HILBERT? What is HILBERT? Matlab Implementation of Adaptive 2D BEM. Dirk Praetorius. Features of HILBERT

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1 Söerhaus-Workshop 2009 October 16, 2009 What is HILBERT? HILBERT Matab Impementation of Adaptive 2D BEM joint work with M. Aurada, M. Ebner, S. Ferraz-Leite, P. Godenits, M. Karkuik, M. Mayr Hibert Is a Lovey Boundary Eement Research Too Matab ibrary for h-adaptive Gaerkin BEM currenty, owest-order eements for 2D Lapacian P 0 for fuxes S 1 for traces research code for FWF project P21732 overview on current state of the art starting point for further investigations free for academic use Vienna University of Technoogy Institute for Anaysis and Scientific Computing Features of HILBERT HILBERT? C impementation of integra operators via MEX interface V (P 0 Γ P0 Γ ) K(S 1 Γ P0 Γ ) W(S 1 Γ S1 Γ ) N(P 1 Ω P0 Γ ) remaining codes in Matab fuy vectorized different error estimators different marking strategies oca mesh-refinement demo fies and adaptive agorithms for weaky-singuar integra equation hypersinguar integra equation symmetric integra formuation of mixed BVP with/without voume force

2 Current Restrictions of HILBERT Γ = Ω piecewise affine boundary ony owest-order eements ony canonica bases characteristic functions for P 0 hat functions for S 1 Weaky-Singuar Integra Equation no matrix compression, i.e., dense matrices direct soution of inear systems by Matab backsash operator Outine Mode Probem 1 HILBERT? 2 Weaky-Singuar Integra Equation 3 Introduction to HILBERT 4 Concusion & Outook Dirichet Probem u = 0 in Ω u = g on Γ Weaky-Singuar Integra Equation V φ = (K + 1/2)g H := H 1/2 (Γ) energy space equivaent energy norm on H induced by V

3 Variationa Formuation Continuous Formuation Seek φ H := H 1/2 (Γ) s.t. V φ,ψ = (K + 1/2)g,ψ for a ψ H T partition of Γ Gaerkin Formuation Seek Φ X := P 0 (T ) s.t. V Φ,Ψ = (K + 1/2)g,Ψ for a Ψ X Perturbed Gaerkin Formuation 2/2 Matrix Formuation of Perturbed Gaerkin Formuation Compute coefficient vector x of Φ by soving Vx = where g is noda vector of g (K M ) g mass-type matrix M anayticay computabe and sparse integra operator matrices V, K anayticay computabe drawback: RHS hard to compute in genera Perturbed Gaerkin Formuation 1/2 Contro Data Approximation Reminder: Continuous Probem assume additiona reguarity g H 1 (Γ) G S 1 (T ) noda interpoant of g Perturbed Gaerkin Formuation Seek Φ X s.t. V φ = (K + 1/2)g Reminder: Perturbed Continuous Probem V φ = (K + 1/2)G V Φ,Ψ = (K + 1/2)G,Ψ for a Ψ X Lemma (Aurada, Godenits, P. 09) φ φ g G H 1/2 (Γ) h1/2 (g G ) L 2 (Γ) =: osc

4 Contro Gaerkin Error Loca Refinement Indicators Φ Gaerkin soution w.r.t. T Φ Gaerkin soution w.r.t. T L 2 -projection Π onto X = P 0 (T ) Lemma (Ferraz-Leite, P. 08) There hods η η µ µ η for error estimators η = Φ Φ η = Φ Π Φ µ = h 1/2 ( Φ Φ ) L 2 (Γ) µ = h 1/2 ( Φ Π Φ ) L 2 (Γ) ρ (T) 2 := h 1/2 ( Φ Π Φ ) 2 L 2 (T) + h1/2 (g G ) 2 L 2 (T) φ Φ 2 (η + osc ) 2 µ 2 + osc2 = T T ρ (T) 2 everything works aso with µ instead of µ Contro Gaerkin and Data Approximation Error Saturation Assumption for Non-Perturbed Probem Φ Gaerkin soution w.r.t. T and non-perturbed RHS Φ Gaerkin soution w.r.t. T and non-perturbed RHS φ Φ q φ Φ Coroary (Aurada, Godenits, P. 09) η φ Φ + osc saturation assumption = φ Φ η + osc Adaptive Mesh-Refinement initia mesh T 0 adaptivity parameter 0 < θ < 1 Adaptive Agorithm 1 compute discrete soution Φ X = P 0 ( T ) 2 for a T T, compute ρ (T) 3 find (minima) set M T s.t. θ T T ρ (T) 2 T M ρ (T) 2 4 refine (at east) marked eements T M to obtain T +1 ensure κ(t +1 ) 2 κ(t 0 ) 5 increase counter + 1 and iterate

