CENTRAL School on Analysis and Numerics for PDEs November 09-12, 2015 Adaptive Boundary Element Methods Part 2: ABEM Dirk Praetorius TU Wien Institute for Analysis and Scientific Computing
Outline 1 Boundary Element Method 2 2D Example 3 Stability (A1) & Reduction (A2) 4 Discrete Reliability (A3) 5 Linear Convergence 6 Optimality 7 3D Examples http://www.asc.tuwien.ac.at/ praetorius/central02 praetorius.pdf Dirk Praetorius (TU Wien) 1 / 75
Boundary Element Method Dirk Praetorius (TU Wien)
Boundary Element Method Newton kernel G(z) := Fundamental solution 1 2π + 1 4π log z for d = 2 1 z for d = 3 note S 2 2 = 2π and S3 2 = 4π for unit sphere Sd 2 Rd j G(z) = 1 S d 2 z j z d jk G(z) = 1 δ jk z 2 d z j z k S2 d z d+2 = G(z) = 0 for z 0 polar coordinates = G L 2 loc (Rd ), j G L 1 loc (Rd ) Dirk Praetorius (TU Wien) 2 / 75
Boundary Element Method Representation formula Proposition (Representation formula) Ω R d bounded u C 2 (Ω) define f := u define φ := n u ˆ ˆ = u(x) = G(x y)f(y) dy + G(x y) φ(y) ds y Ωˆ Γ n(y) G(x y) u(y) ds y for all x Ω Γ proof by integration by parts for Ω ε := Ω\B ε (x) and then ε 0 v(y) := G(x y) C (Ω) = u ; v L 2 (Ω ε) + n u ; v L 2 (Γ ε) = v ; u L 2 (Ω ε) + n v ; u L 2 (Γ ε) if trace u Γ and normal derivative n u known on Γ = solution of u = f can be computed in Ω Dirk Praetorius (TU Wien) 3 / 75
Boundary Element Method ˆ Ṽ φ(x) := Γ Potential operators G(x y) φ(y) ds y G(z) = 0 = Ṽ φ(x) = 0 in Ω ˆ Ku(x) := Γ n(y) G(x y) u(y) ds y G(z) = 0 = Ku(x) = 0 in Ω ˆ Ñf(x) := Ω G(x y)f(y) dy single-layer potential double-layer potential Newton potential representation formula = u = Ñ( u) + Ṽ ( nu) K(u Γ ) in Ω Dirk Praetorius (TU Wien) 4 / 75
Boundary Element Method Double-layer potential representation formula u = Ñ( u) + Ṽ ( nu) K(u Γ ) for u = 1 = K1 = 1 in Ω, since u = 0 = n u same argument applied for Ω ε := Ω B ε (x) = K1 = 0 in R d \Ω Ku has a jump ˆ across Γ Ku(x) := n(y) G(x y) u(y) dy for x R d \Ω Γ note critical singularity order (d 1) of the integral kernel! ˆ Consider Ku(x) := C = K1 = 1/2 a.e. on Γ Γ n(y) G(x y) u(y) dy for x Γ can show (K 1/2)u = ( Ku) Γ a.e. on Γ Dirk Praetorius (TU Wien) 5 / 75
Boundary Element Method Newton potential ˆ Ñf(x) := Ω G(x y)f(y) dy = G f is convolution f Cc (Ω) = Ñ C (R d ) with f = Ñ( f) = (Ñf) f = Ñ( f) follows from representation formula Ñ( f) = (Ñf) follows from convolution + integration by parts therefore: G is called fundamental solution of Dirk Praetorius (TU Wien) 6 / 75
Boundary Element Method Sobolev spaces 1/3 each f L 2 (Ω) defines linear continuous function on H 1 (Ω) f(v) f ; v L 2 (Ω) are these all linear continuous functionals on H 1 (Ω)? essentially yes! Lemma (idea of Gelfand triple) X, Y real Hilbert spaces with continuous inclusion X Y Riesz mapping J Y : Y Y, J Y y := y ; Y = J Y L(Y, X ) with J Y (Y ) X dense X = H 1 (Ω), Y = L 2 (Ω) extended L 2 scalar product becomes duality bracket Dirk Praetorius (TU Wien) 7 / 75
Boundary Element Method Sobolev spaces 2/3 Lemma (idea of Gelfand triple) X, Y real Hilbert spaces with continuous inclusion X Y Riesz mapping J Y : Y Y, J Y y := y ; Y = J Y L(Y, X ) with J Y (Y ) X dense Proof. x X, y Y = y ; x Y y Y x Y y Y x X = y ; Y X y Y = J Y L(Y, X ) J Y (Y ) X dense V := J 1 X J Y (Y ) X dense X = V V = goal: V = {0} x V = 0 = x ; J 1 X J Y x X = (J Y x)(x) = x ; x Y = x 2 Y Dirk Praetorius (TU Wien) 8 / 75
