i. In (3.18) of assumption QA3, we require te exact same property to old for G i. Remark 3.5 It is instructive to illustrate te assumption QA3, i.e., (3.18) in simpler situations. Let Ω R 2 and k = 1. Considering p(x 1, x 2 ) = x 1 in (3.18), we get G i (x 1 ) = φ i dx ω i x 1 n 1 φ i ds = 0, i N, (3.31) ω i Γ were n = (n 1, n 2 ). Similarly, considering p(x 1, x 2 ) = x 2 in (3.18), we get G i (x 2 ) = φ i dx ω i x 2 n 2 φ i ds = 0, i N (3.32) ω i Γ Tus for k = 1, te quadrature must satisfy te two conditions (3.31) and (3.32) for eac i N. In particular, te quadrature must satisfy ω i φ i dx = 0, i N. (3.33) We illustrate now (3.18) for k = 2. Considering p(x 1, x 2 ) = x 2 1 in (3.18), we get [ G i (x 2 1) = 2 ω i x 1 φ i x 1 dx + ω i φ i dx ] x 1 n 1 φ i ds = 0, i N. ω i Γ (3.34) 17

19 Similarly, considering p(x 1, x 2 ) = x 1 x 2 and p(x 1, x 2 ) = x 2 2 in (3.18), we get ( ) G φ i φ i i (x 1 x 2 ) = x 2 + x 1 dx (x 2 n 1 +x 1 n 2 )φ i ds = 0, (3.35) x 1 x 2 ω i Γ ω i and [ G i (x 2 2) = 2 ω i x 2 φ i x 2 dx + ω i φ i dx ] x 2 n 2 φ i ds = 0, (3.36) ω i Γ for i N. Tus for k = 2, te quadrature must satisfy (3.34) (3.36) in addition to te assumptions (3.31) and (3.32). We will present quadrature scemes satisfying assumption QA3 later in tis paper. 4 Effect of Numerical Integration In tis section, we will study te effect of quadrature on te mesless metod. In particular, we will compare u u wit u u, were u, u, and u are defined in problems (2.5), (2.23), and (3.8) respectively. We will assume u to be smoot; in particular, u C k+1 (Ω). Tis assumption will allow us to present te main ideas simply and effectively. We will first prove te so called Strang lemma. Lemma 4.1 Suppose (u, λ ) and (u, λ ) are te solutions of problems (2.23) and (3.8) respectively. Let (w, λ) V be arbitrary. Ten tere exists C, independent of, suc tat [ (u u, λ λ ) V C (u w, λ λ) V + sup (v,µ) V [B(w, v) B (w, v)] + [L (v) L(v)] ]. (4.1) (v, µ) V Proof: We first note tat (u w, λ λ) V. Terefore, from te inf-sup condition (3.15) we get Now, (u w, λ λ) V C sup (v,µ) V B (u w, λ λ; v, µ) (v, µ) V. (4.2) B(u w, λ λ; v, µ) = B (u w, λ λ; v, µ) + B (w, λ; v, µ) B (w, λ; v, µ) + B(u, λ ; v, µ) B (u, λ ; v, µ) = B (u w, λ λ; v, µ) + B(w, v) B (w, v) + L (v) L(v). 18

20 Terefore from (4.2) and (3.14), we get (u w, λ λ) V C sup (v,µ) V 1 (v, µ) V B (u w, λ λ; v, µ) + B(w, v) B (w, v) + L (v) L(v) C (u w, λ λ) V +C sup (v,µ) V B(w, v) B (w, v) + L (v) L(v) (v, µ) V. Finally, using te triangle inequality and te above, we get (u u, λ λ ) V (u w, λ λ) V + (u w, λ λ) V [ C (u w, λ λ) V + sup (v,µ) V [B(w, v) B (w, v)] + [L (v) L(v)] ], (v, µ) V wic is te desired result. In te te analysis presented of tis section, we will apply (4.1) wit w = I u and estimate eac term on te rigt-and side of (4.1). Recall tat I u is te V -interpolant of u, as defined in (2.24). From te interpolation error estimate (2.25), we ave I u W k+1, (Ω) u W k+1, (Ω) + u I u W k+1, (Ω) C u W k+1, (Ω). (4.3) For a smoot function v and i N, let T k i v α k D α v( x i ) α! (x x i ) α (4.4) be te k t degree Taylor polynomial of v centered at x i, were x i is te center of te ball o i ω i (recall tat ω i is star-saped wit respect to o i ). It is well known tat ([7]) v Ti k v W j, (ω i) Ck+1 j (k + 1 j)! v W k+1, (ω i), j = 0, 1,...,k + 1. (4.5) Now consider Ti ki u te k t degree Taylor polynomial of I u centered at x i. We set R i I u Ti k I u, i N. (4.6) 19

