Convergence Rates for AFEM: PDE on Parametric Surfaces

Transcription

1 Department of Mathematics and Institute for Physical Science and Technology University of Maryland Joint work with A. Bonito and R. DeVore (Texas a&m) J.M. Cascón (Salamanca, Spain) K. Mekchay (Chulalongkorn, Thailand) P. Morin (Santa Fe, Argentina) S. Walker (Lousianna) WSC: The thirty-eighth Woudschoten Conference Woudschoten, October 2-4, 2013

2 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

3 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

4 Electrowetting on Dielectric: Modeling (w. A. Bonito and S. Walker) Mixed Formulation (u velocity, p pressure, H curvature, λ multiplier) α u + βu + p = 0 t in div u = 0 Interface Motion p = H + {z} E + P 0sign (u ν) {z } + D viscu ν {z } electric actuation λ(contact line pinning) viscous damping u ν = tx ν on Γ Ω in Ω on Γ

5 Electrowetting on Dielectric: Modeling (w. A. Bonito and S. Walker) Mixed Formulation (u velocity, p pressure, H curvature, λ multiplier) α u + βu + p = 0 t in div u = 0 Interface Motion p = H + {z} E + P 0sign (u ν) {z } + D viscu ν {z } electric actuation λ(contact line pinning) viscous damping u ν = tx ν on Γ Ω in Ω on Γ

6 Electrowetting on Dielectric: Experiments vs Simulations Moving droplet stirred around by varying voltages

7 Electrowetting on Dielectric: Experiments vs Simulations Splitting of glycerin droplet due to voltage actuation

8 Biomembranes: Modeling (w. A. Bonito and M.S. Pauletti) Bending (Willmore) energy: J(Γ) = 1 2 R Γ H2, H mean curvature Geometric Gradient Flow (with area and volume constraint): v = δ ΓJ = ΓH H3 2κH ν λhν + pν where Γ is the Laplace-Beltrami operator on Γ. Fluid-Membrane Interaction (with area constraint): ρd tv div ( pi + µd(v) ) = b in Ω t, {z } Σ div v = 0 in Ω t, [Σ]ν = δ ΓJ on Γ t

9 Biomembrane: Geometric vs Fluid Red Blood Cell play

10 Director Fields on Flexible Surfaces (w. S. Bartels and G. Dolzmann) Coupling of mean curvature H = div Γ ν with a director field n via J(Γ, n) = 1 Z div Γ ν δ div Γ n 2 dσ + λ Z Γn 2 dσ 2 Γ 2 Γ + 1 Z µ( n 2 1)dσ + 1 Z f(n ν)dσ 2 Γ 2ε Γ µ the Lagrange multiplier for the rigid constraint n = 1 f(x) = (x 2 ξ 2 0) 2 with ξ 0 [0, 1] penalizes the deviation of the angle between ν and n from arccos ξ 0 Spontaneous curvature H 0 = δ div Γ n induced by director field n Relaxation dynamics (L 2 - gradient flow): V normal velocity of Γ V = δ ΓJ(Γ, n), tn = δ nj(γ, n)

11 Coupling of Director Fields and Flexible Surfaces: Simulations Cone-like structure near positive degree-one defects pointing outwards stomatocyte shape

12 The Laplace-Beltrami Operator and Curvature Vector curvature: H = Hν = ΓX, X = identity on Γ (Dziuk 91) Semi-implicit Time Discretization (t n t n+1): explicit geometry (Γ = Γ n, Γ = Γ n, ν = ν n ) Z Z H n+1 Ψ = Γ n Γ nx n+1 : Γ nψ, Γ n X n+1 = X n + τ n V n+1 R Γ n H n+1 Ψ τ n R Γ n Γ nv n+1 : Γ nψ = R Γ n Γ nx n : Γ nψ Mixed Method: operator splitting Velocity (gradient flow or Navier-Stokes) Curvature (Laplace-Beltrami)

