Adaptive boundary element methods with convergence rates Gantumur - PowerPoint PPT Presentation

Adaptive boundary element methods with convergence rates Gantumur Tsogtgerel McGill University CRM-McGill Applied Mathematics Seminar Montréal Monday September 19, 2011

Outline Boundary integral equations Au = f Boundary element methods A n u n = f n A posteriori error estimates E ( A , f , u n ) ∼ � u − u n � Convergence analysis � u − u n � → 0 ? � u − u n � � n − α ? Convergence rates A α = { u : � u − v n � � n − α } Approximation classes � Av n � s � n − a � v n � Inverse-type inequalities Gantumur Adaptive BEM with convergence rates Sep 19 2 / 14

Double layer potential Given ρ continuous on a surface Γ , the double layer potential � ρ ( y ) ∂ � 1 � u ( x ) = K ρ ( x ) : = d Γ y , ∂ n y | x − y | Γ is harmonic in R 3 \ Γ . In 1839, Gauss proposed to use the double layer potential to solve the Dirichlet problem ∆ u = 0 in Ω , u = φ on ∂ Ω , by finding ρ on Γ : = ∂ Ω , so that ( K ρ )( x ) → φ ( y ) as x → y ∈ ∂ Ω from the interior. With x ± → x ∈ ∂ Ω from outside and inside, respectively, we have u ( x + ) − u ( x − ) = 4 πρ ( x ), u ( x + ) + u ( x − ) = 2 u ( x ). From this we deduce ( K − 2 π I ) ρ = φ on ∂ Ω . Gantumur Adaptive BEM with convergence rates Sep 19 3 / 14

Boundary integral equations During 1870-1877 Carl Neumann established solvability of ( I − 1 2 π K ) ρ = φ on Γ , for convex domains (with some exceptions). After over a century of development, we now have the same result for Lipschitz domains, which was proved by Gregory Verchota in 1984. In general, there are many ways to convert (interior or exterior) boundary value problems for Ω into an integral equation Au = f on Γ . Typically, A has a singular kernel, A : H t ( Γ ) → H − t ( Γ ) is self-adjoint and bounded, and satisfies 〈 Au , u 〉 ≥ α � u � 2 H t , with α > 0 and t ∈ {0, ± 1 2 } . In particular, A is invertible. Gantumur Adaptive BEM with convergence rates Sep 19 4 / 14

Boundary element methods People numerically solved boundary integral equations since mid 60’s, but only after the discovery by Leslie Greengard and Vladimir Rokhlin of the fast multipole method in mid 80’s, that it became competitive to direct discretizations of BVPs. BEMs are an adaptation of finite element methods to boundary integral equations. For a triangulation T of Γ , let S = S ( T ) be the space of piecewise constant functions on Γ subordinate to T . Then the Galerkin approximation u T ∈ S of u from the subspace S ⊂ H t ( t < 1 2 ) is the solution of 〈 Au T , v 〉 = 〈 f , v 〉 , ∀ v ∈ S . We have the Galerkin orthogonality � u − u T � 2 +� u T − v � 2 = � u − v � 2 , v ∈ S , and the related best approximation property �·� 2 = 〈 A · , ·〉 . � u − u T � = inf v ∈ S � u − v � , where Gantumur Adaptive BEM with convergence rates Sep 19 5 / 14

Adaptive boundary element methods The best approximation property implies the a priori error estimate τ ∈ T diam( τ ) s � u � H s , � u − u T � ≤ C max ( s ≤ 1). If u �∈ H 1 , the convergence rate is slower than the optimal h ∼ N − 1/2 . Adaptive methods are observed, and in some cases proven to recover this rate. Local a posteriori error indicators, η ( T , τ ) , are supposed to measure how much error the triangle τ contains, e.g., � u − u T � H t ( τ ) . We need a parameter 0 < θ < 1 , and an initial triangulation T 0 . Then we repeat the following for k = 0,1,... . Compute u k = u T k , and the error indicators η ( T k , τ ) , τ ∈ T k . Choose a minimal subset R ⊂ T k , such that � � η ( T k , τ ) ≥ θ η ( T k , τ ). τ ∈ R τ ∈ T k Refine (at least) all triangles in R , to get T k + 1 . Gantumur Adaptive BEM with convergence rates Sep 19 6 / 14

Some prior work on a posteriori error indicators Residual is equivalent to error: � r T � H − t ≡ � f − Au T � H − t ∼ � u − u T � H t . There is a localization issue for t fractional. Recall the Slobodeckij norm | v ( x ) − v ( y ) | 2 � | v | 2 s , ω = | x − y | 2 + 2 s d x d y . ω × ω Faermann ’00-’02: for − 1 < t ≤ 0 , global equivalence � r T � 2 � | r T | 2 H − t ∼ − t , ω ( z ) . z ∈ N T Carstensen, Maischak, Stephan ’01: for − 1 < t ≤ 0 , global upper bound � r T � 2 h 2(1 − t ) | r T | 2 � H − t � 1, τ . τ ∈ T Carstensen, Maischak, Praetorius, Stephan ’04, Nochetto, von Petersdorff, Zhang ’10: for t > 0 , global upper bound � r T � 2 h 2 t | r T | 2 � H − t � 0, τ . τ ∈ T Gantumur Adaptive BEM with convergence rates Sep 19 7 / 14

