Adaptive FEM, Approach with hp- and Goal-Oriented A Posteriori Error - PowerPoint PPT Presentation

Adaptive FEM, Approach with hp- and Goal-Oriented A Posteriori Error Estimator Arezou Ghesmati Dept. of Mathematics Texas A&M University 5 th deal.II Users and Developers Workshop Aug. 6, 2015 Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 1 / 48

Outline 1 Motivation 2 hp-Adaptive Finite Element Method(hp-AFEM) Adaptivity A Posteriori Error Estimator Reliability and Efficiency of Estimator hp-Adaptive Refinement Loop Contraction Convergence Results 3 Numerical Results 4 Ongoing/Future Research Direction Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 2 / 48

Motivation Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 3 / 48

Motivation The Boussinesq Equations: − ν ∆ u + ∇ ̺ = f ( T, g, C ) ∇ · u = 0 ∂u ∂t + u · ∇ T − ∇ · κ ∇ T = γ https://images.google.com https://aspect.dealii.org Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 4 / 48

Stokes or Creeping Flow Given f ∈ L 2 (Ω) d , { d = 2 , 3 } , ν ≥ 1, consider the Stokes equations as our model problem: Find u : Ω → R d and ̺ : Ω → R such that − ν ∆ u + ∇ ̺ = f in Ω ∇ · u = 0 in Ω u = 0 on Γ . Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 5 / 48

Weak Formulation We denote The standard weak formulation of problem; Seek [ u, ̺ ] ∈ H such that L ([ u, ̺ ]; [ v, q ]) = ( f, v ) Ω ∀ [ v, q ] ∈ H . Where 0 (Ω) d × L 2 H := H 1 0 (Ω) . the bilinear form L : H × H → R is defined as: L ([ u, ̺ ]; [ v, q ]) := ( ν ∇ u, ∇ v ) Ω − ( ̺, ∇· v ) Ω − ( ∇· u, q ) Ω ∀ [ u, ̺ ] , [ v, q ] ∈ H . Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 6 / 48

Discretization Due to the continuous inf-sup condition L ([ u, ̺ ]; [ v, q ]) [ u,̺ ] ∈H sup inf ( �∇ u � Ω + � ̺ � Ω ) ( �∇ v � Ω + � q � Ω ) ≥ κ > 0 , [ v,q ] ∈H we define the finite element spaces V p u ( T ) and V p ̺ ( T ) by � � � � ˆ V p u ∈ H 1 u ( T ) := 0 (Ω) : u | K ◦ T K ∈ Q p K K for all K ∈ T and � � � � ˆ V p ̺ ∈ L 2 ̺ ( T ) := 0 (Ω) : ̺ | K ◦ T K ∈ Q p K − 1 K for all K ∈ T , the discrete approximation is obtained by finding [ u FE , ̺ FE ] ∈ V p ( T ) u ( T ) d × V p such that : V p ( T ) := V p ̺ ( T ) ∀ [ v FE , q FE ] ∈ V p ( T ) . L ([ u FE , ̺ FE ] ; [ v FE , q FE ]) = ( f, v FE ) Ω Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 7 / 48

hp-Adaptive Finite Element Method Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 8 / 48

Adaptivity h- Adaptive FEM p- Adaptive FEM hp- Adaptive FEM Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 9 / 48

A Posteriori Error Estimator The idea behind a posteriori error estimation is to access the error between the exact solution and its finite element approximation, in terms of known quantities only! Reliability: �∇ ( u − u FE ) � Ω + � ̺ − ̺ FE � Ω ≤ C rel η ( u FE , ̺ FE , f ) . Local error estimators: � η 2 ( u FE , ̺ FE , f ) = η 2 K ( u FE , ̺ FE , f ) , K ∈T Computational Efficiency η K ( u FE , ̺ FE , f ) ≤ C eff ( �∇ ( u − u FE ) � K + � ̺ − ̺ FE � K ) ∀ K ∈ T Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 10 / 48

Residual Based A Posteriori Error Estimator A posteriori error estimator η shall be the sum of local error indicators η K : η 2 := � η 2 K K ∈T Local Error Estimator: The local a posteriori error estimator η K can be decomposed into cell contribution and interface contribution: η 2 K := η 2 K ; R + η 2 K ; B , where η K ; R denotes the residual-based term and η K ; B indicates the jump-based term. These are defined by K ; R := h 2 � 2 K + � ( ∇ · u FE ) � 2 η 2 K � I K �� �� p K f + ν ∆ u FE − ∇ ̺ FE p 2 K K and 2 � �� h e � ν ∂u FE η 2 � � � K ; B := . � � 2 p e ∂n K � � e e ∈E ( K ) Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 11 / 48