13 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

14 The Laplace-Beltrami Problem γu = f on γ, u = 0 on γ. γ is a parametric Lipschitz surface, piecewise smooth, with Lipschitz boundary γ. f L 2 (γ) (see Cohen-DeVore-Nochetto for H 1 (γ) data). F k γu = ũ ( ũ)ν ν, γ = div γ γ. Weak Formulation Z K Seek u H 1 0 (γ) : Z Z γu γv = f v, Z γu γv = (u F k ) T G 1 K (v F p K) det(g K), bk γ γ bk v H0 1 (γ). G K = DFKDF T K

15 The Finite Element Method on Surfaces Γ 0 = n i=1γ i 0 is an initial polyhedral domain with nodes lying on γ P i : Γ i 0 γ i parametrization of γ (globally Lipschitz, C 1 on Γ i 0). Assume, for simplicity, one such Γ i 0 (i.e. n = 1) and drop the index i. A sequence of refinements of Γ 0 is obtained as follows: (1) subdivide Γ 0 onto b T k ; (2) Determine T k using map P to place the new nodes on γ. Hence, the new polygonal approximation is given by Γ k = Range(I ctk P ). I ctk : Lagrange interpolant onto continuous piecewise linears P γ T 1 Γ 1 T 0 = b T 0, Γ 0 b T1 V( b T k ) is the space of continuous piecewise linear polynomials subordinate to bt k and V k := V(T k ) is its lift using I btk P. Discrete Formulation: the geometry changes with iteration counter k Z Z Seek U k V k : Γk U k Γk V = f q V, V V k. Γ k Γ k Q k

16 Standard Adaptive Algorithm SOLVE : ESTIMATE : MARK : Compute the solution U k V k := V(T k ) of the discrete problem. Compute a local estimator η k (U k, K), K T k, for the error in terms of the discrete solution U k and given data. Use the estimator to mark a subset M k T k for refinement P K M k η k (U k, K) 2 θ 2 P K T k η k (U k, K) 2. (Dörfler marking) REFINE : Quasi-Optimal Algorithm Refine the marked subset M k to obtain T k+1, conforming or with hanging nodes, increment k and go to step SOLVE. If f L 2 (Ω) and the decay rate for the best approximation of u is inf #T #T 0 N inf γ(u V ) L 2 (γ) C 1N s, 0 < s 1/d V V(T ) then the finite element method delivers the same rate γ(u U k ) L 2 (γ) C 2(#T k ) s

17 Standard Adaptive Algorithm SOLVE : ESTIMATE : MARK : Compute the solution U k V k := V(T k ) of the discrete problem. Compute a local estimator η k (U k, K), K T k, for the error in terms of the discrete solution U k and given data. Use the estimator to mark a subset M k T k for refinement P K M k η k (U k, K) 2 θ 2 P K T k η k (U k, K) 2. (Dörfler marking) REFINE : Quasi-Optimal Algorithm Refine the marked subset M k to obtain T k+1, conforming or with hanging nodes, increment k and go to step SOLVE. If f L 2 (Ω) and the decay rate for the best approximation of u is inf #T #T 0 N inf γ(u V ) L 2 (γ) C 1N s, 0 < s 1/d V V(T ) then the finite element method delivers the same rate γ(u U k ) L 2 (γ) C 2(#T k ) s

18 Standard Adaptive Algorithm SOLVE : ESTIMATE : MARK : Compute the solution U k V k := V(T k ) of the discrete problem. Compute a local estimator η k (U k, K), K T k, for the error in terms of the discrete solution U k and given data. Use the estimator to mark a subset M k T k for refinement P K M k η k (U k, K) 2 θ 2 P K T k η k (U k, K) 2. (Dörfler marking) REFINE : Quasi-Optimal Algorithm Refine the marked subset M k to obtain T k+1, conforming or with hanging nodes, increment k and go to step SOLVE. If f L 2 (Ω) and the decay rate for the best approximation of u is inf #T #T 0 N inf γ(u V ) L 2 (γ) C 1N s, 0 < s 1/d V V(T ) then the finite element method delivers the same rate γ(u U k ) L 2 (γ) C 2(#T k ) s