Boundary Element Method Sobolev spaces 3/3 H 1/2 (Γ) := { v L 2 (Γ) : v 2 H 1/2 (Γ) = v 2 L 2 (Γ) + v 2 H 1/2 (Γ) < } v 2 H 1/2 (Γ) :=ˆ Γ ˆ Γ v(x) v(y) 2 x y d ds y ds y Sobolev-Slobodeckij seminorm fact: H 1/2 (Γ) = { u Γ : u H 1 (Ω) } is trace space of H 1 (Ω) H 1/2 (Γ) is dual space of H 1/2 (Γ) w.r.t. L 2 (Γ)-scalar product since H 1/2 (Γ) L 2 (Γ) continuous Dirk Praetorius (TU Wien) 9 / 75
Boundary Element Method Single-layer potential & double-layer potential Theorem (single-layer potential) Ṽ L(H 1/2 (Γ), H 1 (Ω)) s.t. Ṽ φ = 0 for all φ H 1/2 (Γ) V φ := (Ṽ φ) Γ = V L(H 1/2 (Γ), H 1/2 (Γ)) (K + 1/2)φ := n (Ṽ φ) = K L(H 1/2 (Γ), H 1/2 (Γ)) Theorem (double-layer potential) K L(H 1/2 (Γ), H 1 (Ω)) s.t. Kv = 0 for all v H 1/2 (Γ) (K 1/2)v := ( Kv) Γ = K L(H 1/2 (Γ), H 1/2 (Γ)) W := n ( Kv) = W L(H 1/2 (Γ), H 1/2 (Γ)) Dirk Praetorius (TU Wien) 10 / 75
Boundary Element Method Single-layer integral operator Theorem (single-layer integral operator) V L(H 1/2 (Γ), H 1/2 (Γ)) is symmetric ˆ ˆ φ ; V ψ L 2 (Γ) = φ(x)g(x y)ψ(y) ds y ds x = ψ ; V φ L 2 (Γ) and elliptic (for diam(ω) < 1 in 2D) Γ Γ φ ; V φ L 2 (Γ) φ 2 H 1/2 (Γ) in particular, isomorphism Idea for ellipticity. u := Ṽ φ satisfies [ nu] := n u n extu = φ n u 2 H 1/2 (Γ) u 2 L 2 (Ω) = nu ; u Γ ext n u 2 H 1/2 (Γ) u 2 = L 2 (R d \Ω) extu ; u Γ = φ 2 H 1/2 (Γ) [ nu] ; u L 2 (Γ) = φ ; V φ L 2 (Γ) n Dirk Praetorius (TU Wien) 11 / 75
Boundary Element Method Weakly-singular integral equation solve u = 0 in Ω with u = g on Γ representation formula u = Ṽ ( nu) Kg in Ω trace g = V ( n u) (K 1/2)g on Γ = weakly-singular integral equation V φ = (K + 1/2)g on Γ has unique solution φ = n u discretization of weakly-singular IE = approximation φ Φ l = approximation u U l := Ṽ Φ l Kg Dirk Praetorius (TU Wien) 12 / 75
Boundary Element Method Comments BEM requires fundamental solution essentially requires homogeneous forces, e.g., u = 0 BEM leads to dense matrices BEM allows for higher convergence rates BEM only requires surface discretization BEM can treat unbounded domains Steinbach: Teubner, 2003 (German), 2008 (English) Sauter, Schwab: Teubner, 2004 (German), Springer 2011 (English) Dirk Praetorius (TU Wien) 13 / 75
Boundary Element Method What is HILBERT? Hilbert Is a Lovely Boundary Element Research Tool Matlab library for h-adaptive Galerkin BEM lowest-order elements for 2D Laplacian P 0 for normal derivatives S 1 for traces research code for FWF project P21732 overview on current state of the art starting point for further investigations http://www.asc.tuwien.ac.at/abem/hilbert/ Dirk Praetorius (TU Wien) 14 / 75
Boundary Element Method Example: Point Errors 10 0 10 2 N 2/3 estimators and point error in Ω 10 4 10 6 10 8 10 10 10 12 10 14 η D,l (unif., nodal) err Ω,l (unif., nodal) η D,l (unif., L 2 -projection) err Ω,l (unif., L 2 -projection) η D,l (adap., nodal) err Ω,l (adap., nodal) η D,l (adap., L 2 -projection) err Ω,l (adap., L 2 -projection) N 3 N 2 N 3/2 N 4/3 10 0 10 1 10 2 10 3 10 4 10 5 number N = #E l of boundary elements Dirk Praetorius (TU Wien) 15 / 75
