Five Recent Trends in Computational PDEs

Transcription

1 Five Recent Trends in Computational PDEs Carsten Carstensen Center Computational Science Adlershof and Department of Mathematics, Humboldt-Universität zu Berlin C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 1 / 44

2 Contributors Wolfgang Boiger Carsten Carstensen Dietmar Gallistl Joscha Gedicke Christian Merdon Daniel Peterseim Hella Rabus Mira Schedensack C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 2 / 44

3 Contents & Overview 1 Comparison of FEMs 2 Stokes Equations 3 Eigenvalue Problems 4 Variational Inequalities 5 Computational Microstructures C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 3 / 44

4 FEM Competition in PMP CFEM seeks u C P 1 (T ) C 0 () s.t. u C v C dx = CR-NCFEM seeks u CR CR 1 0(T ) s.t. NC u CR NC v CR dx = fv C dx for all v C P 1 (T ) C 0 () fv CR dx for all v CR CR 1 0(T ) RT-MFEM seeks p RT RT 0 (T ) and u RT P 0 (T ) s.t. p RT q RT dx + u RT div q RT dx = 0 for all q RT RT 0 (T ) Π 0 f + div p RT = 0 C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 4 / 44

5 Comparison Results in PMP Theorem [Braess (2010), C-Peterseim-Schedensack (2011)]: u u C u NC u CR u u C + osc(f, T ) u NC u CR hf + u p RT u NC u CR + osc(f, T ) But, in general, u NC u CR u p RT + osc(f, T ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 5 / 44

6 Counterexample PMP with f 2 in regular polygon M error RT MFEM CR NCFEM ndof M R M R 2 T s.t. ( ) M u p RT + osc(f, T ) u NC u CR C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 6 / 44

7 FEM Competition in 2D Elasticity CFEM seeks u C P 1 (T ; R 2 ) C 0 (; R 2 ) s.t. ε(u C ) : Cε(v C ) dx = f v C dx for all v C P 1 (T ; R 2 ) C 0 (; R 2 ) KS-NCFEM seeks u KS KS(T ) := ( P 1 (T ) C 0 () ) CR 1 0(T ) s.t. ε NC (u KS ) : Cε NC (v KS ) dx = f v KS dx for all v KS KS(T ) CR-NCFEM seeks u CR CR 1 0(T ; R 2 ) s.t. D NC u CR : CD NC v CR dx = f v CR dx for all v CR CR 1 0(T ; R 2 ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 7 / 44

8 Comparison Results in Elasticity σ σ C λ σ σ KS λ ( σ σ C + osc(f, T ) ) and σ σ KS + osc(f, T ) σ σ CR + osc(f, T ) [C-Rabus (2011): Adaptive CR-NCFEM in elasticity is robust as λ ] C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 8 / 44

9 Convergence Rates in Elasticity f = (1, 0) 10 1 f = (0, 0) /2 P1 10 1! (unif., "=0.3)! Av (unif., "=0.3)! (unif., "=0.49)! Av (unif., "=0.49)! (unif., "=0.499)! Av (unif., "=0.499)! (adapt., "=0.3, #=0.5)! Av (adapt., "=0.3, #=0.5)! (adapt., "=0.3, #=0.1)! Av (adapt., "=0.3, #=0.1)! (adapt., "=0.49, #=0.5)! Av (adapt., "=0.49, #=0.5)! (adapt., "=0.49, #=0.1)! Av (adapt., "=0.49, #=0.1)! (adapt., "=0.499, #=0.5)! Av (adapt., "=0.499, #=0.5) ! (adapt., "=0.499, #=0.1)! Av (adapt., "=0.499, #=0.1) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 9 / 44 1/2 1/ /2 CR

10 Contents & Overview 1 Comparison of FEMs 2 Stokes Equations 3 Eigenvalue Problems 4 Variational Inequalities 5 Computational Microstructures C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 10 / 44

11 Stokes Equations Given f L 2 (; R 2 ), g H 1 (; R 2 ), seek u H 1 (; R 2 ) with u = g on and p L 2 0() s.t. u + p = f and div u = 0 in C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 11 / 44

12 CR-NCFEM Stokes CR-FEM seeks u CR g h + Z NC with a NC (u CR, v CR ) := D NC u CR : D NC v CR dx = F (v CR ) enforced by Lagrange multiplier p CR P 0 (T ) for all v CR Z NC := {v CR CR 1 0(T ; R 2 ) div NC v CR = 0} Energy error u u CR NC := a NC (u u CR, u u CR ) 1/2 =? C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 12 / 44