19 Standard Adaptive Algorithm SOLVE : ESTIMATE : MARK : Compute the solution U k V k := V(T k ) of the discrete problem. Compute a local estimator η k (U k, K), K T k, for the error in terms of the discrete solution U k and given data. Use the estimator to mark a subset M k T k for refinement P K M k η k (U k, K) 2 θ 2 P K T k η k (U k, K) 2. (Dörfler marking) REFINE : Quasi-Optimal Algorithm Refine the marked subset M k to obtain T k+1, conforming or with hanging nodes, increment k and go to step SOLVE. If f L 2 (Ω) and the decay rate for the best approximation of u is inf #T #T 0 N inf γ(u V ) L 2 (γ) C 1N s, 0 < s 1/d V V(T ) then the finite element method delivers the same rate γ(u U k ) L 2 (γ) C 2(#T k ) s

20 Standard Adaptive Algorithm SOLVE : ESTIMATE : MARK : Compute the solution U k V k := V(T k ) of the discrete problem. Compute a local estimator η k (U k, K), K T k, for the error in terms of the discrete solution U k and given data. Use the estimator to mark a subset M k T k for refinement P K M k η k (U k, K) 2 θ 2 P K T k η k (U k, K) 2. (Dörfler marking) REFINE : Quasi-Optimal Algorithm Refine the marked subset M k to obtain T k+1, conforming or with hanging nodes, increment k and go to step SOLVE. If f L 2 (Ω) and the decay rate for the best approximation of u is inf #T #T 0 N inf γ(u V ) L 2 (γ) C 1N s, 0 < s 1/d V V(T ) then the finite element method delivers the same rate γ(u U k ) L 2 (γ) C 2(#T k ) s

21 Quasi-Optimal Adaptive Finite Element Methods (AFEM) Babuska, Vogelius, 1986 (1D problem). Binev, Dahmen, DeVore, 2004 (2D problem, coarsening). Stevenson, 2006 (marking by oscillation). Kreuzer, Cascon, Nochetto, Siebert, Bonito, Nochetto, 2010 (dg). Cohen, DeVore, Nochetto, 2011 (H 1 data and approximation classes). Diening, Kreuzer, Stevenson, 2013 (maximum strategy). Sufficient Condition (for best approximation of u to decay with rate N 1/d ) sob (W 2 p ) > sob (H 1 ) 2 d p > 1 d 2 Main Ingredients Orthogonality (Pythagoras): p > 2d 2 + d (u U k ) 2 L 2 (Ω) = (u U k 1 ) 2 L 2 (Ω) (U k U k 1 ) 2 L 2 (Ω). Upper and Lower Bounds: (u U k ) 2 L 2 (Ω) η 2 k(u k, T k ) (u U k ) 2 L 2 (Ω) + osc 2 (U k, T k ). Monotonicity of Estimator: η k+1 (U k, T k+1 ) η k (U k, T k ).

22 C 1,0.4 surface with Lipschitz boundary C 1,0.4 Surface Re-entrant Corner

23 Standard AFEM Error indicator: ηt 2 (U, T ) := h 2 T f 2 L 2 (T ) + P S side h S [ Γk U] 2 L S T 2 (S) Dörfler parameter in MARK: θ = 10% (quite conservative). 1 Energy Error Optimal Order -1/2 Order -1/ suboptimal decay rate

24 Residual Based A-Posteriori Estimators on Surfaces Refs: Demlow-Dziuk (2007), Demlow (2009), Mekchay-Morin-Nochetto (2011) Upper Bound For U V a discrete Galerkin solution then γ(u U) 2 L 2 (γ) C 1 η(u, T ) 2 + Λ 1λ(T ) 2 where Λ 1 depends on T 0 and λ(t ) is the geometric estimator λ(t ) = max (P T T IT P ) L ((I T P ) 1 (T ) {z } Lower Bound C 2η(U, T ) 2 γ(u U) 2 L 2 (γ) + X h 2 T f q Q f q 2 +Λ 1λ(T ) 2. Q L T T 2 (T ) {z } data oscillation: osc(f,t ) 2 Quasi-Orthogonality For U V(T ), U V(T ) two Galerkin solutions, T a refinement of T, then γ(u U ) 2 L 2 (γ) γ(u U) 2 L 2 (γ) 1 2 γ(u U ) 2 L 2 (γ) + Λ 2λ(T ) 2. = b T ) P I T P

25 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

26 Adaptive Finite Element Method (AFEM) Let T 0 be the initial triangulation of γ, k = 0. Given ɛ 0 > 0 and ω > 0 T + k = GEOMETRY(T k, ωɛ k ) : Refine surface until max T T λ T (T ) ω ɛ k [T k+1, U k+1 ] = PDE(T + k, ɛ k) : Use the Dörfler marking and refine until η(u k+1, T k+1 ) ɛ k Set ɛ k+1 = ɛ k /2 and repeat. Equidistribute the errors, similar to R. Stevenson (2006) for oscillations.

