Oversampling for the partition of unity parallel finite element algorithm

Transcription

1 International conference of Computational Mathematics and its application Urumqi, China, July 25-27, 203 Oversampling for the partition of unity parallel finite element algorithm HAIBIAO ZHENG School of Mathematics and Statistics, Xi an Jiaotong University 9

2 Oversampling for the partition of unity parallel finite element algorithm Model Two-grid methods and parallel finite element algorithms The partition of unity parallel finite element algorithm Error Estimate Numerical Simulation Conclusions 2 9

3 Model The elliptic boundary value problem u + b u = f in Ω, u = 0 on Ω. The variational formulation For any f H (Ω), find u V = H0(Ω) such that () 3 9 B(u, v) := ( u, v) + (b u, v) + (cu, v) = (f, v) v V. (2)

4 a(u, v) = ( u, v), N(u, v) = (b u, v) + (cu, v). A sufficient and necessary condition for the well-posedness of (2) is w,ω C sup φ V the variational problem (2) has a unique solution. B(w, φ) B(φ, w), sup w V, (3) φ,ω φ V φ,ω 4 9

5 2 Two-grid methods and parallel algorithms Two-grid methods Low frequency components of a solution could be approximated on a relatively coarse grid and high frequency components are computed on a fine grid. See Marion and Xu 995, Xu 996, Layton et al 998, Li and Hou 200, He 2003 and so on. Parallel finite element algorithms The local behavior of a solution is governed by the high frequency components. Bank and holst 2000,2003, Xu and Zhou 2000, He et al 2006,2008, 200, Shang 20,

6 Two-grid methods for the elliptic problem Find u h = u H + e h with u h Vh 0 = V h V, u H VH 0, eh Vh 0 such that B(u H, v) = (f, v) v V 0 H. (4) a(e h, v) = (f, v) B(u H, v) v V 0 h. (5) Error estimate (u u h ) 0,Ω C(h + H 2 ) u 2,Ω. (6) 6 9

7 3 The partition of unity parallel finite element algorithm The partition of unity and Oversampling Foe the coarse mesh τ H, for each vertex x i τ H, i =, 2,, N, let ω i Ω denote the union of triangles possessing x i, and denote φ i by the continuous, lagrangian basis function such that φ i (x m ) = δ i,m. Let, ω i = φ i Ω. For k N, denote nodal patches of kth order ω i,k about x i τ H by ω i, := ω i = φ i Ω, ω i,k := ω m,, k = 2, 3,. x m ω i,k 7 9

8 ALGORITHM PUPA: Step. Find a global coarse grid solution u H V 0 H such that B(u H, v) = (f, v) v V 0 H. (7) Step 2. Correct the residual on a fine grid in each overlapping sub-domain ω i,km, namely, for each i =,, N, solve e h i,km V H,0 h,i,km by a(e h i,km, v) = (f, v) B(u H, v) v V H,0 h,i,km. (8) Step 3. Update: u i = u H + e h i,km in ω i,km. Step 4. Construct the finite element solution ũ h = N φ i u i. i= 8 9 In Step 3, localized FE spaces Vh H(ω i,mk) := {v V h v Ω\ωi,mk = 0}, V H,0 h,i,mk = V h H(ω i,mk) V.

9 The most interesting features. The partition of unity technique introduces a framework for domain decomposition; 2. A series of local linearized residual problems are solved on these subdomains in parallel, meanwhile require very a little communication; 3. A global continuous finite element solution is constructed by combining all the local solutions together with the partition of unity functions. 9 9

10 4 Error Estimate Cutoff function Definition For x i τ H and d < D N, let η d,d i continuous function such that : Ω [0, ] be a linear and η d,d i ωi,d =, η d,d i Ω\ωi,D = 0, T τ H, η d,d i L (T ) C, T τ H, η d,d i L (T ) C G (D d)h, where some constant C, C G that only depend on the regularity parameter ρ of the mesh τ H but not on H. 0 9

11 Dual result The local residual solution of e h i,k V H,0 h,i,k satisfies a(e h i,k, v) = (f, v) B(u H, v) v V H,0 h,i,k. (9) Lemma x i τ H, d < D k, and corresponding cut-off function η d,d i, for e h i,k V H,0 h,i,k be the solution of (9), there stands that η d,d i e h i,k 0,ωi,D CH (η d,d i e h i,k) 0,ωi,D. (0) 9 Specially, e h i,k 0,ωi,k CH e h i,k 0,ωi,k. ()

12 Truncation error The local residual solution of e h i,km V H,0 h,i,km = V H h (ω i,km) V satisfies a(e h i,km, v) = (f, v) B(u H, v) v V H,0 h,i,km, (2) Lemma 2 For all x i τ H, j, m, k N, let θ j := η (j )m,jm i, j k, as in Definition. For the solution of (2), we have the following estimate e h i,km 2 0,ω i,m C( C m 2)k (θ k e i ) 2 0,ω i,km. (3) holds with constant C, C that only depend on ρ but not on x i, m, k, H. 2 9

13 Convergence analysis Theorem For ALGORITHM, by selections of τ H and τ h such that V H V h, h H 2 and the oversampling parameter k O(log H ), and under the assumption u H 2 (Ω), we have the following results (u h ũ h ) 0,Ω CH 2 u 2,Ω, (4) (u ũ h ) 0,Ω C(h + H 2 ) u 2,Ω. (5) 3 9

14 5 Numerical Simulation Convergence rate In our example, the domain is the unit square domain Ω = [0, ] [0, ]. we take b = (.0,.0) T, the true solution u(x, y) = 00(x 2 2x 3 + x 4 )(y 3y 2 + 2y 3 ). Then we can get f(x, y) in (). 4 9

15 H error for PUPA without oversampling. H h k (u ũ h ) 0,Ω Order H error for PUPA with oversampling k 2 log(/h) H h k (u ũ h ) 0,Ω Order

16 0.5 H=/4 H=/24 log( e 0 ) k Let e = u ũ h, set h = H 2, we give the errors for PUPA at H = /4, /24 with different number of layers oversampling. It shows that for H = /4, /24, both log( e 0 ) decrease almost linearly respect to k, except k =. 6 9

17 6 Conclusions Construct the partition of unity parallel finite element algorithm; On a uniform coarse mesh τ H, patches of diameter H log(/h) are sufficient to preserve the optimal convergence order; The error of this algorithm decays exponentially with respect to the number of layers oversampling. 7 9

18 Thank you 8 9