Axioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates)

Transcription

1 Axioms of Adaptivity (AoA) in Lecture 3 (sufficient for optimal convergence rates) Carsten Carstensen Humboldt-Universität zu Berlin 2018 International Graduate Summer School on Frontiers of Applied and Computational Mathematics Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

2 Contents L3 1 Jump Control 2 Proof of (A1) 3 Proof of (A2) 4 Quasiinterpolation 5 Proof of (A3) 6 Proof of (A4) 7 Outlook at Applications Poisson model problem (PMP) in 2D lowest-order conforming FEM w.r.t. triangles in shape regular triangulations lead to R = T \ T, Λ 3 = 0 Open-Access Reference: C-Feischl-Page-Praetorius: AoA. Comp Math Appl 67 (2014) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

3 Jump Control Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

4 Λ 1 Comes from Discrete Jump Control Given g P k (T ) for T T, set { (g T+ ) E (g T ) E for E E(Ω) with E = T + T, [g] E = g E for E E( Ω) E(K). Lemma (discrete jump control) For all k N 0 there exists 0 < Λ 1 < s.t., for all g P k (T ) and T T, K 1/2 [g] E 2 L 2 (E) Λ 1 g L 2 (Ω). K T E E(K) Proof with discrete trace inequality on E E(K) for K T K 1/4 g K L 2 (E) C dtr g L 2 (K). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

5 Compute Λ 1 in Proof of Discrete Jump Control The contributions to LHS of interior edge E = T + T with edge-patch ω E := int(t + T ) read ( T + 1/2 + T 1/2 ) [g] E 2 L 2 (E) ( T + 1/2 + T 1/2 ) ( g T+ L 2 (E) + g T L 2 (E) ( Cdtr 2 ( T + 1/2 + T 1/2 ) T + 1/4 g L 2 (T + ) + T 1/4 g L 2 (T ) Cdtr 2 ( T + 1/2 + T 1/2 )( T + 1/2 + T 1/2 ) g 2 }{{} L 2 (ω E ) Csr 2 C 2 dtr C2 sr g 2 L 2 (ω E ). The same final result holds for boundary edge E = T + Ω with ω E := int(t + ). The sum of all those edges proves the discrete jump control lemma with Λ 1 := 3 C dtr C sr. ) 2 ) 2 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

6 Proof of (A1) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

7 Proof of (A1) with δ(t, ˆT ) = P P L2 (Ω) Recall that ˆT is an admissible refinement of T with respective discrete flux approximations P := û h P 0 ( ˆT ; R 2 ) and P := u h P 0 (T ; R 2 ). Given any T T ˆT, set η(t ) := for α T := T 1/2 f L 2 (T ) and β 2 T := T 1/2 E E(T ) αt 2 + β2 T and η(t ) := [P ] E 2 L 2 (E) resp. βt 2 := T 1/2 α 2 T + β T 2 E E(T ) [ P ] E 2 L 2 (E) Convention for conforming P 1 FEM: Jumps on boundary edges vanish. Then, η(t ˆT ) := T T ˆT η2 (T ) and η(t ˆT ) := T T ˆT η 2 (T ) are Euclid norms of vectors in R J for J := 2 T ˆT. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

8 Proof of (A1) with δ(t, ˆT ) = P P L2 (Ω) The reversed triangle inequality in R J bounds the LHS in (A1), namely η( ˆT, T ˆT ) η(t, T ˆT ) = η(t ˆT ) η(t ˆT ), from above by η(t ) η(t ) 2 = α 2T + β 2 T αt β2 T }{{} T T ˆT T T ˆT β T β T 2 (triangle ineq. in R 2 ) The reversed triangle inequality in R 3 and L 2 (E) show β T β T = T 1/4 [ P ] E 2 L 2 (E) T 1/4 E E(T ) E E(T ) [ P P ] E 2 L 2 (E). Altogether, η(t ˆT ) η(t ˆT ) E E(T ) [P ] E 2 L 2 (E) T 1/2 [ P P ] E 2 L 2 (E) T T ˆT E E(T ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