27 The Module GEOMETRY: Greedy Algorithm Let T := T k T + = GEOMETRY (T, τ) while (max T T λ T (T ) > τ) Refine all T T such that λ T (T ) > τ Update T end Module GEOMETRY independent of the PDE Complexity of GEOMETRY?

28 The Module PDE [T, U] = PDE (T +, ɛ) while (η(u, T ) > ɛ) SOLVE ESTIMATE MARK REFINE end Update T Quasi-monotonicity of geometric estimator λ(t ) = max T T (P IT P ) L (T ) max T T (P I T +P ) L (T ) + max T T IT (P I T +P ) L (T ) 2 max T T + (P I T +P ) L (T ) = 2λ(T + ) Relation between geometric and PDE errors λ(t ) 2λ(T + k ) 2ωɛ 2ωη(U, T ) the geometric error λ(t ) is small relative to η(u, T ) and can be controlled

29 Conditional Contraction Property of Module PDE Theorem If λ(t j) 2ωη(U j, T j) for ω ω, then there exists constants 0 < α < 1 and β > 0 such that the inner iterates of the module PDE satisfy γ(u U j+1) 2 L 2 (γ)+βη(u j+1, T j+1) 2 α 2 γ(u U j) 2 L 2 (γ)+βη(u j, T j) 2. Moreover, the number J of inner iterates of PDE is uniformly bounded. The proof proceeds as in Cascón, Kreuzer, Nochetto, and Siebert (2008) and Bonito and Nochetto (2010), with the additional information λ(t ) 2ωη(U, T ) in the inner loops of PDE for ω ω sufficiently small. Reduction of error estimator: there exist constants 0 < ξ < 1 and Λ 2, Λ 3 > 0 such that for all δ > 0 η(u, T ) 2 (1 + δ)`η(u, T ) 2 ξη(u, M) 2 + (1 + δ 1 )`Λ 3 γ(u U) 2 L 2 (γ) + Λ 2λ(T ) 2.

30 Contracting Quantities of AFEM (in flat domains) Energy error: U k u Ω is monotone, but not strictly monotone (e.g. U k+1 = U k ). Ω = (0, 1) 2, A = I, f = 1 U 0 = U 1 = 1 φ0, U2 U1 12 Residual estimator: η k (U k, T k ) is not reduced by AFEM, and is not even monotone. But, if U k+1 = U k, then η k (U k, T k ) decreases strictly η 2 k+1(u k+1, T k+1 ) = η 2 k+1(u k, T k+1 ) η 2 k(u k, T k ) ξη 2 k(u k, M k ) Heuristics: the quantity U k u 2 Ω + βη2 k(u k, T k ) might contract! Additional term λ(t k ) for Laplace-Beltrami, but λ(t k ) 2ωη k (U k, T k ).

31 Contracting Quantities of AFEM (in flat domains) Energy error: U k u Ω is monotone, but not strictly monotone (e.g. U k+1 = U k ). Ω = (0, 1) 2, A = I, f = 1 U 0 = U 1 = 1 φ0, U2 U1 12 Residual estimator: η k (U k, T k ) is not reduced by AFEM, and is not even monotone. But, if U k+1 = U k, then η k (U k, T k ) decreases strictly η 2 k+1(u k+1, T k+1 ) = η 2 k+1(u k, T k+1 ) η 2 k(u k, T k ) ξη 2 k(u k, M k ) Heuristics: the quantity U k u 2 Ω + βη2 k(u k, T k ) might contract! Additional term λ(t k ) for Laplace-Beltrami, but λ(t k ) 2ωη k (U k, T k ).