21 Ten from (4.3) and (4.5) wit v = I u, we get R i W j, (ω i) C k+1 j I u W k+1, (ω i) C k+1 j u W k+1, (Ω). (4.7) We will use tis estimate later for j = 1, 2 in te next lemma. Lemma 4.2 For i N, let G i and G i be te linear functionals as defined (3.29) and (3.17) respectively. Ten tere exists a positive constant C, independent of i and, suc tat { C(η + τ) G i (I u) G k+d 1 i (I u u) W k+1, (Ω) i N ; Cη k+d 1 u W k+1, (Ω) i N. Proof: For eac i N, we write I u as I u = T k i I u + R i, were R i is te remainder defined in (4.6). Since Ti ki u is a polynomial of degree k, we ave G i (Ti ki u) = G i (T i ki u) = 0 from (3.30) and (3.18) and terefore, G i (I u) G i (I u) = G i (T k i I u + R i ) G i (T k i I u + R i ) = G i (R i ) G i (R i ). (4.8) Let i N. Ten, using (3.10), (3.11), and te assumption A2, G i (R i ) G i (R i) R i φ i dx R i φ i dx ω i ω i + R i φ i dx R i φ i dx ω i ω i + R i n φ i ds R i n φ i ds ω i Γ ω i Γ η ω i R i φ i L (ω i) + η ω i R i φ i L (ω i) +τ ω i Γ R i n φ i L ( ω i Γ) Cη d 1 R i W 1, (ω i) + Cη d R i W 2, (ω i) + Cτ d 1 R i W 1, (ω i) C(η + τ) k+d 1 u W k+1, (Ω), (4.9) were we used (4.7) to obtain te last inequality. For i N, we ave ω i Ω and terefore φ ωi i = 0. Now following te arguments leading to (4.9), we get G i (R i ) G i (R i) Cη k+d 1 u W k+1, (Ω), i N. Tus from (4.8) and (4.9), we get te desired result. We now prove te main result of tis paper. 20

22 Teorem 4.1 Suppose te approximating subspace V and te numerical integration sceme satisfy conditions A1 A5 and QA1 QA3, respectively. Ten for small η, tere is a positive constant C, independent of u, η, τ, and, suc tat u u H 1 (Ω) C [ k + (η + τ) k + η k 1] u W k+1, (Ω). Proof: Let I u be te V -interpolant of u. We note tat B(D, v) = 0 and recall tat B (D, v) = 0 for an arbitrary constant D (see (3.6)). We ten substitute (w, λ) = (I u + D, λ ) in Lemma 4.1 to get (u u, λ λ ) V C + sup (v,µ) V [ u I u D H 1 (Ω) [B(I u, v) B (I u, v)] + [L (v) L(v)] ]. (v, µ) V We will now estimate te rigt-and side of (4.10). Since te solution u is smoot, we ave for i N, f φ i dx = u φ i dx ω i ω i = (I u)φ i dx + (I u u)φ i dx ω i ω i and g φ i ds = ω i Γ = ω i Γ ω i Γ (4.10) u n φ i ds (I u) n φ i ds + (u I u) nφ i ds. ω i Γ Terefore using te definition of te linear functional G i (see (3.29)) we get, B(I u, φ i ) L(φ i ) = B(I u, φ i ) f φ i dx g φ i ds ω i ω i Γ = (I u) φ i dx + (I u)φ i dx (I u) nφ i ds ω i ω i ω i Γ + (u I u)φ i dx (u I u) n φ i ds ω i ω i Γ = G i (I u) + e I φ i dx e I n φ i ds, (4.11) ω i ω i Γ were e I u I u. Likewise, repeating te argument leading to (4.11) wit replaced by ffl, we get for i N, B (I u, φ i ) L (φ i ) = G i (I u) + ω i e I φ i dx 21 e I nφ i ds, (4.12) ω i Γ