9 Proof of (A1) with δ(t, ˆT ) = P P L2 (Ω) Recall η(t ˆT ) η(t ˆT ) T T ˆT T 1/2 E E(T ) [ P P ] E 2 L 2 (E) and apply the discrete jump control lemma for each component of the piecewise polynomial vector field P P P 0 ( ˆT ; R 2 ). This concludes the proof of (A1). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

10 Proof of (A2) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

11 Proof of (A2) with ϱ 2 = 2 1/4 and Λ 2 = Λ 1 Recall that ˆT is an admissible refinement of T with respective discrete fluxes P P 0 ( ˆT ; R 2 ) and P P 0 (T ; R 2 ) as before. Given any refined triangle T ˆT (K) := {T ˆT : T K} for K T \ ˆT, recall α T := T 1/2 f L 2 (T ) and βt 2 := T 1/2 [P ] F 2 L 2 (F ) resp. β2 T := T 1/2 [ P ] F 2 L 2 (F ) F E(T ) LHS in (A2) reads η( ˆT \ T ) = K T \ ˆT T ˆT (K) K T \ ˆT T ˆT (K) (αt 2 + βt 2 ) + } {{ } (i) F E(T ) (α 2 T + β 2 T ) (triangle ineq. in l2 ) K T \ ˆT T ˆT (K) ( β T β T ) 2. } {{ } (ii) Observe [P ] F = 0 for F Ê(int(K)) and T K /2 for T ˆT (K). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

12 Proof of (A2) with ϱ 2 = 2 1/4 and Λ 2 = Λ 1 Since [P ] F = 0 for F Ê(int(K)) and T K /2 for T ˆT (K) and K T \ ˆT, (i) := (αt 2 + βt 2 ) K 2 f 2 L 2 (K) + K 1/2 [P ] E 2 2 L 2 (E). T ˆT (K) E E(K) Reversed triangle inequalities in the second term prove β T β T = T 1/4 [ P ] F 2 L 2 (F ) [P ] F 2 L 2 (F ) F E(T ) F E(T ) T 1/4 [ P P ] F 2 and so lead to L 2 (F ) (ii) := K T \ ˆT T ˆT (K) F E(T ) (β T β T ) 2 K T \ ˆT T ˆT (K) T 1/2 [ P P ] F 2 L 2 (F ) F E(T ) This and the discrete jump control lemma conclude the proof. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

13 Quasiinterpolation Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

14 Discrete Quasiinterpolation Notation := L 2 (Ω) and := := H 1 (Ω) Theorem (approximation and stability). 0 < C = C(min T) < T T ˆT T (T ) ˆV S 1 0( ˆT ) V S 1 0(T ) V = ˆV on ˆT T and h 1 T ( ˆV V ) + V C ˆV. Proof. Define V S0 1 (T ) by linear interpolation of nodal values ˆV (z) if z N (Ω) N (T ) for some T T ˆT V (z) := ω z ˆV dx/ ωz if z N (Ω) and T (z) ˆT (z) = 0 if z N ( Ω) Since V and ˆV are continuous at any vertex of any T T ˆT, the first case applies in the definition of V (z) = ˆV (z) for all z N (T ). This proves V = ˆV on T T ˆT. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

15 Given any node z N in the coarse triangulation, let ω z = int( T (z)) denotes its patch of all triangles T (z) in T with vertex z. Lemma A. There exists C(z) diam(ω z ) with ˆV V (z) L 2 (ω z) C(z) ˆV L 2 (ω z). Proof4Case II: z N (Ω) and T (z) ˆT (z) = with V (z) = ω zˆv dx/ ωz. Then, the assertion is a Poincare inequality with C(z) = C P (ω z ). Proof4Case III: z N ( Ω) and V (z) = 0. Since ˆV V vanishes along the two edges along Ω of the open boundary patch ω z with vertex z. Hence the assertion is indeed a Friedrichs inequality with C(z) = C F (ω z ). Proof4Case I: T T (z) ˆT (z) for z N (Ω) and V = ˆV on T. This leads to homogenous Dirichlet boundary conditions on the two edges of the open patch ω z \ T with vertex z and ˆV V allows for a Friedrichs inequality (on the open patch as in Case III for a patch on the boundary) ˆV V L 2 (ω z) C F (ω z \ T ) ( ˆV V ) L 2 (ω z) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