32 Contracting Quantities of AFEM (in flat domains) Energy error: U k u Ω is monotone, but not strictly monotone (e.g. U k+1 = U k ). Ω = (0, 1) 2, A = I, f = 1 U 0 = U 1 = 1 φ0, U2 U1 12 Residual estimator: η k (U k, T k ) is not reduced by AFEM, and is not even monotone. But, if U k+1 = U k, then η k (U k, T k ) decreases strictly η 2 k+1(u k+1, T k+1 ) = η 2 k+1(u k, T k+1 ) η 2 k(u k, T k ) ξη 2 k(u k, M k ) Heuristics: the quantity U k u 2 Ω + βη2 k(u k, T k ) might contract! Additional term λ(t k ) for Laplace-Beltrami, but λ(t k ) 2ωη k (U k, T k ).

33 Contracting Quantities of AFEM (in flat domains) Energy error: U k u Ω is monotone, but not strictly monotone (e.g. U k+1 = U k ). Ω = (0, 1) 2, A = I, f = 1 U 0 = U 1 = 1 φ0, U2 U1 12 Residual estimator: η k (U k, T k ) is not reduced by AFEM, and is not even monotone. But, if U k+1 = U k, then η k (U k, T k ) decreases strictly η 2 k+1(u k+1, T k+1 ) = η 2 k+1(u k, T k+1 ) η 2 k(u k, T k ) ξη 2 k(u k, M k ) Heuristics: the quantity U k u 2 Ω + βη2 k(u k, T k ) might contract! Additional term λ(t k ) for Laplace-Beltrami, but λ(t k ) 2ωη k (U k, T k ).

34 Proof of Contraction: Step 1 e j = γ(u U j) L 2 (γ), E j = γ(u j+1 U j) L 2 (γ), η j = η(u j, T j), λ j = λ(t j). Combine, quasi-orthogonality of energy error e 2 j+1 e 2 j 1 2 E2 j + Λ 2λ 2 j with reduction of residual error estimator: ηj+1 2 (1 + δ)`ηj 2 ξη 2 j(m j) + (1 + δ 1 )`Λ 3Ej 2 + Λ 2λ 2 j to get e 2 j+1 + βηj+1 2 e 2 j β(1 + δ 1 )Λ 3 Ej 2 + Λ β(1 + δ 1 ) λ 2 j + β(1 + δ) ηj 2 ξη j(m j) 2. Choose β, depending on δ, so that β(1 + δ 1 )Λ 3 = 1 β(1 + δ) = δ. 2 2Λ 3 This implies e 2 j+1 + βηj+1 2 e 2 j + Λ β(1 + δ 1 ) λ 2 j + β(1 + δ) ηj 2 ξη j(m j) 2.

35 Proof of Contraction: Step 2 Use Dörfler marking η j(m j) θη j to derive η 2 j ξη j(m j) 2 `1 ξθ 2 η 2 j. Recall λ j 2λ + and properties λ + ωɛ and ɛ < η j to write e 2 j+1 + βη 2 j+1 e 2 j β(1 + δ) ξθ2 2 η2 j + β `1 + δ 1 ξθ2 2 + Λ ω 2 2Λ 3 β Employ upper bound e 2 j C 1 η j + Λ 1λ 2 j C 1`1 + 4ω 2 Λ 1 η2 j = C 3ηj 2 to deduce e 2 j+1+βηj δ ξθ2 2Λ {z 3C 3 } =α 1 (δ) e 2 j+ η 2 j. (1 + δ) 1 ξθ2 + Λ ω 2 2 2Λ 3 β {z } =α 2 (δ) η 2 j Choose δ = ξθ2 ξθ and β = 2 4 2ξθ 2 2Λ 3 to obtain α1, (4 ξθ 2 α2 < 1. )

36 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

37 Optimal Decay Rates Assumptions: Let 0 < s 1/d. The solution u and forcing f are of class A s, namely given an error tolerance ɛ > 0 there exists a refinement T ɛ of T 0 such that γ(u U ɛ) L 2 (γ) + osc(f, T ɛ) ɛ, #T ɛ #T 0 u, f As ɛ 1 s. The surface is of class B s and T + = GEOMETRY (T, τ) is s-optimal, i.e. #M + γ Bs τ 1 s. Theorem Assume that (u, f) are of class A s, that γ is of class B s, and that GEOMETRY is s-optimal. Then for θ θ and ω ω sufficiently small, we have γ(u U k ) L 2 (γ) + ω 1 λ Γk + osc(f, T k ) u, f As + γ Bs (#T k #T 0) s.