Boundary Element Method Features of HILBERT C implementation of integral operators via MEX interface V (P 0 Γ P0 Γ ) K(S 1 Γ P0 Γ ) W (S 1 Γ S1 Γ ) N(P 0 Ω P0 Γ ) remaining codes in Matlab fully vectorized different error estimators (h-h/2, 2-Level, residual, Faermann) different marking strategies local mesh-refinement (1D bisection, 2D NVB) demo files and adaptive algorithms for weakly-singular integral equation hypersingular integral equation symmetric integral formulation of mixed BVP with/without volume force Dirk Praetorius (TU Wien) 16 / 75
2D Example Dirk Praetorius (TU Wien)
2D Example 2D model problem weakly-singular integral equation V u(x) := 1 ˆ log x y u(y) dy = f(x) 2π Γ Ω with Ω R 2 bounded and Lipschitz Variational formulation Find solution u H := H 1/2 (Γ) of Γ for x Γ u, v := V u ; v L 2 (Γ) = f ; v L 2 (Γ) for all v H, is scalar product induced norm v := v, v 1/2 v H 1/2 (Γ) Dirk Praetorius (TU Wien) 17 / 75
2D Example Galerkin BEM T l partition of Γ into affine line segments local mesh-width h l L (Γ), h l T := diam(t ) for T T l X l := P p (T l ) discrete subspace of H = H 1/2 (Γ) Galerkin formulation Find solution U l X l of U l, V l = f ; V l L 2 (Γ) for all V l X l Dirk Praetorius (TU Wien) 18 / 75
2D Example Weighted-residual error estimator Poincaré inequality w H 1 (Γ) with w P 0 (T l ) = w L 2 (Γ) h l w L 2 (Γ) w := f V U l P p (T l ) P 0 (T l ) Reliability (Carstensen, Stephan 95, 96, & Maischak 01) u U l f V U l H 1/2 (Γ) h1/2 l (f V U l ) L 2 (Γ) =: η l proof (later!) requires regularity of T l for localization 2D: uniformly bounded local mesh-ratio 3D: uniform γ-shape regularity Dirk Praetorius (TU Wien) 19 / 75
2D Example Adaptive algorithm initial mesh T 0 adaptivity parameter 0 < θ 1 For all l = 0, 1, 2,..., iterate 1 compute discrete solution U l for mesh T l 2 compute refinement indicators η l (T ) for all T T l 3 find (essentially minimal) set M l T l s.t. θ η l (T ) 2 η l (T ) 2 T T l T M l 4 refine (at least) marked elements T M l to obtain T l+1 Dirk Praetorius (TU Wien) 20 / 75
2D Example 1D example 1D boundary piece Γ = ( 1, 1) {0} u(x, y) = 2x/ 1 x 2 solution to f(x, y) = x singularities at ±(1, 0) Dirk Praetorius (TU Wien) 21 / 75
2D Example Solve on coarse mesh T 0 Dirk Praetorius (TU Wien) 22 / 75
2D Example Compute residual on T 0 η 0 (T ) = diam(t ) 1/2 (f V U 0 ) L 2 (T ) 1 η 0 (T ) 2 η 0 (T ) 2 4 T T 0 T M 0 Dirk Praetorius (TU Wien) 23 / 75
2D Example Mark elements M 0 T 0 for refinement η 0 (T ) = diam(t ) 1/2 (f V U 0 ) L 2 (T ) 1 η 0 (T ) 2 η 0 (T ) 2 4 T T 0 T M 0 Dirk Praetorius (TU Wien) 24 / 75
2D Example Solve on mesh T 1 Dirk Praetorius (TU Wien) 25 / 75
2D Example Compute residual on T 1 η 1 (T ) = diam(t ) 1/2 (f V U 1 ) L 2 (T ) 1 η 1 (T ) 2 η 1 (T ) 2 4 T T 1 T M 1 Dirk Praetorius (TU Wien) 26 / 75
2D Example Mark elements M 1 T 1 for refinement η 1 (T ) = diam(t ) 1/2 (f V U 1 ) L 2 (T ) 1 η 1 (T ) 2 η 1 (T ) 2 4 T T 1 T M 1 Dirk Praetorius (TU Wien) 27 / 75
2D Example Solve on mesh T 2 etc. Dirk Praetorius (TU Wien) 28 / 75
2D Example ABEM results in... 10 1 10 0 O(N 1/2 ) uniform adaptive 10 1 error estimator 10 2 10 3 10 4 O(N 3/2 ) 10 5 10 6 10 0 10 1 10 2 10 3 10 4 number of elements Dirk Praetorius (TU Wien) 29 / 75
