Error Analyses of Stabilized Pseudo-Derivative MLS Methods

Transcription

1 Department of Mathematical Sciences Revision: April 11, 2015 Error Analyses of Stabilized Pseudo-Derivative MLS Methods Don French, M. Osorio and J. Clack Moving least square (MLS) methods are a popular type of meshfree approximation scheme. These MLS schemes, when combined with pseudo derivatives (PDs), are simpler and these PDs retain a similar structure to the MLS functions. PDMLS methods have been developed and implemented in both Galerkin and least squares (LS) frameworks for elasticity and fluids problems. Galerkin PDMLS implementations have low levels accuracy. The LS approach yields more appropriate rates of convergence. The existing literature on PDMLS schemes is entirely computational. In this talk we describe our theoretical error analysis results for PDMLS methods using both the Galerkin and LS approaches on a range of prototype differential equation problems as well as a simple fractional derivative problem. We introduce a special stabilization that remedies the Galerkin low accuracy issue and seems crucial for the effectiveness of PDMLS methods as well. Midwest Numerical Analysis Day 2015 Wright State University (April 25, 2015). 1

2 The Moving Least Squares Approach: Let Λ = {x 1, x 2,..., x N } in Ω = [0, 1] have distinct points, p(z) = {1, z, z 2,..., z m } T m ( ) y x k ( ) y x PU (x, y) = a k (x) = p T a(x) and U R (x) = PU (x, x) (m << N). R R k=0 N ( ) a(x) = {a 0 (x), a 1 (x),..., a m (x)} T xi x (PU ) 2 minimizes W (x, x i ) U(x i ). R i=1 with U = [U(x 1 ), U(x 2 ),..., U(x N )] T. Weight function W is smooth, nonnegative, max s R W (s) = 1 and Supp(W ) [ 1, 1]. N ( ) ( ) ( ) a(x) = M 1 xk x xk x (x)b(x) U where M(x) = W p p T xk x, R R R k=1 ( ) ( ) xi x xi x ith Column of B(x) = W p. R R (M(x) is m + 1 m + 1 and B(x) is m + 1 N.) Approximability of U R to U: There exist constants C 1 and C 2 independent of R such that U U R 0,2,Ω C 1 R m+1 and U U R 0,2,Ω C 2 R m (U smooth.) Note: There are several mesh and weight function assumptions. (E.g. For each x Ω there exists at least m + 1 distinct points from Λ in B R/2 (x). Often N = 2m/R.) 2

3 Small Sample of Literature on MLS Methods: Overviews, Computational Analysis & Engineering Applications: T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139 (1996), 3-47 (And more...). S. Li and W.K. Liu, Meshfree Particle Methods, Springer (2004) (And many more...). Error Analyses: W. Han and X. Meng, Error analysis of the reproducing kernel particle method, Comput. Methods Appl. Math., 190 (2001), W. Han and X. Meng, On a meshfree method for singular problems, CMES, 3 (2002), M.G. Armentano and R.G. Duran, Error estimates for moving least square approximations, Appl. Num. Math., 37 (2001), M.G. Armentano and R.G. Duran, Error in Sobolev spaces for moving least square approximations, SIAM J. Num. Anal., 39 (2001),

4 Essentials of the MLS Approach: Nodes Λ = {x 1, x 2,..., x N } in Ω = [0, 1] and p(z) = {1, z, z 2,..., z m } T. P U (x, y) = m k=0 ( ) y x k a k (x), U R (x) = P U R (x, x) = a 0(x), (m << N). a(x) = M 1 (x)b(x) U and U = [U(x1 ), U(x 2 ),..., U(x N )] T. 5 4 o Nodes (x k, U(x k )) 3 2 P U (x j,y) 1 0 x j y axis 1 U R (y) U(y)