38 Ingredients of the Proof Localized upper bound (to the refined set) Minimality of set M in Dörfler marking Explicit restriction of Dörfler parameter θ < θ < 1 Explicit restriction of surface parameter ω ω < 1 Conditional contraction property of PDE Complexity of REFINE (Binev-Dahmen-DeVore (d = 2), Stevenson (d > 2), for conforming meshes, and Bonito-Nochetto for non-conforming meshes (d 2)).

39 Greedy Algorithm for GEOMETRY T + = GEOMETRY(T, τ) Theorem (Quasi-optimality of GEOMETRY for continuous pw linears) Let γ be a surface of dimension d and piecewise of class W 2 p, p > d (over the initial partition T 0). Then the greedy algorithm terminates and is 1/d-optimal #M + γ d W 2 p τ d. Therefore, γ W 2 p for p > d is of class B 1 d. Note that sob (W 2 p ) > sob (W 1 ) 2 d p > 1 d p > d but NOT γ W 2.

40 The Role of ω for Convergence Rates Consider the example γu = 1, in γ, u = 0, on γ, where γ is the graph of class C 1,α given by z(x, y) = `0.75 x 2 y 2 1+α +, over the flat domain Ω = (0, 1) 2, and consider two cases α = 3/5 and α = 2/5. It turns out that α = 3/5 : z B 1/2 α = 2/5 : z B t, t < 2/5.

41 The Role of ω for Convergence Rates: Case α = 3/ ω = 0.1 ω = 1 ω = 10 N ω = 0.1 ω = 1 ω = 10 N η+λ/ω η+λ Figure: η k + λ k /ω (left) and η k + λ k (right) versus the number of elements in logarithmic scale for ω = 0.1, 1, 10. We observe that η k + λ k /ω decays as N 0.5 right from the beginning, whereas η k + λ k shows the same decay after the meshes have some refinement, depending on the value of ω. Our theory predicts the decay of N 0.5 for both notions of total error if ω is sufficiently small, but the best relation between the error η k + λ k and #DOFs is obtained for w = 1, which is not so small.

42 10 1 N N N 0.5 The Role of ω for Convergence Rates: Case α = 3/ η+λ/ω η λ/ω 10 0 η+λ/ω η λ/ω 10 1 η+λ/ω η λ/ω ω = 0.1 ω = 1 ω = 10 Figure: η k, λ k /ω and η k + λ k /ω for ω = 0.1 (left) ω = 1 (middle) and ω = 10 (right). Figure: Meshes after 10, 20 and 30 refinements have been performed, C 1,0.6 -surface, with ω = 1. They are composed of 192, 1216 and 5564 elements, respectively.

43 The Role of ω for Convergence Rates: Case α = 2/ ω = 0.1 ω = 1 ω = 10 N ω = 0.1 ω = 1 ω = 10 N η+λ/ω η+λ Figure: η k + λ k /ω (left) and η k + λ k (right) versus the number of elements in logarithmic scale for ω = 0.1, 1, 10. We observe that η k + λ k /ω decays as N 0.4 right from the beginning, whereas η k + λ k shows the same decay after the meshes have some refinement, depending on the value of ω. Our theory predicts the decay of N 0.4 for both notions of total error if ω is sufficiently small. The best relation between the error η k + λ k seems to occur for ω = 1 and ω = 10.

44 10 2 N N N 0.5 The Role of ω for Convergence Rates: Case α = 2/ η+λ/ω η λ/ω N η+λ/ω η λ/ω N η+λ/ω η λ/ω N ω = 0.1 ω = 1 ω = 10 Figure: η k, λ k /ω and η k + λ k /ω for ω = 0.1 (left) ω = 1 (middle) and ω = 10 (right).