23 were G i is te linear functional defined in (3.17). Terefore combining (4.11) and (4.12), we get for i N, B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) = G i (I u) G (I u) + e I φ i dx e I φ i dx ω i ω i e I n φ i ds + e I nφ i ds. (4.13) ω i Γ ω i Γ Let i N. Ten using (3.10), (3.11), Lemma 4.2, (2.25), and assumption A2 in (4.13), we ave B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) G i (I u) G i (I u) + η ω i e I φ i L (ω i) +τ ω i Γ e I n φ i L ( ω i) C(η + τ) k+d 1 u W k+1, (Ω), i N (4.14) Now let i N, so φ i ωi = 0 and using (3.10), Lemma 4.2, (2.25), and assumption A2 in (4.13), we ave B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) G i (I u) G i (I u) + η ω i e I φ i L (ω i) Cη k+d 1 u W k+1, (Ω), i N (4.15) We now estimate te second term of te RHS of (4.10). Let v = i N v i φ i be an arbitrary element in V. Ten from (4.14), (4.15), (2.16), (2.17), (2.20), and a trace-inequality, we ave B(I u, v) B (I u, v) + L (v) L(v) [ v i B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) ] i N + [ v i B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) ] i N ( i N v 2 i ( + i N ) 1/2 ( v 2 i i N ) 1/2 ( i N Cη k+d 1 N 1/2( B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) 2) 1/2 B(I u, φ i ) B (I u, φ i ) + L (φ i ) L(φ i ) 2) 1/2 i N v 2 i ) 1/2 u W k+1, (Ω) +C(η + τ) k+d 1 N 1/2( vi 2 ) 1/2 u W k+1, (Ω) i N C k 1[ η v L2(Ω) + (η + τ) v L2(Γ)] u W k+1, (Ω) C k 1[ η + (η + τ) ] u W k+1, (Ω) v H1 (Ω). (4.16) 22

24 Ten, from (4.10) and te Poincaré inequality, we get u u H 1 (Ω) (u u, λ λ ) V C inf u I u D H1 (Ω) D R +C k 1[ η + (η + τ) ] u W k+1, (Ω) sup (v,µ) V v H1 (Ω) (v, µ) V C u I u H 1 + C [ (η + τ) k + η k 1] u W k+1, (Ω). (4.17) Finally, from (2.27) and (2.28) u u H 1 (Ω) u u H 1 (Ω) + u u H 1 (Ω) wic is te desired result. C [ k + (η + τ) k + η k 1] u W k+1, (Ω), For k = 1, we only require te quadrature to satisfy a reduced form of QA3, namely, we assume tat (3.18) of QA3 is satisfied only for i N. Teorem 4.2 Suppose te approximating subspace V satisfies conditions A1 A5 wit k = 1. We consider numerical integration sceme satisfying QA1 QA3, but (3.18) of QA3 is satisfied only for i N. Ten for small η, tere is a constant C, independent of u, η, τ, and, suc tat u u H 1 (Ω) C [ + η + τ ] u W 2, (Ω) Te proof of tis result can be obtained by sligtly modifying te proof of Teorem 4.1; we do not provide te details ere. Remark 4.1 It is clear from Teorem 4.1 tat we do not ave optimal order of convergence, i.e., u u H 1 (Ω) = O ( k 1 [+η] ). But if we consider η C, ten we get u u H 1 (Ω) = O( k ). Tis means tat if we increase te accuracy of te quadrature as becomes smaller, we restore te optimal order of convergence. Tis effect of numerical integration in mesless metod is very different from te effect of numerical integration in FEM. 5 Numerical Results In tis section, we present computational data illuminating te results in Section 4 in one dimension. We will also develop numerical integration rules satisfying (3.18) of assumption QA3. We consider te one dimensional version of te problem (2.5) wit Ω = (0, 1). Let u(x) = e x (e 1), satisfying Φ(u) = 1 u dx = 0, be te exact solution of 0 23