5 Pseudo-Derivatives: In PD schemes only the polynomial in P U is differentiated. MLS (Real) Derivative: U R (x) = d dx P U(x, x) = p T (0) d dx a(x) = pt (0) d dx Pseudo (Diffuse) Derivative: δu R (x) = lim y x Approximability of δ : PU (x, y) y = lim y x ( M 1 (x)b(x) U ) (= a 0 (x)). ( ) y x y pt a(x) = 1 R R a 1(x). C independent of R so D k V δ k V 0,2,Ω CR m+1 k, 0 k m (V smooth.) Sampling of Papers on Pseudo-Derivatives: (1) Y. Krongauz and T. Belytschko, Comput. Methods Appl. Mech. Engrg. (1997); (2) Y. Vidal, P. Villon and A. Huerta Comm. Num. Meth. Eng. (2003); (3) Y. Vidal, P. Villon and A. Huerta, Comp. Meth. Appl. Math. Engrg. (2004); (4) A. Huerta, T. Belytschko, S. Fernández-Méndez and T. Rabczuk, Encyclopedia of Computational Mechanics (2004); (5) S.-H. Lee and Y.-C. Yoon, Int. J. Num. Meth. Engrg., (2004); (6) Y.-C. Yoon, S.-H. Lee, and T. Belytschko, Comp. Math. Appl. (2006); (7) D. W. Kim, W.K. Liu, Y.-C. Yoon, T. Belytschko and S.-H. Lee, Comput. Mech. (2007); and many more. 5

6 Villon et al (2003) Div-Free PD Method: Incompressible Stokes Problem: ν U + P = f in Ω U = 0 in Ω U = 0 on Ω. Find div-free U which is zero on Ω such that U V da = f V da Ω Ω for all div-free V which are zero on Ω. Pseudo-Divergence Free Spaces: m = 1 case; they define ( ) ( ) P1 ( x, y) a00 ( x) + a 10 ( x)(y 1 /R) + a 01 ( x)(y 2 /R) ( u( x) ) ( P1 ( x, x) ) P 2 ( x, y) = b 00 ( x) + b 10 ( x)(y 1 /R) + b 01 ( x)(y 2 /R) and v( x) = P 2 ( x, x) and require, after a pseudo-divergence computation, y1 P 1 ( x, y) + y2 P 2 ( x, y) = 0 a 10 ( x) + b 01 ( x) = 0. This restricted solution form is then used in a Galerkin MLS formulation of the Incompressible Stokes Problem... 6

7 Yoon, Lee, Belytschko PDMLS X -Methods (2004, 2006, 2007): Suppose F S (z) exhibits a singular behavior in a PDE problem. scheme from with Yoon et al form an MLS type P U (x, y) = p T( y x) a(x) + c(x)fs (y) and u(x) = P U (x, x) R δu(x) = lim y x y P u (x, y) = 1 R a 1(x) + c(x)f S (x). A collocation method is formed for the problem by requiring the full (strong form) of the PDE with all Pseudo-Derivatives to hold at node points. Remark: Belytschko and co-authors state that Galerkin formulations with pseudo-derivatives have low accuracy. 7

8 Function Spaces V R : For nodal values v 1,..., v N associated with grid Λ = {x 1, x 2,..., x N } and parameter R we can develop local polynomials P v (x, y) and MLS function v(x) = P v (x, x) where ( ) y x P v (x, y) = p T b(x) with b(x) = M 1 (x)b(x) v. R Stabilization: For MLS functions v and z let P(v, z) = Ω [ l Λ(x;R) where Λ(x; R) = {l : x l [x R, x + R]}. (P v (x, x l ) v l )(P z (x, x l ) z l ) Lemma: There exists a constant C, independent of R, such that v δv 2 0,2,Ω C P R 2 v (x, x k ) v k 2 dx = C R2P(v, v). Ω k Λ(x;R) Lemma: There exists constant C independent of R such that ] dx (P(U R, U R )) 1/2 CR m+1 (U is smooth). (P and lemmas based on theory in Armentano & Duran (2001)). 8

9 BVP Stabilized Diffuse Galerkin Method (SDGM): { (au ) + ku = f on Ω = (0, 1), a(x) a 0 > 0, k > 0 U (0) = 0, U (1) = 0. Weak Form: Find U H 1 (Ω) such that (au, V ) + (ku, V ) = (f, V ) V H 1 (Ω) where (Z, W ) = (SDGM) Find MLS function u so that (a δu, δβ) + (ku, β) + R 2γ P(u, β) = (f, β). for all MLS functions β. Find MLS function u that minimizes Ω Z(x)W (x) dx. (γ > 0) J(v) = (a δv, δv) + (kv, v) 2(f, v) + R 2γ P(v, v) (γ > 0) over MLS functions. Theorem: There exists a constant C independent of R such that, δu δu R 0,2,Ω CR m/2, (γ = m/2 + 1 and U smooth.). 9