45 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

46 Discontinuous Coefficients (w. A. Bonito and R. DeVore) Consider elliptic PDE of the form div(a u) = f with A = (a ij(x)) d i,j=1 uniformly positive definite and bounded λ min(a) y 2 y t A(x)y λ max(a) y 2 x Ω, y R d ; The discontinuities of A are not match by the sequence of meshes T ; The forcing f W 1 p (Ω) for some p > 2. Goal: Design and study an AFEM able to handle such an A. Difficulty: PDE perturbation results hinge on approximation of A in L u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L (Ω) f H 1 (Ω)

47 Discontinuous Coefficients (w. A. Bonito and R. DeVore) Consider elliptic PDE of the form div(a u) = f with A = (a ij(x)) d i,j=1 uniformly positive definite and bounded λ min(a) y 2 y t A(x)y λ max(a) y 2 x Ω, y R d ; The discontinuities of A are not match by the sequence of meshes T ; The forcing f W 1 p (Ω) for some p > 2. Goal: Design and study an AFEM able to handle such an A. Difficulty: PDE perturbation results hinge on approximation of A in L u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L (Ω) f H 1 (Ω)

48 Discontinuous Coefficients (w. A. Bonito and R. DeVore) Consider elliptic PDE of the form div(a u) = f with A = (a ij(x)) d i,j=1 uniformly positive definite and bounded λ min(a) y 2 y t A(x)y λ max(a) y 2 x Ω, y R d ; The discontinuities of A are not match by the sequence of meshes T ; The forcing f W 1 p (Ω) for some p > 2. Goal: Design and study an AFEM able to handle such an A. Difficulty: PDE perturbation results hinge on approximation of A in L u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L (Ω) f H 1 (Ω)

49 Perturbation Argument Theorem (perturbation). Let p 2, q = 2p/(p 2) [2, ] and u L p (Ω). Then u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L q (Ω) u L p (Ω) Question: can we guarantee that u L p (Ω) with p > 2? Proposition (Meyers). Let e K > 0 be so that the solution eu of the Laplacian satisfies eu L p (Ω) e K f W 1 p (Ω). Then the solution u of div(a u) = f satisfies if 2 p < p and K = u L p (Ω) K f W 1 p (Ω) 1 η(p)` ek η(p) λ max(a) 1 K e 1 λ min (A) λmax(a) with η(p) = p p

50 Perturbation Argument Theorem (perturbation). Let p 2, q = 2p/(p 2) [2, ] and u L p (Ω). Then u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L q (Ω) u L p (Ω) Question: can we guarantee that u L p (Ω) with p > 2? Proposition (Meyers). Let e K > 0 be so that the solution eu of the Laplacian satisfies eu L p (Ω) e K f W 1 p (Ω). Then the solution u of div(a u) = f satisfies if 2 p < p and K = u L p (Ω) K f W 1 p (Ω) 1 η(p)` ek η(p) λ max(a) 1 K e 1 λ min (A) λmax(a) with η(p) = p p

51 Perturbation Argument Theorem (perturbation). Let p 2, q = 2p/(p 2) [2, ] and u L p (Ω). Then u bu H 1 0 (Ω) λ 1 min( A) b f f b H 1 (Ω) + A A b L q (Ω) u L p (Ω) Question: can we guarantee that u L p (Ω) with p > 2? Proposition (Meyers). Let e K > 0 be so that the solution eu of the Laplacian satisfies eu L p (Ω) e K f W 1 p (Ω). Then the solution u of div(a u) = f satisfies if 2 p < p and K = u L p (Ω) K f W 1 p (Ω) 1 η(p)` ek η(p) λ max(a) 1 K e 1 λ min (A) λmax(a) with η(p) = p p

52 DISC: AFEM for Discontinuous Diffusion Matrices Given ω > 0 explicit and β < 1, let DISC(T 0, ɛ 1) k = 1 LOOP [T (f) k, f k ] = RHS(T k 1, f, ωε k ) [T (A) k, A k ] = COEFF(T k (f), A, ωε k ) [T k, U k ] = PDE(T (A) k, A k, f k, ε k /2) ɛ k+1 = βɛ k k k + 1 END LOOP END DISC [T (f) k, f k ] = RHS(T k 1, f, ωε k ) gives a mesh T k (f) T k 1 and a pw polynonial approximation f k of f on T F k such that f f k H 1 (Ω) ωɛ k ; [T (A) k, A k ] = COEFF(T k (f), A, ωε k ) gives a mesh T k (A) T k (f) and a pw polynomial approximation A k of A on T k (A) such that A A k L q (Ω) ωɛ k and its eigenvalues satisfy uniformly in k C 1 λ min(a) λ(a k ) Cλ max(a).