25 (2.5) wit L(v) = 1 0 ex v(x)dx + e v(1) v(0). To approximate tis solution by te mesless metod (2.23), we first construct a C 2 (R), symmetric, RKP basic sape function φ(x) wit support [ R, R], satisfying φ(x j) = 1 and j φ(x j) = x, x R, j Z j Z wit R = 1.8 (see [2, 18, 19]). We ten consider a natural number N > 1, and for = 1/N, we let N = {x i = i : i = 1, 0, 1,, N, N + 1}. For eac x i N, we define te sape function ( x ) φ i (x) φ i, x Ω (0, 1). (5.1) Ten for i N, ω i (α i, β i ) = (i R, i + R) Ω and supp φ i (x) = ω i = [α i, β i ] Ω. We note tat for i = 2, 3,...,N 2, we ave (i R, i + R) Ω, α i = i R, β i = i + R and φ i (α i ) = φ i (β i ) = 0. (5.2) Tus N = {2, 3,..., N 2} and N = { 1, 0, 1, N 1, N, N + 1}. It can be easily sown tat te sape functions {φ i } N+1 i= 1 reproduce polynomials of degree k = 1, i.e., N+1 i= 1 p(x i )φ i (x) = p(x), p P 1 (Ω). We next sow a procedure to obtain a quadrature sceme tat satisfies te condition (3.18) of te assumption QA3. Suppose f(x) is smoot in [α i, β i ] and β let I i (f) i α i f(x)dx. To approximate I i (f), we seek a quadrature rule of te form Q i gc(f) p w k f( z k ) ( z k [α i, β i ] and w k depend on i) (5.3) k=1 wit te property tat Q i gc(φ i) = 0, i N. (5.4) Tis is precisely te condition (3.18) in 1-d for k = 1 (see (3.33) in Remark 3.5). We start wit a p-point quadrature rule for te interval [α i, β i ] of te form Q i g(f) p w k f(z k ). (5.5) k=1 24

26 We now define z k z k and w k w k + θ i w k φ i(z k ) (5.6) in (5.3), and coose θ i suc tat (5.4) is satisfied. We first note tat Q i gc (φ i ) = = p p w k φ i ( z k) = [w k + θ i w k φ i (z k)] φ i (z k) k=1 k=1 p w k φ i(z k ) + θ i k=1 k=1 p w k [φ i(z k )] 2. Tus imposing condition (5.4), we get p w k φ i(z k ) + θ i k=1 p w k [φ i(z k )] 2 = 0 k=1 or, θ i = p k=1 w kφ i (z k) p k=1 w k [φ i (z k)] 2. (5.7) Tus Q i gc(f) satisfies te condition (5.4) and we refer to Q i gc(f) as te p-point corrected quadrature. We now consider te quadrature rule Q i g (f) in (5.5) to be te p-point Gauss quadrature rule. It is well known tat te points {z k } p k=1 are symmetrically placed in te interval (α i, β i ) about te mid-point m i (α i +β i )/2; te weigts {w k } p k=1 are also symmetric, i.e., w k = w p+1 k, k = 1, 2,, p. We next recall tat te sape functions φ i (x), defined in (5.1) are symmetric in te interval (α i, β i ) about te mid-point m i. Consequently, φ i (x) is anti-symmmetric in te interval (α i, β i ) about m i. Terefore, it is clear tat Q i g(φ i) = p k=1 w k φ i(z k ) = 0, i N. Tus te Gaussian quadrature Q i g(f) satisfies (5.4); in fact, θ i = 0 in tis situation and Q i g(f) = Q i gc(f). We now present numerical experiments to illuminate te results in Teorem 4.1 for k = 1; in particular we illuminate te result in Teorem 4.2. We considered u = e x 1 (e 1) to be exact solution of (2.5) wit Φ(v) v dx. Te 0 function u was approximated by u V = span{φ i (x)} N+1 i= 1 - te solution of te mesless metod wit numerical integration (3.8), were φ i is defined in (5.1). We recall tat te linear functional ( ) was used in te mesless metod (3.8) to compute u. We employed te p-point Gauss quadrature rule Qi g(f) to numerically integrate te relevant terms, e.g., γij and l i (see (3.2) and (3.3)). We note tat we did not approximate te boundary term in (3.3) in our 1-d example and so ave τ = 0. 25