2D Example Questions? can we prove convergence for adaptive mesh-refinement? can we prove linear convergence η 2 l+n C qn η 2 l? constant C > 0 reflects pre-asymptotic convergence rates can we prove optimal convergence rates? at least asymptotically does the problem fit into the axioms framework? Dirk Praetorius (TU Wien) 30 / 75
2D Example Optimal rates in adaptive BEM Tsogtorel: Numer. Math. (2013) general integral kernels smooth boundaries lowest-order BEM Feischl, Karkulik, Melenk, Praetorius: SINUM (2013) weakly-singular integral equation for 2D / 3D Laplace polygonal boundaries lowest-order BEM Feischl, Führer, Karkulik, Melenk, Praetorius: Calcolo (2014) Feischl, Führer, Karkulik, Melenk, Praetorius: ETNA (2015) weakly-singular / hyper-singular IE for 2D / 3D Laplace polygonal boundaries fixed-order BEM with approximation of RHS Dirk Praetorius (TU Wien) 31 / 75
2D Example Main results on rate optimality 1 linear convergence η l+n C q n η l for 0 < θ 1 arbitrary with 0 < q = q(θ) < 1, C = C(θ) > 0 2 optimal convergence η l (#T l #T 0 ) s for 0 < θ 1 sufficiently small for each possible s > 0 3 optimality constrained by estimator and isotropic refinement! ok for 2D suboptimal in 3D w.r.t. error! Dirk Praetorius (TU Wien) 32 / 75
2D Example Road map Reduction (A2) Estimator reduction Discrete reliability (A3) Stability (A1) Reliability Optimality of Dörfler marking Linear convergence of η l Quasi-orthogonality (A4) Discrete reliability (A3) Optimal convergence of η l Closure Overlay Efficiency Optimal convergence of U l Dirk Praetorius (TU Wien) 33 / 75
Stability (A1) & Reduction (A2) Dirk Praetorius (TU Wien)
Stability (A1) & Reduction (A2) Model problem weakly-singular integral equation V u = f in 2D/3D u H := H 1/2 (Γ) u, v := V u ; v L 2 = f ; v L 2 for all v H =, 1/2 H 1/2 (Γ) T regular and γ-shape regular triangulation 2D: partition into affine line segments, local mesh ratio γ 3D: partition into affine surface triangles, no hanging nodes max T T U P p (T ) =: X unique solution of diam(t ) 2 γ T U, V = f ; V L 2 for all V P p (T ) Dirk Praetorius (TU Wien) 34 / 75
Stability (A1) & Reduction (A2) Residual error estimator for model problem Reliability and efficiency u U OK η??? u U + osc ( η = η (T ) 2 T T ) 1/2 η (T ) 2 = T 1/(d 1) (f V U ) 2 L 2 (T ) h T := T 1/(d 1) diam(t ) reliability 2D: [Carstensen, Stephan 95, 96] reliability 3D: [Carstensen, Maischak, Stephan 01] efficiency 2D: [Carstensen 96], [Aurada, Feischl, et al. 13] only direct BEM & closed boundaries & smooth data Dirk Praetorius (TU Wien) 35 / 75
Stability (A1) & Reduction (A2) Local inverse estimate for non-local operators Theorem (Feischl, Karkulik, Melenk, P. 13; Tsogtorel 13) h 1/2 V W L 2 (Γ) C inv W H 1/2 (Γ) for all W P p (T ) C inv = C inv (Γ, p, γ) sketch of proof later (lecture 3) is classical inv.est. for non-classical space V P p (T ) Ψ LHS = h 1/2 Ψ L 2 weighted H 1 -seminorm (stronger) RHS Ψ H 1/2 (Γ) (weaker norm) lowest-order case W P 0 (T l ) Feischl, Karkulik, Melenk, P. 13 Tsogtorel 13 for smooth boundary, but general kernels Dirk Praetorius (TU Wien) 36 / 75
Stability (A1) & Reduction (A2) Local inverse estimate: uniform meshes aim to prove: h 1/2 V W L 2 (Γ) W for all W P 0 (T ) stability V : L 2 (Γ) H 1 (Γ) provides h 1/2 V w L 2 (Γ) h 1/2 V w H 1 (Γ) h 1/2 w L 2 (Γ) = h 1/2 w L 2 (Γ) local inverse estimate of Graham, Hackbusch, Sauter 05 h 1/2 W L 2 (Γ) W H 1/2 (Γ) W Dirk Praetorius (TU Wien) 37 / 75
