Numerical analysis and a-posteriori error control for a new nonconforming quadrilateral linear finite element

Transcription

1 Numerical analysis and a-posteriori error control for a new nonconforming quadrilateral linear finite element M. Grajewski, J. Hron, S. Turek Matthias.Grajewski@math.uni-dortmund.de, jaroslav.hron@math.uni-dortmund.de, stefan.turek@math.uni-dortmund.de Institut für Angewandte Mathematik und Numerik, LS3, Universität Dortmund, Germany p.1/21

2 structure of the talk 1. Introduction of the P 1 -element 2. Numerical analysis of P 1 3. Dual-weighted a-posteriori error-control p.2/21

3 basic observations By connecting the midpoints of an arbitrary quadrilateral, one gets a parallelogram. p.3/21

4 basic observations By connecting the midpoints of an arbitrary quadrilateral, one gets a parallelogram. u P 1 u(m 1 ) + u(m 3 ) = u(m 2 ) + u(m 4 ) u m1 + u m3 = u m2 + u m4!u P 1 : u(m i ) = u mi p.3/21

5 definition of P 1 v j : vertex, m i : edge midpoint M(j) := {i N Γ T : mi Γ v j Γ} v j p.4/21

6 definition of P 1 v j : vertex, m i : edge midpoint M(j) := {i N Γ T : mi Γ v j Γ} v j Φ j (m i ) = { 1, i M(j) 0, else,φ j P 1 (T) p.4/21

7 definition of P 1 v j : vertex, m i : edge midpoint M(j) := {i N Γ T : mi Γ v j Γ} v j Φ j (m i ) = { 1, i M(j) 0, else,φ j P 1 (T) P 1 (T) := { v : Ω R F Γ (v) = v(m Γ ) (continuity constraint) v T P 1 (T) T T vcont. w.r.t. F Γ } p.4/21

8 P 1 basis function Φ (0,0) p.5/21

9 theorem (Park) Ω simply connected with polygonal boundary, T triangulation of Ω with N vertices, 0 < ĵ N p.6/21

10 theorem (Park) Ω simply connected with polygonal boundary, T triangulation of Ω with N vertices, 0 < ĵ N dim P 1 (T) = N 1. p.6/21

11 theorem (Park) Ω simply connected with polygonal boundary, T triangulation of Ω with N vertices, 0 < ĵ N dim P 1 (T) = N 1. {Φ j j {1..., N} \ {ĵ}} basis of P 1 (T) p.6/21

12 theorem (Park) Ω simply connected with polygonal boundary, T triangulation of Ω with N vertices, 0 < ĵ N dim P 1 (T) = N 1. {Φ j j {1..., N} \ {ĵ}} basis of P 1 (T) ( u, ϕ) Ω + (cu, ϕ) Ω + (u, ϕ) Ω = (f, ϕ) Ω + (g, ϕ) Ω ϕ H 1 (Ω). p.6/21

13 theorem (Park) Ω simply connected with polygonal boundary, T triangulation of Ω with N vertices, 0 < ĵ N dim P 1 (T) = N 1. {Φ j j {1..., N} \ {ĵ}} basis of P 1 (T) ( u, ϕ) Ω + (cu, ϕ) Ω + (u, ϕ) Ω = (f, ϕ) Ω + (g, ϕ) Ω ϕ H 1 (Ω). u u h 0 Ch 2 u 2, if g H1( Ω), f L 2 (Ω) 2 p.6/21

14 Dirichlet boundary cond. I (Park) u h (m j ) = c j + c j+1 ( ) p.7/21

15 Dirichlet boundary cond. I (Park) u h (m j ) = c j + c j+1 ( ) and dim P 1 (T) = N 1 p.7/21

16 Dirichlet boundary cond. I (Park) u h (m j ) = c j + c j+1 ( ) and dim P 1 (T) = N 1 explicit boundary treatment : fix one arbitrary coefficient per boundary component compute recursively the coefficients on the boundary according to (*) p.7/21