Results on a posteriori error indicators Gantumur ’11: Lower bounds and local results. Example of a local result for t = 0 : Lemma Let T ′ be a refinement of T , and let γ = � τ . Then we have τ ∈ T \ T ′ α � u T − u T ′ � ≤ � r T � L 2 ( γ ) ≤ β � u T − u T ′ �+ 2 � r T − v � L 2 ( γ ) for any function v ∈ S T ′ . Proof of the first inequality. Let v = u T ′ − u T , and let v T ∈ S T be the L 2 -orthogonal projection of v onto S T . Then we have 〈 Av , v 〉 = 〈 r T , v 〉 = 〈 r T , v − v T 〉 ≤ � r T � γ � v − v T � γ ≤ � r T � γ � v � γ where we have used that v = v T outside γ . Gantumur Adaptive BEM with convergence rates Sep 19 8 / 14

Oscillation The second inequality. Let v ∈ S T ′ be supported in γ . Then we have � v � 2 � � γ = 〈 v , v 〉 = 〈 v − r T , v 〉+〈 A ( u T ′ − u T ), v 〉 ≤ � v − r T � γ +� A ( u T ′ − u T ) � γ � v � γ implying that � r T � γ ≤ � r T − v � γ +� v � γ ≤ 2 � r T − v � γ +� A ( u T ′ − u T ) � . Suppose r T is piecewise H r . Then v ∈ S T ′ � r T − v � 2 γ ≤ C 2 h 2 r τ | r T | 2 � inf r , τ . J τ ∈ T \ T ′ Define � 1 � 2 h 2 r τ | f − Au T | 2 � osc( T , ω ) : = , r , τ τ ∈ T , τ ⊂ ω for ω ⊆ Γ and v ∈ S T , so that we have α � u T − u T ′ � ≤ � r T � γ ≤ β � u T − u T ′ �+ 2 C J osc( T , γ ). Gantumur Adaptive BEM with convergence rates Sep 19 9 / 14

Other works on convergence analysis Symm’s integral equation ( t = − 1 2 ). Ferraz-Leite, Ortner, Praetorius ’10: With ˜ T the uniform refinement of T , use error estimators of the type η ( T , τ ) = h 1/2 τ � u T − u ˜ T � L 2 ( τ ) . Assume saturation (1985-): � u − u ˜ T � ≤ α � u − u T � , ( α < 1). Then � u − u k � ≤ C ρ k with ρ < 1 . Aurada, Ferraz-Leite, Praetorius ’11: Estimator convergence � τ η ( T k , τ ) → 0 without saturation. Feischl, Karkulik, Melenk, Praetorius ’11: Weighted residual estimator from [CMS01], geometric error reduction and convergence rate, without saturation. Gantumur Adaptive BEM with convergence rates Sep 19 10 / 14

Geometric error reduction Assume � h 2 r τ | Av | 2 r , τ ≤ C A � v � 2 , v ∈ S T , τ ∈ T for all admissible T . Let T , T ′ be admissible partitions with T ′ being a refinement of T , and let γ = � τ ∈ T \ T ′ τ . Suppose, for some θ ∈ (0,1] that γ + osc( T , γ ) 2 ≥ θ � r T � 2 � r T � 2 Γ + osc( T , Γ ) 2 � � . Then there exist constants δ ≥ 0 and ρ ∈ (0,1) such that � u − u T ′ � 2 + δ osc( T ′ , Γ ) 2 ≤ ρ � u − u T � 2 + δ osc( T , Γ ) 2 � � . Proof sketch: � u − u T � � � r � Γ � � r � γ � � u T − u T ′ � . � u − u T � 2 = � u T − u T ′ � 2 +� u − u T ′ � 2 . Gantumur Adaptive BEM with convergence rates Sep 19 11 / 14

Convergence rates We know � u − u k � ≤ C ρ k with ρ < 1 . How fast does # T k grow? Define approximation classes A s = { u ∈ L 2 : inf � u − v � ≤ CN − s }. # T ≤ N inf v ∈ S T 2 , and that W 2 s , p is much larger It is known that W 2 s , p ⊂ A s with 1 p = s + 1 than H 2 s , and friendlier to solutions of BVP and BIE. Define A r , s by replacing � u − v � with � u − v �+ osc . We expect A r , s to be close to A s . Assume h 2 r τ | Av | 2 r , τ ≤ C A � v � 2 , � v ∈ S T , τ ∈ T for all admissible T . Let θ ∈ (0, θ ∗ ) . Let f be piecewise H r in the initial triangulation, and u ∈ A r , s for some s > 0 . Then � u − u k � ≤ C | u | A r , s (# T k ) − s . Gantumur Adaptive BEM with convergence rates Sep 19 12 / 14

Inverse-type inequalities h 2 r τ | Av | 2 r , τ ≤ C A � v � 2 , � v ∈ S T . τ ∈ T If A = I or multiplication by a smooth function, then it is the standard inverse inequality. Validity of this inequality depends on how A shifts low frequencies to high frequencies locally, and how it moves frequencies around in space. We decompose L 2 = S T ⊕ H T and correspondingly, Av = ( Av ) S + ( Av ) H . The low frequency component poses no problem: r , τ � � ( Av ) S � 2 ≤ � Av � 2 � � v � 2 . h 2 r τ | ( Av ) S | 2 � τ ∈ T For each triangle τ ∈ T , we decompose v as v = v τ + ( v − v τ ) , where v τ is the part of v near τ . Then the high frequency component of Av locally decomposes into near-field interactions and far-field interactions: ( Av ) H | τ = ( Av τ ) H | τ + ( A ( v − v τ )) H | τ . For boundary integral operators, the far-field part is harmless, and the near-field part is ok if the underlying surface is regular (e.g., C 1,1 ). Gantumur Adaptive BEM with convergence rates Sep 19 13 / 14