53 DISC: AFEM for Discontinuous Diffusion Matrices Given ω > 0 explicit and β < 1, let DISC(T 0, ɛ 1) k = 1 LOOP [T (f) k, f k ] = RHS(T k 1, f, ωε k ) [T (A) k, A k ] = COEFF(T k (f), A, ωε k ) [T k, U k ] = PDE(T (A) k, A k, f k, ε k /2) ɛ k+1 = βɛ k k k + 1 END LOOP END DISC [T (f) k, f k ] = RHS(T k 1, f, ωε k ) gives a mesh T k (f) T k 1 and a pw polynonial approximation f k of f on T F k such that f f k H 1 (Ω) ωɛ k ; [T (A) k, A k ] = COEFF(T k (f), A, ωε k ) gives a mesh T k (A) T k (f) and a pw polynomial approximation A k of A on T k (A) such that A A k L q (Ω) ωɛ k and its eigenvalues satisfy uniformly in k C 1 λ min(a) λ(a k ) Cλ max(a).

54 Optimality of DISC Theorem (optimality). Assume that the right side f is in B s f (H 1 (Ω)) with 0 < s f S, and that the diffusion matrix A is positive definite, in L (Ω) and in M s A (L q(ω)) for q := 2p and 0 < sa S. Let T0 be the initial subdivision p 2 and U k V(T k ) be the Galerkin solution obtained at the kth iteration of the algorithm. Then, whenever u A s (H0 1 (Ω)) for 0 < s S, we have for k 1 u U k H 1 0 (Ω) ɛ k, and #T k #T 0 A 1/s Ms (Lq(Ω)) + f 1/s + B (H 1 (Ω)) u 1/s A (H1 0 (Ω)) with s = min(s, s A, s f ). ɛ 1/s k, Counterexample: s cannot be achieved if s A, s f < s.

55 Optimality of DISC Theorem (optimality). Assume that the right side f is in B s f (H 1 (Ω)) with 0 < s f S, and that the diffusion matrix A is positive definite, in L (Ω) and in M s A (L q(ω)) for q := 2p and 0 < sa S. Let T0 be the initial subdivision p 2 and U k V(T k ) be the Galerkin solution obtained at the kth iteration of the algorithm. Then, whenever u A s (H0 1 (Ω)) for 0 < s S, we have for k 1 u U k H 1 0 (Ω) ɛ k, and #T k #T 0 A 1/s Ms (Lq(Ω)) + f 1/s + B (H 1 (Ω)) u 1/s A (H1 0 (Ω)) with s = min(s, s A, s f ). ɛ 1/s k, Counterexample: s cannot be achieved if s A, s f < s.

56 Outline Motivation: Geometric PDE The Laplace-Beltrami Operator and Standard Adaptivity Adaptive Finite Element Methods (AFEM) on Parametric Surfaces Convergence Rates of AFEM Discontinuous Coefficients Comments and Conclusions

57 Comments and Conclusions Coupling PDE-Geometry: This is a new feature in adaptivity and leads to separate handling of geometry and PDE resolution with specific relative tolerances. Convergence rates: We show optimal convergence rates in the energy norm (u U k ) L 2 (γ) (#T k ) s provided this is the rate of the best approximation of u in H 1 and that of γ in W 1. Weaker conditions on f: We refer to Cohen, DeVore, Nochetto (2011) for convergence rates of elliptic PDE in flat domains with f H 1 and A piecewise constant: div(a u) = f. We show that approximability of u is sufficient for a complete theory. Weaker conditions on γ: We assume γ is Wp 2 with p > d, which implies γ is C 1. In the flat case, this corresponds to piecewise continuous A. We refer to Bonito, DeVore, Nochetto (2013) for convergence rates with weaker assumptions on A.