Guaranteed and robust discontinuous Galerkin a posteriori error - PowerPoint PPT Presentation

I Abstract framework A posteriori estimates C Previous results on a posteriori error estimation in DG DG, pure diffusion case Karakashian and Pascal (2003), Becker, Hansbo, and Larson (2003), residual-based estimates Rivière and Wheeler (2003), L 2 -estimates Ainsworth (2007), reconstruction of side fluxes Kim (2007), Cochez-Dhondt and Nicaise (2008), Lazarov, Repin, and Tomar (numerical experiments, 2008), our works, reconstruction of equilibrated H ( div , Ω) -conforming fluxes DG, convection–diffusion–reaction case Sun and Wheeler (2006), L 2 -estimates Schötzau and Zhu (2008, preprint), res.-based estimates A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Previous results Equilibrated fluxes estimates Prager and Synge (1947) Ladevèze and Leguillon (1983) Repin (1997) Destuynder and Métivet (1999) Luce and Wohlmuth (2004) Convection–diffusion problems Verfürth (1998, 2005), conforming finite elements Sangalli (2008), conforming finite elements A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Previous results Equilibrated fluxes estimates Prager and Synge (1947) Ladevèze and Leguillon (1983) Repin (1997) Destuynder and Métivet (1999) Luce and Wohlmuth (2004) Convection–diffusion problems Verfürth (1998, 2005), conforming finite elements Sangalli (2008), conforming finite elements A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Motivations and key points Motivations establish an optimal abstract framework for a posteriori error estimation in potential- and flux-nonconforming methods derive estimates for DG satisfying as many as possible of the five optimal properties Key points focus on inhomogeneous and anisotropic diffusion case of nonmatching meshes singular regimes of dominant convection or reaction A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Motivations and key points Motivations establish an optimal abstract framework for a posteriori error estimation in potential- and flux-nonconforming methods derive estimates for DG satisfying as many as possible of the five optimal properties Key points focus on inhomogeneous and anisotropic diffusion case of nonmatching meshes singular regimes of dominant convection or reaction A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A model convection–diffusion–reaction problem A model convection–diffusion–reaction problem in Ω , −∇· ( K ∇ u ) + β ·∇ u + µ u = f 0 on ∂ Ω u = Bilinear form u , v ∈ H 1 ( T h ) B ( u , v ) := ( K ∇ u , ∇ v ) + ( β ·∇ u , v ) + ( µ u , v ) , Weak solution Find u ∈ H 1 ∀ v ∈ H 1 0 (Ω) such that B ( u , v ) = ( f , v ) 0 (Ω) . Energy norm Decompose B into B = B S + B A , where �� � � µ − 1 B S ( u , v ) := ( K ∇ u , ∇ v ) + 2 ∇· β u , v , � � β ·∇ u + 1 B A ( u , v ) := 2 ( ∇· β ) u , v . B S is symmetric on H 1 ( T h ) ; put ||| v ||| 2 := B S ( v , v ) B A is skew-symmetric on H 1 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A model convection–diffusion–reaction problem A model convection–diffusion–reaction problem in Ω , −∇· ( K ∇ u ) + β ·∇ u + µ u = f 0 on ∂ Ω u = Bilinear form u , v ∈ H 1 ( T h ) B ( u , v ) := ( K ∇ u , ∇ v ) + ( β ·∇ u , v ) + ( µ u , v ) , Weak solution Find u ∈ H 1 ∀ v ∈ H 1 0 (Ω) such that B ( u , v ) = ( f , v ) 0 (Ω) . Energy norm Decompose B into B = B S + B A , where �� � � µ − 1 B S ( u , v ) := ( K ∇ u , ∇ v ) + 2 ∇· β u , v , � � β ·∇ u + 1 B A ( u , v ) := 2 ( ∇· β ) u , v . B S is symmetric on H 1 ( T h ) ; put ||| v ||| 2 := B S ( v , v ) B A is skew-symmetric on H 1 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A model convection–diffusion–reaction problem A model convection–diffusion–reaction problem in Ω , −∇· ( K ∇ u ) + β ·∇ u + µ u = f 0 on ∂ Ω u = Bilinear form u , v ∈ H 1 ( T h ) B ( u , v ) := ( K ∇ u , ∇ v ) + ( β ·∇ u , v ) + ( µ u , v ) , Weak solution Find u ∈ H 1 ∀ v ∈ H 1 0 (Ω) such that B ( u , v ) = ( f , v ) 0 (Ω) . Energy norm Decompose B into B = B S + B A , where �� � � µ − 1 B S ( u , v ) := ( K ∇ u , ∇ v ) + 2 ∇· β u , v , � � β ·∇ u + 1 B A ( u , v ) := 2 ( ∇· β ) u , v . B S is symmetric on H 1 ( T h ) ; put ||| v ||| 2 := B S ( v , v ) B A is skew-symmetric on H 1 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A model convection–diffusion–reaction problem A model convection–diffusion–reaction problem in Ω , −∇· ( K ∇ u ) + β ·∇ u + µ u = f 0 on ∂ Ω u = Bilinear form u , v ∈ H 1 ( T h ) B ( u , v ) := ( K ∇ u , ∇ v ) + ( β ·∇ u , v ) + ( µ u , v ) , Weak solution Find u ∈ H 1 ∀ v ∈ H 1 0 (Ω) such that B ( u , v ) = ( f , v ) 0 (Ω) . Energy norm Decompose B into B = B S + B A , where �� � � µ − 1 B S ( u , v ) := ( K ∇ u , ∇ v ) + 2 ∇· β u , v , � � β ·∇ u + 1 B A ( u , v ) := 2 ( ∇· β ) u , v . B S is symmetric on H 1 ( T h ) ; put ||| v ||| 2 := B S ( v , v ) B A is skew-symmetric on H 1 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Properties of the weak solution 1.8 1.5 exact solution -exact flux 1.6 1.0 1.4 1.2 0.5 1.0 0.8 0.0 0.6 -0.5 0.4 0.2 -1.0 0.0 -0.2 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Solution u is in H 1 Flux − K ∇ u is in H ( div , Ω) 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the energy norm Theorem (Optimal abstract estimate, energy norm (Vohralík ’07, Ern & Stephansen ’08)) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then � inf ||| u − u h ||| ≤ ||| u h − s ||| s ∈ H 1 0 (Ω) � � ( f − ∇· t − β ·∇ s − µ s , ϕ ) inf sup + t ∈ H ( div , Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � �� � �� µ − 1 � − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ ≤ 2 ||| u − u h ||| . Properties Guaranteed upper bound, quasi-exact, and robust. Holds uniformly for any mesh (anis.) and pol. degree. Not computable (infimum over an infinite-dim. space). A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the energy norm Theorem (Optimal abstract estimate, energy norm (Vohralík ’07, Ern & Stephansen ’08)) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then � inf ||| u − u h ||| ≤ ||| u h − s ||| s ∈ H 1 0 (Ω) � � ( f − ∇· t − β ·∇ s − µ s , ϕ ) inf sup + t ∈ H ( div , Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � �� � �� µ − 1 � − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ ≤ 2 ||| u − u h ||| . Properties Guaranteed upper bound, quasi-exact, and robust. Holds uniformly for any mesh (anis.) and pol. degree. Not computable (infimum over an infinite-dim. space). A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the energy norm Theorem (Optimal abstract estimate, energy norm (Vohralík ’07, Ern & Stephansen ’08)) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then � inf ||| u − u h ||| ≤ ||| u h − s ||| s ∈ H 1 0 (Ω) � � ( f − ∇· t − β ·∇ s − µ s , ϕ ) inf sup + t ∈ H ( div , Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � �� � �� µ − 1 � − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ ≤ 2 ||| u − u h ||| . Properties Guaranteed upper bound, quasi-exact, and robust. Holds uniformly for any mesh (anis.) and pol. degree. Not computable (infimum over an infinite-dim. space). A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Approximate solution and approximate flux 1.8 1.5 exact solution -exact flux approximate solution -approximate flux 1.6 1.0 1.4 1.2 0.5 1.0 0.8 0.0 0.6 -0.5 0.4 0.2 -1.0 0.0 -0.2 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Approximate solution u h is not Approximate flux − K ∇ u h is not in H 1 in H ( div , Ω) 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Approximate solution and approximate flux 1.8 1.5 exact solution -exact flux approximate solution -approximate flux 1.6 postprocessed solution -postprocessed flux 1.0 1.4 1.2 0.5 1.0 0.8 0.0 0.6 -0.5 0.4 0.2 -1.0 0.0 -0.2 -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Construct a postprocessed Construct a postprocessed flux approx. solution s h in H 1 t h in H ( div , Ω) 0 (Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A first computable estimate in the energy norm Theorem (A first computable estimate, energy norm) Let u be the weak solution and let u h ∈ H 1 ( T h ) be arbitrary. Take any t h ∈ H ( div , Ω) and any s h ∈ H 1 0 (Ω) . Then � � C 1 / 2 F , Ω h Ω 1 ||| u − u h ||| ≤||| u h − s h ||| + min , min T ∈T h c 1 / 2 min T ∈T h c 1 / 2 K , T β ,µ, T × � f − ∇· t h − β ·∇ s h − µ s h � � � 2 � 1 � � � � �� � 1 2 ∇ u h + K − 1 1 � 2 + µ − 1 2 ( u h − s h ) 2 � K 2 t h + 2 ∇· β . Properties Guaranteed upper bound ( C F , Ω ≤ 1). � � 1 2 ∇ u h + K − 1 � results from − K ∇ u h �∈ H ( div , Ω) . � K 2 t h ||| u h − s h ||| results from u h �∈ H 1 0 (Ω) . Advantage: scheme-independent (promoted by Repin). Disadvantage: no info. from computation used, too rough . A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the augmented norm Theorem (Optimal abstract estimate, augmented norm) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then ||| u − u h ||| ⊕ � � ≤ 2 inf inf sup ||| u h − s ||| + ( f − ∇· t − β ·∇ s − µ s , ϕ ) s ∈ H 1 t ∈ H ( div , Ω) 0 (Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 ��� �� � µ − 1 sup − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � � inf { ( f − ∇· t − β ·∇ u h − µ u h , ϕ ) − ( K ∇ u h + t , ∇ ϕ ) }−B D ( u h , ϕ ) t ∈ H ( div , Ω) ≤ 5 ||| u − u h ||| ⊕ . Comments only the highlighted terms are new their form is similar to the energy estimate necessary for robustness in the convection-dominated case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the augmented norm Theorem (Optimal abstract estimate, augmented norm) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then ||| u − u h ||| ⊕ � � ≤ 2 inf inf sup ||| u h − s ||| + ( f − ∇· t − β ·∇ s − µ s , ϕ ) s ∈ H 1 t ∈ H ( div , Ω) 0 (Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 ��� �� � µ − 1 sup − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � � inf { ( f − ∇· t − β ·∇ u h − µ u h , ϕ ) − ( K ∇ u h + t , ∇ ϕ ) }−B D ( u h , ϕ ) t ∈ H ( div , Ω) ≤ 5 ||| u − u h ||| ⊕ . Comments only the highlighted terms are new their form is similar to the energy estimate necessary for robustness in the convection-dominated case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Energy