Stability (A1) & Reduction (A2) Axiom (A1): Stability on non-refined elements (A1) Stability on non-refined elements, T nvb(t l ) ( ) 1/2 ( η (T ) 2 ) 1/2 η l (T ) 2 U U l T T l T T T l T Verification for BEM model problem: η (T ) 2 = T 1/(d 1) (f V U ) 2 L 2 (T ) η (T ) 2 = h 1/2 (f V U ) 2 L 2 ( (T l T )) T T l T inverse triangle inequality + novel inverse estimate LHS h 1/2 (f V U ) h 1/2 (f V U ) L 2 ( (T l T )) h 1/2 V (U l U ) L 2 (Γ) U l U H 1/2 (Γ) l Dirk Praetorius (TU Wien) 38 / 75
Stability (A1) & Reduction (A2) Axiom (A2): Reduction on refined elements (A2) Reduction on refined elements, T nvb(t l ) η (T ) 2 q red η l (T ) 2 + C red U U l 2 T T \T l T T l \T Verification for BEM model problem: η (T ) 2 = T 1/(d 1) (f V U ) 2 L 2 (T ) (T \T l ) = (T l \T ) T 1 2 T for T T T T l triangle inequality + Young ineq. + novel inverse estimate q red 2 1/(d 1) Dirk Praetorius (TU Wien) 39 / 75
Stability (A1) & Reduction (A2) Estimator reduction Stability (A1) + reduction (A2) = estimator reduction 0 < θ 1 0 < q est < 1 C est > 0 l N 0 : η 2 l+1 q est η 2 l + C est U l+1 U l 2 sketch: Young inequality + (A1) + (A2) variable parameter δ > 0 sufficiently small q est = (1 + δ) θ(1 + δ q red ) 1 θ(1 q red ) C est = C 2 stab (1 + δ 1 ) + C red Dirk Praetorius (TU Wien) 40 / 75
Stability (A1) & Reduction (A2) Road map Reduction (A2) Estimator reduction Discrete reliability (A3) Stability (A1) Reliability Optimality of Dörfler marking Linear convergence of η l Quasi-orthogonality (A4) Discrete reliability (A3) Optimal convergence of η l Closure Overlay Efficiency Optimal convergence of U l Dirk Praetorius (TU Wien) 41 / 75
Discrete Reliability (A3) Dirk Praetorius (TU Wien)
Discrete Reliability (A3) Necessity of discrete reliability Suppose BEM model problem Let T nvb(t l+1 ) = U U l 2 u U l 2 Suppose knowledge that η l is reliable + Dörfler marking = u U l 2 η 2 l θ 1 T M l η l (T ) 2 marked elements are refined, i.e., M l T l \T l+1 T l \T = obtain discrete reliability U U l 2 η l (T ) 2 T T l \T Dirk Praetorius (TU Wien) 42 / 75
Discrete Reliability (A3) Axiom (A3): Discrete reliability (A3) Discrete reliability, T nvb(t l ) exists R l T l with T l \T R l #R l #(T l \T ) U U l 2 Crel 2 η l (T ) 2 T R l introduced by Stevenson 07 proof refines usual reliability proof by choice of clever test fct R l = T l \T for FEM R l = patch(t l \T ) for BEM Dirk Praetorius (TU Wien) 43 / 75
Discrete Reliability (A3) Patch S T l set of elements patch(s) γ-shape regularity = #S #patch(s) Dirk Praetorius (TU Wien) 44 / 75
Discrete Reliability (A3) Discrete reliability = reliability 1/2 Uniform refinement yields convergence T l nvb(t 0 ) ε > 0 T nvb(t l ) : u U ε Verification for BEM model problem: Céa lemma + L 2 -projection + density of smooth fcts u U u U H 1/2 (Γ) u Π v H 1/2 (Γ) u v H 1/2 (Γ) + v Π v H 1/2 (Γ) ε + h 1/2 v L 2 (Γ) ε Dirk Praetorius (TU Wien) 45 / 75
Discrete Reliability (A3) Discrete reliability = reliability 2/2 triangle inequality for T nvb(t l ) u U l u U + U U l u U + C rel η l approximation property ε > 0 T nvb(t l ) : u U ε = u U l C rel η l Dirk Praetorius (TU Wien) 46 / 75