27 We used p = 8, 16, and 32 in Q i g (f) and computed te seminorm u u H 1 (Ω). We note tat η decreases as p increases. We presented tese results in Table 1. We also present te log-log grap of u u H 1 (Ω) wit respect to in Figure 1. Table 1. Standard p-point Gauss rule. u u H 1 (Ω) p = 8 p = 16 p = 32 1/ E E E-03 1/ E E E-03 1/ E E E-04 1/ E E E-04 1/ E E E-04 1/ E E E-04 1/ E E E-05 1/ E E E-05 Table 1: Te H 1 -seminorm of te error, u u H 1 (Ω), were u(x) = e x (e 1) and u is te approximate solution obtained using standard Gaussian quadrature. Te sape functions reproduce polynomial of degree k = Gauss rule (uncorrected): k= point u u * H 1 (Ω) point point Figure 1: Te loglog plot of u u H 1 (Ω) wit respect to. u is te approximate solution obtained using p-point standard Gaussian quadrature (symmetric) wit p = 8, 16, and 32. We observe from Table 1 tat te error u u H 1 (Ω) decreases as decreases. Moreover, for p = 16, we observe from Figure 1 tat u u H 1 (Ω) = O() at 26

28 te beginning, but levels off for smaller values of. For p = 32 te pattern is same, but te error is O() for few more smaller values of. Tis pattern suggests tat u u H 1 (Ω) = O( + η). We will now sow tat te error u u H 1 (Ω) is not O( + η) wen te underlying quadrature rule does not satisfy te assumption (5.4); we will sow tat te error increases as becomes smaller. We construct a quadrature rule on (α i, β i ) suc tat te quadrature points are not situated symmetrically about te mid-point m i. Consider te mapping : [ 1, 1] [ 1, 1] given by Clearly, Consider y = (z) = z + 0.1(z 2 1). (z) = z > 0 z [ 1, 1]. I(g) 1 1 g(y)dy = 1 1 g((z)) (z)dz Te integral on te rigt could be approximated by te standard p-point Gauss quadrature, given by p I(g) v k g((ζ k )) (ζ k ), k=1 were {ζ k } and {v k } are standard Gauss points and Gauss weigts respectively for te interval ( 1, 1). Tis induces an associated p-point non-symmetric Gauss quadrature Q ns (g) on ( 1, 1) to approximate I(g) given by were Q ns (g) p k=1 v ns k g(ζns k ), v ns k v k (ζ k ), ζ ns k (ζ k ). Clearly vk ns and ζk ns are not symmetric. It is well known tat te precision of standard p-point Gauss quadrature is (2p 1). It can be easily sown tat te precision of te associated p-point non-symmetric Gauss quadrature is (p 1). Te p-point quadrature Q ns ( ) induces te associated p-point non-symmetric Gauss quadrature Q i ns ( ) for te interval (α i, β i ) given by were We note tat z ns k Q i ns(f) p k=1 = β i α i ζk ns + β i + α i 2 2 w ns k f(z ns k ), and w ns k Q i ns (φ i ) 0, for i N. 27 = β i α i vk ns. (5.8) 2