17 Dirichlet boundary cond. II p.8/21

18 Dirichlet boundary cond. II u h and coefficient vector of u h p.8/21

19 Dirichlet boundary cond. II u h and coefficient vector of u h p.8/21

20 comparison with other elements Test problem: u = 10, u Ω = 0 e h l2 := u u h l2 := ( 1 N ) 1/2 N u(v i ) u h (v i ) 2 i=1 NEL e h l2, Q 1 red. e h l2, Q 1 red. e h l2, P 1 red p.9/21

21 Dirichlet boundary cond. III observation: multiply connected domains and explicit boundary treatment: e h 0 = O(h) p.10/21

22 Dirichlet boundary cond. III observation: multiply connected domains and explicit boundary treatment: e h 0 = O(h) lemma Let us assume that u = α. If for one arbitrary T T holds c 1 + c 3 = c 2 + c 4, then: c 1 + c 3 = c 2 + c 4 T and representation is oscillationfree p.10/21

23 Dirichlet boundary cond. III observation: multiply connected domains and explicit boundary treatment: e h 0 = O(h) lemma Let us assume that u = α. If for one arbitrary T T holds c 1 + c 3 = c 2 + c 4, then: c 1 + c 3 = c 2 + c 4 T and representation is oscillationfree implicit boundary treatment : for one element on the first boundary component, apply c 1 + c 3 = c 2 + c 4 for all other boundary edges, apply u(m j ) = c j + c j+1 p.10/21

24 Dirichlet boundary cond. III observation: multiply connected domains and explicit boundary treatment: e h 0 = O(h) lemma Let us assume that u = α. If for one arbitrary T T holds c 1 + c 3 = c 2 + c 4, then: c 1 + c 3 = c 2 + c 4 T and representation is oscillationfree implicit boundary treatment : remark: for one element on the first boundary component, apply c 1 + c 3 = c 2 + c 4 for all other boundary edges, apply u(m j ) = c j + c j+1 on tensor product grids : Lemma even holds for linear functions with implicit boundary treatment: error of boundary treatment can be O(h 3 ) p.10/21

25 Dirichlet boundary cond. IV L 2 -error of computation of u = x + 1 and reduction rates of the error NEL implicit red. explicit red p.11/21

26 matrix structure standard treatment via elimination (in rows only) in Q 1 and Q 1 (left) P 1 with explicit boundary treatment (middle) P 1 with implicit boundary treatment (right) p.12/21

27 iterative solvers for P 1 problem : implicit boundary treatment affects solver behaviour example : square in the channel, u(x, y) = x(x 1)(1 y)y 2 sin(x + 2y), solver: BiCGStab NEL ILU(0) ILU(0) sort ILU(1) ILU(1) sort >10000 > p.13/21

28 P 1 not LBB-stable problem: P 1 does not fulfill the LBB-condition: Ω = [0,1] 2, T tensor product grid, u(x, y) = (y, 0), p(x, y) = const. p.14/21

29 P 1 not LBB-stable problem: P 1 does not fulfill the LBB-condition: Ω = [0,1] 2, T tensor product grid, u(x, y) = (y, 0), p(x, y) = const. observation: checkerboard oscillation of the pressure problem: P 1 does not fulfill Korn s inequality (analogous to the Q 1 -case according to Knobloch) p.14/21

30 fast matrix assembly advantage: extremely fast matrix assembly possible: 1-point Gauss rule exact nonparametric transformation for P 1 as fast as parametric one par par NEL P 1, G 1 P 1, G 2 P1,G 1 P1, G 2 Qpar 1, G 2 Q 1, G 2 133, s , s ,129, s remark: for Q 1 : 2 2-Gauss rule necessary for Douglas-element (enhanced Q 1 ): even 3 3-Gauss necessary! p.15/21

31 a-posteriori error control (conforming case) a(u, ϕ) := ( u, ϕ) = (f, ϕ) ϕ V := H0(Ω) 1 wanted: upper bound of the error of an output quantity J( ) Becker, Rannacher: dual problem ( ϕ, z) = J(ϕ) ϕ H 1 0(Ω) p.16/21