Discrete Reliability (A3) Proof of discrete reliability 1/5 W = U U l { P l W P 0 W on T T l T (T l ), P l W = 0 on T T l \T R l := { z N l : z (T l \T ) } 1 use Galerkin orthogonality W 2 = V U V U l ; W = ϕ z (f V U l ) ; (1 P l )W z R l (1 P l )W = 0 on T T l T z R l ϕ z = 1 on T T l \T Dirk Praetorius (TU Wien) 47 / 75
Discrete Reliability (A3) Proof of discrete reliability 2/5 W = U U l { P l W P 0 W on T T l T (T l ), P l W = 0 on T T l \T R l := { z N l : z (T l \T ) } 1 W 2 = ϕ z (f V U l ) ; W P l W z R l sum ; W sum H 1/2 (Γ) W H 1/2 (Γ) sum H 1/2 (Γ) W sum ; P l W h 1/2 l sum L 2 (Γ) h 1/2 l P l W L 2 (Γ) h 1/2 l P l W L2 (Γ) = h 1/2 P l W L2 (Γ) h 1/2 W L2 (Γ) W 2 Obtain W 2 sum 2 H 1/2 (Γ) + h 1/2 l sum 2 L 2 (Γ) Dirk Praetorius (TU Wien) 48 / 75
Discrete Reliability (A3) Proof of discrete reliability 3/5 R l := { z N l : z (T l \T ) } sum = z R l ϕ z (f V U l ) 2 U U l 2 sum 2 H 1/2 (Γ) + h 1/2 l sum 2 L 2 (Γ) 3 elementwise Poincaré inequality, since f V U l P 0 (T l ) h 1/2 l sum 2 L 2 (Γ) diam(t ) 1 sum 2 L 2 (T ) T T l diam(t ) 1 f V U l 2 L 2 (T ) T R l T R l T R l diam(t ) (f V U l ) 2 L 2 (T ) η l (T ) 2 Dirk Praetorius (TU Wien) 49 / 75
Discrete Reliability (A3) Proof of discrete reliability 4/5 R l := { z N l : z (T l \T ) } sum = z R l ϕ z (f V U l ) 2 U U l 2 sum 2 H 1/2 (Γ) + h 1/2 l sum 2 L 2 (Γ) 3 h 1/2 l sum 2 L 2 (Γ) η l (T ) 2 T R l 4 coloring argument as in Carstensen, Maischak, Stephan 01 sum 2 H 1/2 (Γ) z R l ϕ z (f V U l ) 2 H 1/2 (Γ) 5 interpolation + Poincaré estimate + scaling arguments ϕ z (f V U l ) 2 H 1/2 (Γ) ϕ z(f V U l ) L 2 (Γ) ϕ z (f V U l ) H 1 (Γ) h 1/2 l (f V U l ) 2 L 2 (supp(ϕ z)) Dirk Praetorius (TU Wien) 50 / 75
Discrete Reliability (A3) Proof of discrete reliability 5/5 R l := { z N l : z (T l \T ) } sum = z R l ϕ z (f V U l ) 2 U U l 2 sum 2 H 1/2 (Γ) + h 1/2 l sum 2 L 2 (Γ) 3 h 1/2 l sum 2 L 2 (Γ) η l (T ) 2 T R l 4 sum 2 H 1/2 (Γ) z R l ϕ z (f V U l ) 2 H 1/2 (Γ) 5 ϕ z (f V U l ) 2 H 1/2 (Γ) h1/2 l (f V U l ) L 2 2 (supp(ϕ z)) = U U l 2 sum 2 H 1/2 (Γ) + h 1/2 l sum 2 L 2 (Γ) η l (T ) 2 T R l Dirk Praetorius (TU Wien) 51 / 75
Discrete Reliability (A3) Road map Reduction (A2) Estimator reduction Discrete reliability (A3) Stability (A1) Reliability Optimality of Dörfler marking Linear convergence of η l Quasi-orthogonality (A4) Discrete reliability (A3) Optimal convergence of η l Closure Overlay Efficiency Optimal convergence of U l Dirk Praetorius (TU Wien) 52 / 75
Linear Convergence Dirk Praetorius (TU Wien)
Linear Convergence Slow convergence? Current state 1 (A1) + (A2) = η 2 l+1 q estη 2 l + C est U l+1 U l 2 2 Céa lemma U l U as l 3 estimator reduction principle = η l 0 4 reliability = u U l η l 0 in particular, u = U question: Can convergence of η l 0 be slow? U l+1 U l 2 0 could be slow? goal: η 2 l+n qn η 2 l Dirk Praetorius (TU Wien) 53 / 75
Linear Convergence Slow convergence? No! 10 1 10 0 O(N 1/2 ) uniform adaptive 10 1 error estimator 10 2 10 3 10 4 O(N 3/2 ) 10 5 10 6 10 0 10 1 10 2 10 3 10 4 number of elements Dirk Praetorius (TU Wien) 54 / 75
Linear Convergence Axiom (A4): Quasi-orthogonality (A4) Quasi-orthogonality, for some small ε > 0 and all l, N N ( Uk+1 U k 2 εηk 2 ) Corth (ε) ηl 2 k=l Verification for model problem Galerkin orthogonality + symmetry = Pythagoras theorem u U k+1 2 + U k+1 U k 2 = u U k 2 telescoping series = quasi-orth. with C orth (ε) = Crel 2, ε = 0 N N U k+1 U k 2 ( = u Uk 2 u U k+1 2) u U l 2 k=l k=l Dirk Praetorius (TU Wien) 55 / 75