13 A Posteriori Error Estimators Stokes Eqns Helmholtz decomposition as in [Ainsworth-Dörfler (2005)] leads to ( ) 2 u u CR 2 NC f T /2 ( mid(t )) L 2 () + 1/π osc(f, T ) ( + min D NC (u CR v) v V L 2 () + div v L 2 () /c 0 + C γ osc( g/ s, E( )) u u CR 2 NC + osc2 (f, T ) + osc 2 ( g/ s, E( )) ) 2 C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 13 / 44

14 Backward Facing Step Example GUB for adaptive and uniform mesh-refinement R (adaptive) A (adaptive) PMA (adaptive) MAred (adaptive) PMred (adaptive) MP1 (adaptive) MP2 (adaptive) MP1red (adaptive) MP1redCG3 (adaptive) MP2CG10 (adaptive) R (uniform) A (uniform) PMA (uniform) MAred (uniform) PMred (uniform) MP1 (uniform) MP2 (uniform) MP1red (uniform) MP1redCG3 (uniform) MP2CG10 (uniform) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 14 / 44

15 Pseudo Stress FEM for Stokes Equations RT-PSFEM seeks (σ PS, u PS ) PS k (T ) P k (T ; R 2 ) s.t. σ PS : dev τ PS dx + u PS div τ PS dx = v PS div σ PS dx = g τ PS νds v PS fdx for all (τ PS, v PS ) PS k (T ) P k (T ; R 2 ) PS k (T ) := (RT k (T ) RT k (T )) H(div, ; R 2 2 ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 15 / 44

16 AFEM for Stokes Equations SOLVE ESTIMATE MARK REFINE CRFEM [Becker-Mao, Hu-Xu, C-Peterseim-Rabus] η 2 (T ) := T f 2 L 2 (T ) + T 1/2 PSFEM [C-Gallistl-Schedensack] E E(T ) [ u CR / s] E 2 L 2 (E) η 2 (T ) := osc 2 (f, T ) + T curl(dev σ PS ) 2 L 2 (T ) + T 1/2 [dev σ PS ] E τ E 2 L 2 (E) E E(T ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 16 / 44

17 Optimal Convergence For σ H(div, ; R 2 2 ), f L 2 (; R 2 ), g H 1 ( ; R 2 ), APSFEM computes (u l, σ l ) with σ σ l 2 L 2 () + osc2 (f, T l ) + osc 2 ( g/ s, E l ( )) (σ, f, g) 2 A s ( T l T 0 ) 2s for ( (σ, f, g) 2 A s := sup N 2s inf σ Π RT0 (T )σ 2 L N N T T 0 N 2 () + osc 2 (f, T ) + osc 2( g/ s, E( ) )) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 17 / 44

18 Convergence History L-Shaped Domain RT-PSFEM CR-NCFEM eta eta 10 1 η adaptive σ σ PS adaptive η uniform σ σ PS uniform η adaptive u u CR adaptive η uniform u u CR uniform ndof ndof C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 18 / 44

19 Contents & Overview 1 Comparison of FEMs 2 Stokes Equations 3 Eigenvalue Problems 4 Variational Inequalities 5 Computational Microstructures C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 19 / 44

20 Can one hear the shape of a drum? [Kac (1966)] λ 1 = left right C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 20 / 44

21 A Posteriori Error Estimators for EVP [Duran-Padra-Rodriguez (2003)] η 2 l := T T l h 2 T λ 2 l u l 2 L 2 (T ) + E E l h E [ u l ] ν E 2 L 2 (E) [Mao-Shen-Zhou (2006)] η 2 MSZ := T T l (h 2 T λ l u l +div(a l ( u l )) 2 L 2 (T ) + A l( u l ) u l 2 L 2 (T ) ) [CC-Gedicke (2011) Numer Math] η 2 CG := E E l h E [ u l ] ν E 2 L 2 (E) and µ 2 CG := T T l A l ( u l ) u l 2 L 2 (T ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 21 / 44