10 Numerical Results: Diffuse Galerkin Method (DGM) vs SDGM δ u δ U R m = R Variable δ u δ U R m = R Variable Method DGM SDGM m = m = δ u δ U R m = R Variable δ u δ U R R Variable m = m = m = 4 Figure: Log/log graphs of errors δu δu R in L 2 (Ω) norm vs R for polynomial degrees m. The - - o - - are for the DGM scheme and the are for the new SDGM. 10

11 Proof of Error Estimate δu δu R 0,2,Ω CR m/2 where γ = m/2 + 1: Let e = U R u and B(β, ζ) = (aδβ, δζ) + k(β, ζ) U u 0,2,Ω e 0,2,Ω + U R U 0,2,Ω and since e is an MLS function B(U R, e) = ( (f, e) [ (au, e ) + k(u, e) ]) + [ (aδu R, δe) + k(u R, e) ] And substituting gives = (f, e) + (au, δe e ) + (a(δu R U ), δe) + k(u R U, e) (f, e) = B(u, e) + R 2γ( ) P(u, U R ) P(u, u) B(e, e) + R 2γ P(u, u) = (au, δu R U R ) (au, δu u ) Since We find + [ (aδu R U, δe) + k(u R U, e) ] + R 2γ P(u, U R ) B(e, e) = δe 2 0,2,Ω + k e 2 0,2,Ω. δe 2 + k e 2 0,2,Ω + R 2γ P(u, u) CR m + C ɛ R 2m + ɛ( δe 2 0,2,Ω + k e 2 0,2,Ω ) +C δu u 0,2,Ω + ɛr 2γ P(u, u) + C ɛ R 2γ P(U R, U R ). Then, from the stabilization lemma, δu u 0,2,Ω C ɛ R 2(γ 1) + ɛr 2γ R 2 δu u 2 0,2,Ω C ɛr 2(γ 1) + ɛr 2γ P(u, u). Choosing γ = m/2+1 and using P(U R, U R ) CR 2m+2 ( 2(γ 1) = m and 2γ+2m+2 = m) provides the error estimate. 11

12 Publications on Galerkin Project: 1-D Results: DF and M. Osorio, A Galerkin meshfree method with diffuse derivatives and stabilization, Computational Mechanics, 50 (2012) D Results: DF and M. Osorio, A Galerkin meshfree method with diffuse derivatives and stabilization: Two-dimensional case, Revista Ingenieria Y Ciencia (J. Engr. Sci.), 9 (2013), Results for SDGM in 2-D are applied to (a U) + cu = f in Ω with a U n = g on Ω. 12

13 Least Squares PDMLS for a 1-D Elliptic BVP: Prototype Elliptic BVP: { LU = U + κu = f in Ω = (0, 1) U(0) = U(1) = 0 with 0 < κ 0 κ(x) κ 1 < and κ 2 = max x Ω κ (x) < κ 0 /2. Least Squares PDMLS: To avoid the integration-by-parts associated with Galerkin, Facts: Find u V R that minimizes Φ( v) = Ω (L δv f) 2 dx + ηr 4 P(v, v) + [ v(0) 2 + v(1) 2] over all v V R with L δ v = δ 2 v + κv. (i) C 1 > 0 and C 2 > 0 such that w 2 2,2,Ω C 1 Lw 2 0,2,Ω + C 2( max{ w(0), w(1) } ) 2. (ii) The minimization problem has a unique solution u V R and Φ( u) Φ(U R ) CR 2m 2. (iii) Stabilization: v (j) δ j v 2 0,2,Ω CR 2j P(v, v) for j = 0, 1, 2. 13

14 Error Analysis for LS PDMLS for a 1-D Elliptic BVP: Let e = u U R And: Le 0,2,Ω Lu L δ u 0,2,Ω + L δ u f 0,2,Ω + f LU 0,2,Ω + LU LU R 0,2,Ω CR 2 P(u, u) + C ( R 2 ηr 2m+2 + e(0) u(0) + U R (0) As well as: Thus C > 0 independent of R so that Φ(u) CR m 1 U m+1,2,ω R 2m 2 + R m 1 U m+1,2,ω ) CR m 1, Φ(u) + Φ(U R ) CR m 1 and similarly e(1) CR m 1. e 2 2,2,Ω C 1 Le 2 0,2,Ω + C 2 max{ e(0), e(1) } 2. e 2,2,Ω CR m 1 u U 2,2,Ω U R U 2,2,Ω + e 2,2,Ω CR m 1. (Facts: w 2 2,2,Ω C 1 Lw 2 0,2,Ω + C 2( max{ w(0), w(1) } ) 2, Φ( u) Φ(UR ) CR 2m 2 and v (2) δ 2 v 0,2,Ω CR 4 P(v, v)). 14