norm A first computable estimate Augmented norm Optimal abstract estimate in the augmented norm Theorem (Optimal abstract estimate, augmented norm) Let u be the weak sol. and let u h ∈ H 1 ( T h ) be arbitrary. Then ||| u − u h ||| ⊕ � � ≤ 2 inf inf sup ||| u h − s ||| + ( f − ∇· t − β ·∇ s − µ s , ϕ ) s ∈ H 1 t ∈ H ( div , Ω) 0 (Ω) ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 ��� �� � µ − 1 sup − ( K ∇ u h + t , ∇ ϕ ) + 2 ∇· β ( s − u h ) , ϕ + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 � � inf { ( f − ∇· t − β ·∇ u h − µ u h , ϕ ) − ( K ∇ u h + t , ∇ ϕ ) }−B D ( u h , ϕ ) t ∈ H ( div , Ω) ≤ 5 ||| u − u h ||| ⊕ . Comments only the highlighted terms are new their form is similar to the energy estimate necessary for robustness in the convection-dominated case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Discontinuous Galerkin method Discontinuous Galerkin method Find u h ∈ P k ( T h ) such that for all v h ∈ P k ( T h ) ( K ∇ u h , ∇ v h ) + (( µ − ∇· β ) u h , v h ) − ( u h , β ·∇ v h ) � − { ( n F ·{ { K ∇ u h } } ω , [ [ v h ] ]) F + θ ( n F ·{ { K ∇ v h } } ω , [ [ u h ] ]) F } F ∈F h � � � (( α F γ K , F h − 1 + + γ β , F )[ [ u h ] ] , [ [ v h ] ]) F + ( β · n F { { u h } } , [ [ v h ] ]) F F F ∈F h = ( f , v h ) . ] = v − − v + jump operator [ [ v ] } = 1 2 ( v − + v + ) average operator { { v } } ω = ( ω − v − + ω + v + ) diffusivity-weighted av. operator { { v } diff.-dep. penalties γ K , F (Ern, Stephansen, and Zunino 08) θ : different scheme types (SIPG/NIPG/IIPG/OBB) γ β , F : upwind-weighting stabilization u h �∈ H 1 0 (Ω) , − K ∇ u h �∈ H ( div , Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Discontinuous Galerkin method Discontinuous Galerkin method Find u h ∈ P k ( T h ) such that for all v h ∈ P k ( T h ) ( K ∇ u h , ∇ v h ) + (( µ − ∇· β ) u h , v h ) − ( u h , β ·∇ v h ) � − { ( n F ·{ { K ∇ u h } } ω , [ [ v h ] ]) F + θ ( n F ·{ { K ∇ v h } } ω , [ [ u h ] ]) F } F ∈F h � � � (( α F γ K , F h − 1 + + γ β , F )[ [ u h ] ] , [ [ v h ] ]) F + ( β · n F { { u h } } , [ [ v h ] ]) F F F ∈F h = ( f , v h ) . ] = v − − v + jump operator [ [ v ] } = 1 2 ( v − + v + ) average operator { { v } } ω = ( ω − v − + ω + v + ) diffusivity-weighted av. operator { { v } diff.-dep. penalties γ K , F (Ern, Stephansen, and Zunino 08) θ : different scheme types (SIPG/NIPG/IIPG/OBB) γ β , F : upwind-weighting stabilization u h �∈ H 1 0 (Ω) , − K ∇ u h �∈ H ( div , Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Discontinuous Galerkin method Discontinuous Galerkin method Find u h ∈ P k ( T h ) such that for all v h ∈ P k ( T h ) ( K ∇ u h , ∇ v h ) + (( µ − ∇· β ) u h , v h ) − ( u h , β ·∇ v h ) � − { ( n F ·{ { K ∇ u h } } ω , [ [ v h ] ]) F + θ ( n F ·{ { K ∇ v h } } ω , [ [ u h ] ]) F } F ∈F h � � � (( α F γ K , F h − 1 + + γ β , F )[ [ u h ] ] , [ [ v h ] ]) F + ( β · n F { { u h } } , [ [ v h ] ]) F F F ∈F h = ( f , v h ) . ] = v − − v + jump operator [ [ v ] } = 1 2 ( v − + v + ) average operator { { v } } ω = ( ω − v − + ω + v + ) diffusivity-weighted av. operator { { v } diff.