5 Convergence of Adaptive Scheme Data Structure Theorem (Aurada, Godenits, P. 09) Use refinement procedure of [AGP 09] Then, concept of estimator reduction appies, whence im ( µ2 + osc2 ) = 0 Saturation assumption for non-perturbed probem impies im φ Φ = 0 = im φ Φ T = {T 1,...,T N } partition N = {z 1,...,z N } nodes coordinates (N 2) doube-array z j = (x,y) R 2 corresponds to coordinates(j,:) = [x, y ] eements (N 2) int-array T i = conv{z j,z k } corresponds to eements(i,:) = [j, k ] Buid Gaerkin Data Introduction to HILBERT buidv assembes V = V (P 0 P 0 ) use of symmetry outer integration over eement with ength(t j ) ength(t k ) outer integration by Gaussian quadrature for far-fied entries buidk assembes K = K(S 1 P 0 ) buidsymmrhs assembes b for perturbed RHS (K + 1/2)G

6 Loca Mesh-Refinement (h h/2)-error Estimators 1/2 markeements returns index vector for marked eements M T from θ T T ρ (T) 2 T M ρ (T) 2 even mutipe trianguations, e.g., FEM-BEM, mixed BVP refineboundarymesh agorithm from [Aurada, Godenits, P. 09] compexity O(N og N) for oca mesh-refinement compexity O(N) for uniform mesh-refinement computeestspeta returns η 2 = Φ Φ 2. computeestspetatide returns η 2 = Φ Π Φ 2. Data Osciations (h h/2)-error Estimators 2/2 computeoscdirichet returns v R N with osc 2 õsc2 = N j=1 v j, where v j ength(t j ) (g G ) 2 L 2 (T j ) computeestspmu returns v R N with µ 2 = N j=1 v j, where v j = ength(t j ) Φ Φ 2 L 2 (T j ) quadrature ocated at nodes and midpoints exact for g S 2 osc = O(h 3/2 ) and osc õsc = O(h 5/2 ) for smooth g computeestspmutide returns ṽ R N with µ 2 = N j=1 ṽj, where ṽ j = ength(t j ) Φ Π Φ 2 L 2 (T j )

7 Adaptive Mesh-Refining Agorithm 1 whie s i z e ( eements,1) < nemax 2 3 % bui d uniformy r ef i ned mesh 4 [ co ordin ates f i ne, eements fine, father2son ]... 5 = refineboundarymesh ( coordinates, eements ) ; 6 7 % compute fine mesh sou t i on 8 V fine = buidv ( c oordinates f ine, e ements f i ne ) ; 9 b f i ne = buidsymmrhs( coo r dina t es f i ne, eements ) ; 10 x f i n e = V fine \ b f i ne ; % compute (h h/2) e r r o r estimator t i de mu 13 mu tide = computeestspmutide( coordinates, eements, father2son, x f i n e ) ; 15 % compute data o s c i a t i o n s 16 o sc f i ne = computeoscdirichet ( coo rd ina t es f in e, eements ) ; 17 osc = o s c f i n e ( father2son ( :, 1) ) + o s c f i n e ( father2son ( :, 2 ) ) ; % mark eements f or refinement 20 marked = markeements ( theta, mu tide + osc ) ; % generate new mesh 23 [ coordinates, eements ] = refineboundarymesh ( coordinates, eements, marked ) ; end estimators Numerica Experiment 2/2 O(N 2/3 ) err + osc (unif) O(N 11/10 ) eµ (unif) osc (unif) err + osc (adap) O(N 3/2 ) eµ (adap) osc (adap) number of eements Numerica Experiment 1/2 Adaptivity Pays 1/ adaptive pays w.r.t. number of eements observe optima convergence behaviour does adaptivity pay w.r.t. computationa time? := unif () (T 0 ) +assembe +sove t (adap) 0 := assembe +sove +estimate +mark +refine t (unif) t (adap) +1 := t (adap) +... known exact soution φ generic singuarity of φ at reentrant corner (green) singuarity of g at uppermost corner (red)

8 Adaptivity Pays 2/2 HILBERT Reease One estimators err + osc (unif) eµ (unif) osc (unif) err + osc (adap) eµ (adap) osc (adap) computationa time [s] reease date: Friday 13. November 2009 avaiabe at: Reease One contains owest-order BEM for 2D Lapacian Dirichet probem Neumann probem mixed Dirichet-Neumann probem with/without voume forces Reease One incudes (h h/2)-error estimators for a probems oca mesh-refinement for boundary/voume meshes adaptive agorithms & demo fies severa visuaization toos fu documentation of a Matab codes (about 60 pages) Short-Term Extensions Concusion & Outook further error estimators, e.g., weighted-residua error estimators two-eve error estimators Faermann residua error estimator Steinbach-Schuz error estimator adaptive FEM-BEM couping Lamé equation

9 Thank You for Listening Vienna University of Technoogy Institute for Anaysis and Scientific Computing