Reliability and Efficiency of Estimator Reliability: Let [ u FE , ̺ FE ] ∈ V p ( T ) be the solution of discrete problem and [ u, ̺ ] ∈ H be solution of weak problem. Further, assume that triangulation T is ( γ h , γ p )-regular. Then, there exists some constant C rel > 0 independent of mesh size vector h and polynomial degree vector p such that � K + h 2 � � 2 �∇ ( u − u FE ) � 2 Ω + � ̺ − ̺ FE � 2 � p 2 K η 2 K � � I K � Ω ≤ C rel p K f − f . p 2 K K K ∈T Efficiency: Let [ u FE , ̺ FE ] ∈ V p ( T ) be the solution of discrete problem, and [ u, ̺ ] ∈ H be solution of weak problem. Further, we assume that triangulation T is ( γ h , γ p )-regular. Then, there exists some constant C eff > 0 independent of mesh size vector h and polynomial degree vector p such that + h 2 � � 2 ν 2 �∇ ( u − u FE ) � 2 � � � � η 2 ω K + � ̺ − ̺ FE � 2 � I K K K ≤ C eff p K p K f − f � � p 2 ω K � ω K K for all K ∈ T . Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 12 / 48

hp-Adaptive Refinement Loop The fully automatic hp-adaptive refinement strategy is based on the standard adaptive loop SOLVE − → ESTIMATE − → MARK − → REFINE . SOLVE and REFINE are almost the same in all adaptive refinement patterns. ESTIMATE and MARK are the two most crucial modules in hp-adaptive method. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 13 / 48

hp- Adaptive Refinement Algorithm Initialization : Set N = 0, a coarse mesh T 0 , θ ∈ [0 , 1] and also tolerance TOL . SOLVE : Find the solution ( u FE , ̺ FE ) of discrete problem. ESTIMATE : Compute a posteriori error estimation. If η < TOL then STOP the algorithm, else , MARK : select set of cells A to be marked either with h- or p-refinement REFINE : Given ( A , ( j K ) K ∈A ), we refine the cells contained in A with refinement patterns j K corresponding to each cell. Then set N = N + 1 and go to step SOLVE. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 14 / 48

Module MARK More information needed in module MARK to choose the best adaptive strategy: Convergence Estimator: k K,j , j ∈ { 1 , 2 , 3 } j=1 j=2 j=3 Efficiency ≈ Workload number: ̟ K,j = n DoF s j ∈ { 1 , 2 , 3 } Choose the best refinement pattern ⇒ find integer j K ∈ { 1 , 2 , 3 } such that: k K,j K k K,j � k 2 K,j k η 2 K ≥ θ 2 η 2 = max , ̟ K,j K ̟ K,j j ∈{ 1 , 2 , 3 } K ∈A Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 15 / 48

Challenges to calculate the Convergence Estimator: k K,j Considering the residual of Stokes problem on the local patch domain ω K , we have: ( ∇ v, ∇ ( w N,j )) ω K + ( q, w N,j ∀ ( v, q ) ∈ V p ) ω K = L ([ v, q ]; [ e, E ]) ω K , K,j ( T N | ω K ) . u ̺ The pair ( w N,j , w N,j ) ∈ V p K,j ( T N | ω K ) is defined to be the Ritz representation of the u ̺ residual. � 1 �� 1 2 2 2 � � � � ∇ w N,j � w N,j k K,j = + � � � � u ̺ η K ( u FE , ̺ FE ) � � ω K ω K non-uniform patch regular patch Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 16 / 48

Build the local triangulation to compute ⇒ k K j non-uniform patch get-cells-at-coarsest- set-FE-Nothing common-level Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 17 / 48

Contraction Convergence Results Contraction for Error in Energy Norm: �∇ ( u − u N +1 FE ) � 2 Ω d + � ̺ − ̺ N +1 FE � 2 �∇ ( u − u N FE ) � 2 Ω d + � ̺ − ̺ N FE � 2 � � Ω ≤ µ Ω Quasi- Convergence : � �∇ ( u − u N +1 FE ) � 2 Ω d + � ̺ − ̺ N +1 FE � 2 Ω + ϑη 2 ( T N +1 ) ≤ µ FE ) � 2 �∇ ( u − u N Ω d + � � ̺ − ̺ N FE � 2 Ω + ϑη 2 ( T N ) for constants ϑ > 0 and µ ∈ (0 , 1) independent of mesh size h and polynomial degree vector p . Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 18 / 48

Important Equivalence Result Total Error: �∇ ( u − u N FE ) � 2 Ω d + � ̺ − ̺ N FE � 2 Ω + osc 2 N Quasi Error: FE ) � 2 FE � 2 Ω + ϑη 2 ( T N ) �∇ ( u − u N Ω d + � ̺ − ̺ N Quasi Error ≈ η 2 ≈ Total Error Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 19 / 48

Numerical Results Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 20 / 48

Example-1 Manufactured solution on L-shaped domain: Let Ω ∈ R 2 be L-shaped domain, ( − 1 , 1) 2 \ ([0 , 1] × [ − 1 , 0]) we enforce appropriate inhomogeneous boundary conditions for velocity u on Γ such that the analytical solution u : Ω → R 2 and ̺ : Ω → R are given by: � − e x ( y cos( y ) + sin( y )) � ̺ = 2 e x sin( y ) − (2(1 − e )(cos(1) − 1)) / 3 . u = , e x y sin( y ) Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 21 / 48

hp- adaptive refinement, cycle = 0 Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 22 / 48

hp- adaptive refinement, cycle = 2 Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 23 / 48

hp- adaptive refinement, cycle = 4 Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 24 / 48

hp- adaptive refinement, cycle = 6 Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM Aug. 6, 2015 25 / 48