Linear Convergence Linear convergence = Quasi-Orthogonality Proposition (Carstensen, Feischl, Page, P. 14) reliability u U l η l linear convergence η l+n C lin qlin n η l = quasi-orthogonality (A4) with ε = 0, C orth (ε) = C orth (0) > 0 N i.e.: U k+1 U k 2 ηl 2 for all l, N k=l sketch: triangle inequality + reliability + linear convergence N N+1 = U k+1 U k 2 u U k 2 ηk 2 ηl 2 k=l k=l k=l Dirk Praetorius (TU Wien) 56 / 75
Linear Convergence Linear convergence = General Quasi-Orthogonality Proposition (Carstensen, Feischl, Page, P. 14) reliability u U l η l estimator reduction for 0 < θ 1, e.g., stab. (A1) + red. (A2) quasi-orthogonality (A4) = linear convergence η l+n C lin q n lin η l sketch: est. reduction + reliability + quasi-orth. (A4) = ηk 2 η2 l k=l+1 is equivalent to linear convergence Dirk Praetorius (TU Wien) 57 / 75
Linear Convergence Contraction axioms (A1) (A4) = linear convergence η 2 l+n qn η 2 l for symmetric problems, slightly stronger result available Theorem (Feischl, Karkulik, Melenk, P. 13) l = φ Φ l 2 + γη 2 l η2 l with 0 < γ < 1 sufficiently small = Exists 0 < κ = κ(θ) < 1 s.t. l+1 κ l hence lim l η l = 0 = lim l φ Φ l Cascon, Kreuzer, Nochetto, Siebert: SINUM (2008) (concept of proof) Dirk Praetorius (TU Wien) 58 / 75
Linear Convergence Proof of contraction theorem 1 Pythagoras φ Φ l+1 2 = φ Φ l 2 Φ l+1 Φ l 2 2 estimator reduction η 2 l+1 q η2 l + C Φ l+1 Φ l 2 3 reliability φ Φ l 2 C 2 rel η2 l use small parameters 0 < γ, ε < 1 to see φ Φ l+1 2 + γη 2 l+1 φ Φ l 2 + γq η 2 l obtain: l+1 κ l 0 < κ < 1 l = φ Φ l 2 + γηl 2 (1 εγ) φ Φ l 2 + γ(q + εc 2 rel )η2 l κ ( φ Φ l 2 + γη 2 l ) Dirk Praetorius (TU Wien) 59 / 75
Linear Convergence Road map Reduction (A2) Estimator reduction Discrete reliability (A3) Stability (A1) Reliability Optimality of Dörfler marking Linear convergence of η l Quasi-orthogonality (A4) Discrete reliability (A3) Optimal convergence of η l Closure Overlay Efficiency Optimal convergence of U l Dirk Praetorius (TU Wien) 60 / 75
Optimality Dirk Praetorius (TU Wien)
Optimality Dörfler marking θ η l (T) 2 η l (T) 2 T T l T M l goal: determine M l T l with minimal cardinality = requires sorting η l (T 1 ) η l (T N ) O(N log N) sufficient: M l T l has essentially minimal cardinality i.e., #M l C # M l if M l has minimal cardinality idea: sorting with binning [Stevenson 07] O(N) 1 exclude all T T l with η l (T ) 2 < θ η 2 l /N with N := #T l the remaining elements will satisfy the Dörfler marking 2 determine M := max T Tl η l (T ) 2 3 determine minimal K N 0 with 2 (K+1) M < θ η 2 l /N 4 fill k = 0,..., K bins with all 2 (k+1) M < η l (T ) 2 2 k M 5 successively take elts from (unsorted) bins until M l is OK = #M l 2 # M l Dirk Praetorius (TU Wien) 61 / 75
Optimality Dörfler marking = (linear) convergence shown: stability (A1) + reduction (A2) + Dörfler marking = estimator reduction estimator reduction + reliability (A3) + quasi-orth. (A4) = linear convergence ηl+n 2 qn lin η2 l i.e.: Dörfler marking sufficient for (linear) convergence question: Dörfler marking also necessary? Dirk Praetorius (TU Wien) 62 / 75