22 Optimal Convergence For any u in V := H 1 0() AFEM computes u l with for u u l 2 + λ λ l u 2 A s ( T l T 0 ) 2s u 2 A s := sup N 2s inf N 0 T N T 0 +N dist2 (u, V (T N )) < P 1 P 2 P 4 [C-Gedicke (2009 preprint), Dai-Xu-Zhou (2008)] C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 22 / 44

23 Optimal Computational Complexity - 3D l 2, l P1 l adaptive 2/3 2 P1 adaptive l 1 P2 adaptive 4/9 l 2 P2 l adaptive 1 P3 adaptive l 2 1 P3 l adaptive P4 l adaptive 2 P4 adaptive l 4/3 P4 uniform 1 l CPU time (sec) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 23 / 44

24 Guaranteed Lower Bounds for Eigenvalues Example: (λ = 2π 2 = ) λ CR 1 + Cλ CR H λ λ < 24 = λ CR λ < 24 = λ CR λ λ CR = [C-Gedicke: Guaranteed lower bounds for eigenvalues (in prep)] [C-Gedicke-Mehrmann-Miedlar: An adaptive homotopy approach for non-selfadjoint eigenvalue problems (2011 Numer Math)] C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 24 / 44

25 Isospectral Eigenvalue Bounds λ 50 = lower bounds N left domain right domain λ 50 = upper bounds N left domain right domain C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 25 / 44

26 Contents & Overview 1 Comparison of FEMs 2 Stokes Equations 3 Eigenvalue Problems 4 Variational Inequalities 5 Computational Microstructures C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 26 / 44

27 Model Obstacle Problem Seek u K := {v V χ v} s.t. a(u, u v) F (u v) for all v K FEM seeks u h K(T ) := {v V (T ) Iχ v} s.t. a(u h, u h v h ) F (u h v h ) for all v h K(T ) Energy error u u h := a(u u h, u u h ) 1/2 =? C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 27 / 44

28 Lagrange Multiplier after Braess Define Λ h P 1 (T ) C() s.t., for all z N, Λ h ϕ z dx = fϕ ζ(z) dx u h ϕ ζ(z) dx Auxiliary Residual [Braess (2005 NumMath)], for v V, Res AUX (v) := (f Λ h )vdx Since P 1 (T ) C 0 () ker(res AUX ), Res AUX all error estimators u h vdx C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 28 / 44

29 Global Upper Bound u u h GUB := a ( a (χ u h )(JΛ h )dx }{{} overhead with a := Res AUX + Λ }{{} h JΛ h }{{} η xyz overhead for conforming χ χ h and positive interpolation Jv = ϕ z vϕ z dx/ ϕ z dx z N ) 1/2 (Efficiency holds for model scenarios [C-Merdon (2011)]) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 29 / 44

30 L-Shaped Domain Example u(r, ϕ) := r 2/3 g(r) sin(2ϕ/3) with s := 2(r 1/4) and g(r) := max{0, min{1, 6s s 4 10s 3 + 1}} Adaptive meshes l = 0, 3, 6 C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 30 / 44

31 L-Shaped Domain Example Efficiency indices GUB/ e vs ndof for uniform (solid) and adaptive (dotted) meshes MP1 A1 B LW MFEM LS CF EQL B(r1) LW(1) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 31 / 44

32 Contents & Overview 1 Comparison of FEMs 2 Stokes Equations 3 Eigenvalue Problems 4 Variational Inequalities 5 Computational Microstructures C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 32 / 44

33 Computational Microstructure Phase Transitions 0 0! YB 2 Cu 3 O 6+x tetrag. monocl. [C-Plecháĉ,SINUM 2001] F ! t=0.1, h=0.1 dash dotted line! t=0.1, h=0.01 dotted line! t=0.05, h=0.01 dashed line! t =0.01, h=0.01 solid line Hysteresis t [C-Plecháĉ,2001] Optimal Design Micromagnetics [C-Plecháĉ,1999] [C-Prohl,Numer Math 2001] C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 33 / 44

34 Minimisation & Relaxation E(v) := E (v) := ( ) W (Dv) + β v g 2 fv dx ( ) W (Dv) + β v g 2 fv dx C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 34 / 44

35 Comput. Microstructure Benchmark microstructure = enforced finer and finer oscillations, weak convergence but no strong convergence of infimising sequences C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 35 / 44