15 Highly Preliminary Evidence for Stabilization Requirement: { Simple IVP: Find Y = Y (t) with Y (0) = 0 such that Y (t) = f(t) LS PDMLS Approximation Method: (Definitions in (t, s) instead of (x, y)). Find y V R such that y minimizes Φ(z) for all z V R with Φ(z) = 1 2 Ω (δz f) 2 ds + 1 2R 2P(z, z) (z(0))2 Condition # s and Errors for m = 2 and R = 0.8, 0.4, 0.2, Condition Numbers 1 Errors No Stabilization o Stabilized o 1.5 No Stabilization o Stabilized o Log10(Condition #) Log10(Errors) R Values R Values (LS Method for IVP has Y (t) = (e t 1)/(e 1 1) for t [0, 1] and Gauss Quadrature with 5 subintervals and 8 nodes per subinterval.) 15

16 Least Squares PDMLS for a Simple Fractional Derivative Problem: For 0 < γ < 1 the Caputo derivative of Y = Y (t) is C 0 Dγ t Y (t) = 1 (t τ) γ Y (τ) dτ Γ(1 γ) 0 { For given f = f(t) find Y = Y (t) such that (F IV P ) LY = Y + k C 0 Dγ t Y = f with Y (0) = 0 (0 < γ < 1/2) on an interval Ω = [0, T ]. Assume Y C m+1 ([0, T ]) with k = k(t) so 0 < k 0 k(t) k 1 and f = f(t). Fractional Pseudo Caputo Derivative for v V R (Definitions in (t, s) instead of (x, y)). C 0 δγ t v(t) = 1 Γ(1 γ) t a t Approximability: For V C m+1 (Ω) C > 0 such that (t τ) γ δv(τ)dτ (0 < γ < 1). C 0 D γ t V (t) C 0 δγ t V R(t) C V δv R 0,2,Ω CR m V m+1,2,ω (0 < γ < 1/2). 16

17 LS PDMLS for Simple FIVP (Continued): LS PDMLS Scheme: Find y V R that minimizes Z(z) = (L δ z f) 2 ds + η R2P(z, z) + z(0)2 Ω where L δ z = δz + k c 0 δγ t z and η > 0. Combining Gronwall estimates, approximability and LS analysis obtain y Y 0,2,Ω CR m 1. 17

18 Preliminary Results: LS PDMLS and Collocation: Simple IVP: { Find Y = Y (t) with Y (0) = 0 such that Y (t) = f(t) t [0, T ] Stabilization at Nodes: (Definitions in (t, s) instead of (x, y)). LS PDMLD Scheme: (P R ) Θ( z) = N i=0 k Λ(t i ;R) [ Pz (t i, t k ) z k ] 2 Find y V R that minimizes Φ( z) = D( z; f) + Θ( z) + (z(0) Y 0 ) 2 where N D( z; f) = i=0 over all z V R. k Λ(t i ;R) [ ( P z / 2)(t i, t k ) P f (t i, t k ) ] 2. Using similar LS techniques as in earlier proofs and equivalence of polynomials in integral and sum norms can show y Y 0,2,Ω CR m 1. 18

19 Thanks/Acknowledgement: DF was partially supported by a Charles Phelps Taft Summer Fellowship at the University of Cincinnati during Mauricio Osorio was supported by a Taft Dissertation Fellowship from the University of Cincinnati where he was a GA in the Department of Mathematical Sciences (Finishing his PhD in 2010.) He is now at Escuela de Matema ticas, Universidad Nacional de Colombia, Apartado Ae reo 3840 Medellı n, Colombia. Jhules Clack was a GA in the Department of Mathematical Sciences (Finishing his PhD in 2014). D.F. Mauricio Osorio Jhules Clack 19

20 Thanks! 20