Optimality Dörfler marking = convergence so far: Dörfler = linear convergence η 2 l+n qn lin η2 l Stab. (A1) + Rel. (A3) = Optimality of Dörfler Marking Exists 0 < θ < 1 and 0 < q < 1 s.t. for all T = nvb(t l ) with η 2 q η 2 l and all 0 < θ < θ and R l from discrete reliability (A3) holds Dörfler marking θ η l (T ) 2 η l (T ) 2 T T l T R l linear convergence = Dörfler marking holds every fixed number n of steps independently of how elements are actually marked! Dirk Praetorius (TU Wien) 63 / 75
Optimality Optimal convergence rates A s = { (u, f) : u, f As = sup ((N + 1) s min η ) < } N N 0 T T N Theorem (Carstensen, Feischl, Page, P. 14) optimal mesh-refinement M l T l has (essentially) minimal cardinality 0 < θ < θ sufficiently small = (u, f) A s η l (#T l #T 0 ) s Carstensen, Feischl, Page, Praetorius: CAMWA (2014) Dirk Praetorius (TU Wien) 64 / 75
Optimality Sketch of proof 1/2 1 exists T nvb(t l ) s.t. #T #T l η 1/s l η 2 q η 2 l 2 optimality of Dörfler marking = R l T l \T satisfies Dörfler marking 3 M l has (essentially) minimal cardinality = #M l #R l #(T l \T ) #T #T l η 1/s l l N 0 4 overlay estimate l 1 l 1 = #T l #T 0 #M j j=0 j=0 η 1/s j Dirk Praetorius (TU Wien) 65 / 75
Optimality Sketch of proof 2/2 l 1 4 obtained: #T l #T 0 j=0 η 1/s j 5 linear convergence η j+n q n η j & geometric series = η l q l j η j = l 1 j=0 η 1/s j ( l 1 j=0 q (l j)/s) η 1/s l η 1/s l 6 combining this, we obtain = #T l #T 0 η 1/s l = η l (#T l #T 0 ) s for all s > 0 with u, f As < Dirk Praetorius (TU Wien) 66 / 75
Optimality Road map Reduction (A2) Estimator reduction Discrete reliability (A3) Stability (A1) Reliability Optimality of Dörfler marking Linear convergence of η l Quasi-orthogonality (A4) Discrete reliability (A3) Optimal convergence of η l Closure Overlay Efficiency Optimal convergence of U l Dirk Praetorius (TU Wien) 67 / 75
3D Examples Dirk Praetorius (TU Wien)
3D Examples L-shaped screen solve V u = 1 on Γ with lowest-order BEM p = 0 Dirk Praetorius (TU Wien) 68 / 75
3D Examples L-shaped screen: errors and estimators 10 0 10 1 1/1 1/2 10 2 10 3 10 4 η 2 l unif. u U l 2 unif. η 2 l adap. u U l 2 adap. 10 1 10 2 10 3 10 4 10 5 number of elements Dirk Praetorius (TU Wien) 69 / 75 1 1/1
Adaptive Boundary Element Methods 3D Examples L-shaped screen: adaptive meshes Dirk Praetorius (TU Wien) 70 / 75
3D Examples L-shaped screen: estimator competition 10 0 10 1 10 2 10 3 10 4 η 2 l adap. u U l 2 η l -adap µ 2 l adap. u U l 2 µ l -adap 10 1 10 2 10 3 10 4 10 5 number of elements Dirk Praetorius (TU Wien) 71 / 75
3D Examples L-shaped screen: estimator competition 10 0 10 1 10 2 10 3 10 4 µ 2 l adap. u U l 2 µ l -adap η 2 l adap. u U l 2 η l -adap 10 1 10 2 10 3 10 4 10 5 10 6 10 7 computational time Dirk Praetorius (TU Wien) 72 / 75
3D Examples Fichera s cube solve V u = 1 on Γ with lowest-order BEM p = 0 Dirk Praetorius (TU Wien) 73 / 75
3D Examples Fichera s cube: errors and estimators 10 1 1/1 10 2 2/3 1 10 3 10 4 η 2 l unif. u U l 2 unif. η 2 l adap. u U l 2 adap. 1/1 10 2 10 3 10 4 10 5 number of elements Dirk Praetorius (TU Wien) 74 / 75
Adaptive Boundary Element Methods 3D Examples Fichera s cube: adaptive meshes Dirk Praetorius (TU Wien) 75 / 75
Thanks for listening! Dirk Praetorius TU Wien Institute for Analysis and Scientific Computing dirk.praetorius@tuwien.ac.at http://www.asc.tuwien.ac.at/ praetorius Dirk Praetorius (TU Wien)