36 Stabilised Relaxed Simulation a l (v l, v l ) := E l (v l ) := E (v l ) a l(v l, v l ) h 1+γ max [Dv l ] h E 2 L 2 (E) for v l H 2 (T l ) E E E l () Convergence for h max 0, σ σ l L p () + β u u l L2 () + a l(u l, u l ) 0 C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 36 / 44

37 A Posteriori Error Estimate η F := min τ RT RT 0 (T ) ( σl τ RT L q () + f 2β(u l g) + div τ RT L q () satisfies for 2 q p ) σ σ l 2 L p () + β u u l 2 L 2 () η F u u l W 1,q () C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 37 / 44

38 Two-Well Benchmark ndof η F unif (h 4/5 ) η L unif (h 1 ) η F F-adapt η F R-adapt η L F-adapt η L R-adapt σ σ l 2 σ σ l 2 p F-adapt p R-adapt σ σ l 2 p unif (h 5/3 ) u u l 2 2 F-adapt u u l 2 2 R-adapt u u l 2 2 unif (h3/2 ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 38 / 44

39 Interface Model Problem Strong Convergence of Gradients in Relaxed Stabilized FE η H (unif) η H (unif,stab) η H (F-adapt,stab) η H (R-adapt,stab) η H (R-adapt) η H (F-adapt) ndof D(u u l ) 2 2 (F-adapt,stab) D(u u l ) 2 2 (R-adapt,stab) D(u u l ) 2 2 (unif,stab) D(u u l ) 2 2 (unif) D(u u l ) 2 2 (R-adapt) D(u u l ) 2 2 (F-adapt) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 39 / 44

40 MFEM in Topology Optimisation µ 1 t 2 ϕ t 1 t 2 t 2 (ε + µ 1 ) t 1 (ε + µ 2 ) t ϕ µ 1 t 2 t (ϕ 2 ε) t t C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 40 / 44

41 Topology Optimisation Minimise [Bartels-C (2007)] E (v) := ϕ ( v ) dx v dx for v V = H0() 1 Maximise [C-Günther-Rabus SINUM (2011)] E (τ) := ϕ ( τ ) dx for τ Q = {τ H(div, ) : 1 + div τ = 0} C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 41 / 44

42 Topology Optimisation Minimise [Bartels-C (2007)] E (v) := ϕ ( v ) dx v dx Maximise [C-Günther-Rabus SINUM (2011)] E (τ) := ϕ ( τ ) dx for v V = H 1 0() u C P 1 (T ) C 0 () for τ Q = {τ H(div, ) : 1 + div τ = 0} σ RT Q(T ) = Q RT 0 (T ) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 41 / 44

43 A Posteriori Error Estimate µ 2 σ σ RT min v V ϕ ( σ RT ) v ( h 1/2 2 E [ ϕ ( σ RT )] E τ E E E + h T curl ϕ ( σ RT ) 2 L 2 (T ) T T L 2 (E) ) 1/2 [C-Günther-Rabus SINUM (2011) accepted] C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 42 / 44

44 Computational Example 10 0 η ; uniform 1 H η H ; adaptive η R ; uniform 0.5 η R ; adaptive / ndof C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 43 / 44

45 Thank You for Your Attention! W. Boiger, C. Carstensen, On the Strong Convergence of Gradients in Stabilised Degenerate Convex Minimisation Problems, SINUM 47(6), 2010 W. Boiger, C. Carstensen, Stabilised FEM for Degenerate Convex Minimisation Problems Without Higher Regularity Assumptions (2011 submitted) C. Carstensen, C. Merdon, Estimator Competition for Poisson Problems, J. Comp. Math., 28, 2010 C. Carstensen, C. Merdon, A Posteriori Error Estimator Competition for Conforming Obstacle Problems (2011 submitted) C. Carstensen, D. Gallistl, M. Schedensack, Quasi Optimal Adaptive Pseudostress Approximation of the Stokes Equations (2011 submitted) C. Carstensen, D. Günther, H. Rabus, Adaptive Mixed Finite Element Method for an Optimal Design Problem (SINUM 2011 accepted) C. Carstensen, D. Peterseim, H. Rabus, Optimal Adaptive Nonconforming FEM for the Stokes Problem (2011 submitted) C. Carstensen, D. Peterseim, M. Schedensack, Comparison Results of Three First-order Finite Element Methods for the Poisson Model Problem (2011 submitted, MATHEON preprint 831) C. Carstensen (CCSA & HU Berlin) 5 Trends in CPDEs 44 / 44