16 However, this is not the claim! The idea is to realize that LHS= w L 2 (ω z) for w := ˆV ˆV (z), which is affine on T and vanishes at vertex z. Hence (as an other inverse estimate or discrete Friedrichs inequality) w L 2 (T ) C df (T ) w L 2 (T ) C df (T ) w L 2 (ω z) E.g. the integral mean w T := T w dx/ T of w := ˆV ˆV (z) on T satisfies w T 2 T C df (T ) 2 w 2 L 2 (ω z) Compare with integral mean w := w dx/ ω z and compute ω z w w T 2 T = T 1 (w w)dx 2 w w 2 L 2 (T ) T w w 2 L 2 (ω C z) P (ω z ) 2 w 2 L 2 (ω z) Consequently, w w T 2 ω z ω z / T }{{} C sr C P (ω z ) 2 w 2 L 2 (ω z) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

17 The orthogonality of 1 and w w in L 2 (ω z ) is followed by Poincare s and geometric-arithmetic mean inequality to verify w 2 L 2 (ω z) = w 2 ω z + w w 2 L 2 (ω z) 2 w w T 2 ω z + 2 w T 2 ω z + C P (ω z ) 2 w 2 L 2 (ω z) The above estimates for w T 2 T and w w T 2 T lead to w 2 L 2 (ω z) ( 2 ω z / T (C df (T ) + C P (ω z ) 2 ) + C P (ω z ) 2) }{{} =:C(z) 2 2 w L 2 (ω z) W.r.t. triangulation T and nodal basis functions ϕ 1, ϕ 2, ϕ 3 in S 1 (T ), let T = conv{p 1, P 2, P 3 } T and Ω T := ω 1 ω 2 ω 3 for ω j := {ϕ j > 0} Lemma B. There exists C(T ) h T with ˆV V L 2 (T ) C(T ) ˆV L 2 (Ω T ). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

18 Proof of Lemma B. N.B. V = 3 j=1 V (P j) ϕ j and 1 = 3 j=1 ϕ j on T Hence ˆV V 2 L 2 (T ) = ( T T 3 ( ˆV V (P j )) ϕ j 2 dx j=1 3 ( ˆV 3 V (P j ) 2 )( j=1 3 ˆV V (P j ) 2 L 2 (T ) j=1 k=1 ϕ 2 k }{{} 1 3 C(P j ) 2 ˆV 2 L 2 (ω j ) (Lemma A) j=1 3 C(P j ) 2 ) ˆV 2 L 2 (Ω T ) j=1 } {{ } C 2 (T ) ) dx (CS in R 3 ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

19 Lemma C. There exists C > 0 (which solely depends on min T) with V L 2 (T ) C ˆV L 2 (Ω T ). Proof. N.B. V = 3 j=1 V (P j) ϕ j and 0 = 3 j=1 ϕ j on T Hence 3 V 2 L 2 (T ) = ( ˆV V (P j )) ϕ j 2 dx T T ( j=1 3 ˆV V (P j ) 2 )( j=1 C(min T ) 2 h 2 T... (as before)... 3 j=1 3 ϕ k 2 k=1 }{{} C(min T ) 2 /h 2 T C(min T ) 2 h 2 T C2 (T ) }{{} ˆV =:C 2 T ˆV V (P j ) 2 dx 2 L 2 (Ω T ) ) dx (CS in R 6 ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

20 Finish of proof of theorem: h 1 T ( ˆV V ) L 2 (Ω) + V ˆV. Lemma B and C show for some generic constant C > 0 hidden in the notation and any T T that h 1 T ( ˆV V ) 2 L 2 (T ) + V 2 L 2 (T ) ˆV 2 L 2 (Ω T ) Notation Ω T := z N (T ) ω x is the interior of the set of T plus one layer of triangles of T around. The sum over all those inequalities for T T concludes the proof because the overlap of (Ω T ) T T is bounded by generic constant C(min T). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