29 Tus Q i ns (f) does not satisfy te assumption (5.4). We computed u te solution of te mesless metod (3.8) wit numerical integration, were we used p-point Q i ns (f) to compute te relevant integrals. We computed te error u u H 1 (Ω) for p = 8, 16, 32, and 64, and presented te data in Table 2. We also present te log-log grap of u u H 1 (Ω) wit respect to in Figure 2. Table 2. Non-Symmetric p-point Gauss rule. u u H 1 (Ω) p = 8 p = 16 p = 32 p = 64 1/ E E E E-03 1/ E E E E-03 1/ E E E E-04 1/ E E E E-04 1/ E E E E-04 1/ E E E E-04 1/ E E E E-05 1/ E E E E-04 Table 2: Te H 1 -seminorm of te error, u u H 1 (Ω), were u = e x (e 1) and u is te approximate solution obtained using non-symmetric Gaussian quadrature ; te quadrature does not satisfy te assumption (5.4). Te sape functions reproduce polynomial of degree k = point Non symmetric Gauss rule (uncorrected): k= point u u * H 1 (Ω) point point Figure 2: Te loglog plot of u u H 1 (Ω) wit respect to. u is te approximate solution obtained using non-symmetric Gaussian quadrature (uncorrected) wit 8, 16, 32, and

30 We observe from Table 2 and Figure 2 tat for p = 8, te error increases as decreases and it levels off for smaller values of. For p = 16, 32, and 64, te error first decreases and ten increases. Te data suggest tat u u H 1 (Ω) is not O( + η). We now consider a quadrature rule Q i nsc(f) for te interval (α i, β i ) given by tat satisfies (5.4), i.e., Q i nsc(f) p k=1 w ns k f( z ns k ) Q i nsc (φ i ) = 0, i N (5.9) Using te ideas presented at te beginning of tis section, specifically, using (5.3) (5.7), we define z k ns zns k and w ns k = w ns k + θ i w ns k φ i(z ns k ), were zk ns and wk ns is defined in (5.8). We coose θ i, as in (5.7), suc tat Q i nsc ( ) satisfies (5.9). We refer to Qi nsc (f) as te corrected non-symmetric Gauss-rule. We again compute u te solution of te mesless metod (3.8) wit numerical integration, were we used p-point Q i nsc (f) to compute te relevant integrals. We present te error u u H 1 (Ω) and te values of in Table 3. We also present te log-log grap of u u H 1 (Ω) wit respect to in Figure 3. Table 3. Corrected non-symmetric Gauss Rule. u u H 1 (Ω) 8 points 16 points 32 points 64 points 1/ E E E E-03 1/ E E E E-03 1/ E E E E-04 1/ E E E E-04 1/ E E E E-04 1/ E E E E-04 1/ E E E E-05 1/ E E E E-05 Table 3: Te H 1 -seminorm of te error, u u H 1 (Ω), were u(x) = e x (e 1) and u is te approximate solution obtained using corrected non-symmetric Gaussian quadrature. Te sape functions reproduce polynomial of degree k = 1. 29

31 10 2 Corrected non symmetric Gauss rule: k=1 8 point u u * H 1 (Ω) point point 64 point Figure 3: Te loglog plot of u u H 1 (Ω) wit respect to. u is te approximate solution obtained using corrected non-symmetric Gaussian quadrature wit 8, 16, 32, and 64 points. We observe from Table 3 and Figure 3 tat te error u u H 1 (Ω) beaves differently tan te error given in Table 2 and Figure 2. Moreover, Figure 3 suggests tat u u H 1 (Ω) = O( + η), wic illuminates te main result of tis paper for k = 1. Tus te data in Tables 2 and 3 strongly suggest tat te assumption QA3 on te numerically quadrature is necessary. 6 Remarks and Conclusions In tis paper we ave developed a matematical framework to analyze te effect of numerical integration on mesless metods employing sape functions tat reproduce polynomials of degree k 1. Te main results are summarized as follows: One of our main assumptions on te numerical quadrature is tat it satisfy a form of Green s teorem, given in (3.18) in QA3. Using numerical integration rules tat satisfy (3.18), we ave proved error estimates. Numerical integration rules, satisfying te assumptions mentioned in tis paper, automatically yield te so called zero row sum condition (see (3.5)). Tis was one of te main assumptions tat was used to obtain a similar error estimate in ([4]) for te case k = 1. Our results indicate tat numerical integration wit increased accuracy is required as 0 to obtain te optimal order of convergence. Te 30

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