32 a-posteriori error control (conforming case) a(u, ϕ) := ( u, ϕ) = (f, ϕ) ϕ V := H0(Ω) 1 wanted: upper bound of the error of an output quantity J( ) Becker, Rannacher: dual problem ( ϕ, z) = J(ϕ) ϕ H 1 0(Ω) J(e h ) = a(e h, z) p.16/21

33 a-posteriori error control (conforming case) a(u, ϕ) := ( u, ϕ) = (f, ϕ) ϕ V := H0(Ω) 1 wanted: upper bound of the error of an output quantity J( ) Becker, Rannacher: dual problem ( ϕ, z) = J(ϕ) ϕ H 1 0(Ω) J(e h ) = a(e h, z) = ( u }{{} + u h, z z h ) T 1 2 ([ nu h ], z z h ) T T T =f p.16/21

34 a posteriori error control (nonconf. case) problem 1: e h V, as u h V J(e h ) =? remedy: dual problem defined by (ϕ, z) = J(ϕ) ϕ L 2 (Ω), z H 2 (Ω) p.17/21

35 a posteriori error control (nonconf. case) problem 1: e h V, as u h V J(e h ) =? remedy: dual problem defined by (ϕ, z) = J(ϕ) ϕ L 2 (Ω), z H 2 (Ω) problem 2: Galerkin-orthogonality : V V h too small for z h remedy: element enhacing by (parametric) bulb function ϕ B (x, y) = xy z C h p.17/21

36 a posteriori error control (nonconf. case) problem 1: e h V, as u h V J(e h ) =? remedy: dual problem defined by (ϕ, z) = J(ϕ) ϕ L 2 (Ω), z H 2 (Ω) problem 2: Galerkin-orthogonality : V V h too small for z h remedy: element enhacing by (parametric) bulb function ϕ B (x, y) = xy z C h J(e h ) = T T { (f + u N h, z zh C }{{} ) T 1 2 ([ nu N h ], z zc h ) T =0 1 } 2 ([un h ], nz) T + a h (e h, zh C ) p.17/21

37 practical aspects observation: consistency error a h (e h, z C h ) = O(h4 ) for Laplace equation u N h P 1 (T) : u N h = 0 replace z (unknown) by z I obtained by patchwise biquadratic interpolation of z h in nonconformity term ([u N h ], nz) T : n z replaced by n z := 0.5( n z h T1 + n z h T2 ) p.18/21

38 practical aspects lemma: observation: consistency error a h (e h, z C h ) = O(h4 ) for Laplace equation u N h P 1 (T) : u N h = 0 replace z (unknown) by z I obtained by patchwise biquadratic interpolation of z h in nonconformity term ([u N h ], nz) T : n z replaced by n z := 0.5( n z h T1 + n z h T2 ) T T : ([u N h ], nz N h ) T = 0 p.18/21

39 practical aspects lemma: observation: consistency error a h (e h, z C h ) = O(h4 ) for Laplace equation u N h P 1 (T) : u N h = 0 replace z (unknown) by z I obtained by patchwise biquadratic interpolation of z h in nonconformity term ([u N h ], nz) T : n z replaced by n z := 0.5( n z h T1 + n z h T2 ) error estimation: T T : ([u N h ], nz N h ) T = 0 J(e h ) η(e h ) := (f, z I zh C ) T 1 2 ([ nu N h ], z I zh C ) T T T p.18/21

40 numerical test I square in a channel, analytical solution, point error in (0.35, 0.5) NEL J r (0.35,0.5) (e h ) η(e h ) I eff p.19/21

41 numerical test II DFG-benchmark-grid cylinder in channel, u = 10, u Ω = 0, J(ϕ) = Ω c u n ds NEL J Γ (e h ) η(e h ) I eff p.20/21

42 conclusion advantages of P 1 simplest nonconforming quadrilateral element optimal approximation order very fast matrix assembly (for parametric and non-parametric trafo) fast solvers possible rigorous a-posteriori error control open problems multigrid stabilization for mixed formulation problems (CFD) realistic applications p.21/21