21 Proof of (A3) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

22 Proof of (A3) Given discrete solution U (resp. Û) of CFEM in PMP w.r.t. T (resp. refinement ˆT ), set ê := Û U S1 0 ( ˆT ) with quasiinterpolant e S 1 0 (T ) as above. Then, v := ê e satisfies δ 2 (T, ˆT ) = ê 2 = a(ê, v) = F (v) a(u, v) }{{} Res(v) A piecewise integration by parts with a careful algebra with the jump terms for appropriate signs shows a(u, v) = v [ U/ ν E ] E ds E E(Ω) E E(Ω) E E 1 v 2 L 2 (E) E E(Ω) E [ U/ ν E ] E 2 L 2 (E) Recall trace inequality E 1 v 2 L 2 (E) C tr(h 2 ω E v 2 L 2 (ω E ) + v 2 L 2 (ω E ) ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

23 to estimate E E(Ω) E 1 v 2 L 2 (E) Finish of Proof of (A3) E E(Ω) (h 2 ω E v 2 L 2 (ω E ) + v 2 L 2 (ω E ) ) h 1 T v 2 L 2 (Ω) + v 2 ê 2 with the approximation and stability of the quasiinterpolation. A weighted Cauchy inequality followed by approximation property of quasiinterpolation show F (v) h T f L 2 (Ω) h 1 T v L 2 (Ω) C h T f L 2 (Ω) ê All this plus shape-regularity (e.g. T h 2 T h2 E ) lead to reliability δ 2 (T, ˆT ) = ê 2 Λ 3 η(t ) ê The extra fact v = 0 on T ˆT and a careful inspection on disappearing integrals in the revisited analysis prove the asserted upper bound in (A3), δ(t, ˆT ) Λ 3 η(t, T \ ˆT ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

24 Proof of (A4) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

25 (A4) follows from (A3) for CFEM with Λ 4 = Λ 2 3 The pairwise Galerkin orthogonality in the CFEM allows for the (modified) LHS in (A4) the representation l+m k=l δ 2 (T k, T k+1 ) = δ 2 (T l, T l+m+1 ) for m N 0. (A3) shows that this is bounded from above by Λ 2 3 η2 l. Since m N 0 is arbitrary, this implies k=l l+m δ 2 (T k, T k+1 ) = lim δ 2 (T k, T k+1 ) Λ 2 m 3ηl 2. k=l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

26 Outlook at Applications Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

27 Elastoplasticity Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

28 An Optimal Adaptive FEM for Elastoplasticity [Carstensen-Schröder-Wiedemann: An optimal adaptive FEM for elastoplasticity, Numer. Math. (2015)] Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

29 Computational Benchmark in Elastoplasticity Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

30 Convergence History in Computational Benchmark Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

31 Affine Obstacle Problem Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

32 An Optimal Adaptive FEM for an Obstacle Problem Reference: An optimal Adaptive FEN for an obstacle problem. Carstensen-Hu Jun, CMAM [Online Since 13/06/2015] Given RHS F H 1 (Ω) (dual to H0 1 (Ω) w.r.t. energy scalar product a) and affine obstacle χ P 1 (Ω) s.t. K := {v H 1 0 (Ω) : χ v a.e. in Ω}, the obstacle problem allows for a unique weak solution u K to F (v u) a(u, v u) for all v K. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

33 An Optimal Adaptive FEM for an Obstacle Problem Reference: An optimal Adaptive FEN for an obstacle problem. Carstensen-Hu Jun, CMAM [Online Since 13/06/2015] Conforming discretization leads to discrete solution u l and a posteriori error control via for any interior edge E. η 2 E := h E [ u l ] E ν E 2 L 2 (E) + Osc2 (f, ω E ) Theorem (Carstensen-Hu 2015). AFEM leads to optimal convergence rates. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

34 Eigenvalue Problems Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

35 SIAM Student Paper Prize 2013 for Joscha Gedicke Eigenvalue Problem u = λu in Ω and u = 0 on Ω Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36

36 Optimal Computational Complexity - 3D [Carstensen-Gedicke (2013) SINUM] l 2, l P1 l adaptive 2/3 2 P1 l adaptive 1 P2 adaptive 4/9 l 2 P2 l adaptive 1 P3 l adaptive 2 1 P3 l adaptive P4 adaptive l 2 P4 adaptive l 4/3 P4 1 l uniform CPU time (sec) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 3 Shanghai / 36