-dep. penalties γ K , F (Ern, Stephansen, and Zunino 08) θ : different scheme types (SIPG/NIPG/IIPG/OBB) γ β , F : upwind-weighting stabilization u h �∈ H 1 0 (Ω) , − K ∇ u h �∈ H ( div , Ω) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Potential- and flux-conforming reconstructions Choice of s h : the Oswald interpolate of u h I Os : P k ( T h ) → P k ( T h ) ∩ H 1 0 (Ω) prescribed at Lagrange nodes by arithmetic averages � 1 I Os ( v h )( V ) = v h | T ( V ) #( T V ) T ∈T V one can also use diffusivity-weighted averages (Ainsworth ’05) Choice of t h : a new H ( div , Ω) flux reconstruction Ern, Nicaise & Vohralík ’07 (matching meshes) the present work (nonmatching meshes) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Potential- and flux-conforming reconstructions Choice of s h : the Oswald interpolate of u h I Os : P k ( T h ) → P k ( T h ) ∩ H 1 0 (Ω) prescribed at Lagrange nodes by arithmetic averages � 1 I Os ( v h )( V ) = v h | T ( V ) #( T V ) T ∈T V one can also use diffusivity-weighted averages (Ainsworth ’05) Choice of t h : a new H ( div , Ω) flux reconstruction Ern, Nicaise & Vohralík ’07 (matching meshes) the present work (nonmatching meshes) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

Pure di�usion Intro du tion ADR: semi-robust estimat e s La m�tho de GD ADR: robust estima t e s Estimat i ons d'erreur a p osterio ri Numeri al results Lo ally omputable estimate I I I l Ravia rt�Thomas FE spa es of degree l : RT h l = 0 l = 1 l Constru tion of t RT ( l p o r p 1) h h I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction dof 's fo r no rmal omp onent on ea h fa e: q F , h l t RTN l ( T h ) : Raviart–Thomas–Nédélec spaces of degree l t n q n K u u q F F F F h h F h h h h d dof 's in ea h element: r T , h l 1 t t r K u r n Kr u h h T h h T T F F h h F F T Construction of t h ∈ RTN l ( T h ) , l = k or l = k − 1 Alexan d re Ern Estimat i ons d'erreur a p osterio ri p our les m�tho des GD normal components on each side: ∀ q h ∈ P l ( F ) , � � } ω + α F γ K , F h − 1 ( t h · n F , q h ) F = − n F ·{ { K ∇ u h } F [ [ u h ] ] , q h F on each element (only for l ≥ 1): ∀ r h ∈ P d l − 1 ( T ) , � ( t h , r h ) T = − ( K ∇ u h , r h ) T + θ ω T , F ( n F · Kr h , [ [ u h ] ]) F F ∈F T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

Pure di�usion Intro du tion ADR: semi-robust estimat e s La m�tho de GD ADR: robust estima t e s Estimat i ons d'erreur a p osterio ri Numeri al results Lo ally omputable estimate I I I l Ravia rt�Thomas FE spa es of degree l : RT h l = 0 l = 1 l Constru tion of t RT ( l p o r p 1) h h I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction dof 's fo r no rmal omp onent on ea h fa e: q F , h l t RTN l ( T h ) : Raviart–Thomas–Nédélec spaces of degree l t n q n K u u q F F F F h h F h h h h d dof 's in ea h element: r T , h l 1 t t r K u r n Kr u h h T h h T T F F h h F F T Construction of t h ∈ RTN l ( T h ) , l = k or l = k − 1 Alexan d re Ern Estimat i ons d'erreur a p osterio ri p our les m�tho des GD normal components on each side: ∀ q h ∈ P l ( F ) , � � } ω + α F γ K , F h − 1 ( t h · n F , q h ) F = − n F ·{ { K ∇ u h } F [ [ u h ] ] , q h F on each element (only for l ≥ 1): ∀ r h ∈ P d l − 1 ( T ) , � ( t h , r h ) T = − ( K ∇ u h , r h ) T + θ ω T , F ( n F · Kr h , [ [ u h ] ]) F F ∈F T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction property Crucial diffusive flux reconstruction property note that all the terms of the DG scheme are used in the construction of t h denote by Π l the L 2 -orthogonal projection onto P k ( T h ) the above construction yields ∇· t h = Π l ( f ) Proof: ( ∇· t h , ξ h ) T = − ( t h , ∇ ξ h ) T + � t h · n , ξ h � ∂ T = B h ( u h , ξ h ) = ( f , ξ h ) T diffusive flux as in the Raviart–Thomas–Nédélec mixed finite element method of order l , by local postprocessing A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction property Crucial diffusive flux reconstruction property note that all the terms of the DG scheme are used in the construction of t h denote by Π l the L 2 -orthogonal projection onto P k ( T h ) the above construction yields ∇· t h = Π l ( f ) Proof: ( ∇· t h , ξ h ) T = − ( t h , ∇ ξ h ) T + � t h · n , ξ h � ∂ T = B h ( u h , ξ h ) = ( f , ξ h ) T diffusive flux as in the Raviart–Thomas–Nédélec mixed finite element method of order l , by local postprocessing A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction property Crucial diffusive flux reconstruction property note that all the terms of the DG scheme are used in the construction of t h denote by Π l the L 2 -orthogonal projection onto P k ( T h ) the above construction yields ∇· t h = Π l ( f ) Proof: ( ∇· t h , ξ h ) T = − ( t h , ∇ ξ h ) T + � t h · n , ξ h � ∂ T = B h ( u h , ξ h ) = ( f , ξ h ) T diffusive flux as in the Raviart–Thomas–Nédélec mixed finite element method of order l , by local postprocessing A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction property Crucial diffusive flux reconstruction property note that all the terms of the DG scheme are used in the construction of t h denote by Π l the L 2 -orthogonal projection onto P k ( T h ) the above construction yields ∇· t h = Π l ( f ) Proof: ( ∇· t h , ξ h ) T = − ( t h , ∇ ξ h ) T + � t h · n , ξ h � ∂ T = B h ( u h , ξ h ) = ( f , ξ h ) T diffusive flux as in the Raviart–Thomas–Nédélec mixed finite element method of order l , by local postprocessing A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Diffusive flux reconstruction property Crucial diffusive flux reconstruction property note that all the terms of the DG scheme are used in the construction of t h denote by Π l the L 2 -orthogonal projection onto P k ( T h ) the above construction yields ∇· t h = Π l ( f ) Proof: ( ∇· t h , ξ h ) T = − ( t h , ∇ ξ h ) T + � t h · n , ξ h � ∂ T = B h ( u h , ξ h ) = ( f , ξ h ) T diffusive flux as in the Raviart–Thomas–Nédélec mixed finite element method of order l , by local postprocessing A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Towards an a posteriori error estimate recall that the energy norm framework gives � � � ( f −∇· t h , ϕ ) − ( K ∇ u h + t h , ∇ ϕ ) � sup ||| u − u h ||| ≤ ||| u h − s h ||| + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 note that, by the Cauchy–Schwarz inequality: 2 ∇ u h + K − 1 1 2 t h � T ||| ϕ ||| T ( K ∇ u h + t h , ∇ ϕ ) T ≤ � K local conservativity of t h : ( f − ∇· t h , ϕ ) T = ( f − ∇· t h , ϕ − Π 0 ( ϕ )) T Poincaré inequality ( C P = 1 /π 2 ), energy norm definition: P h T �∇ ϕ � T ≤ C 1 / 2 1 h T 2 P � ϕ − Π 0 ( ϕ ) � T ≤ C ||| ϕ ||| T c 1 / 2 K , T notation: c K , T ( C K , T ) is the min (max) eigenvalue of K on T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Towards an a posteriori error estimate recall that the energy norm framework gives � � � ( f −∇· t h , ϕ ) − ( K ∇ u h + t h , ∇ ϕ ) � sup ||| u − u h ||| ≤ ||| u h − s h ||| + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 note that, by the Cauchy–Schwarz inequality: 2 ∇ u h + K − 1 1 2 t h � T ||| ϕ ||| T ( K ∇ u h + t h , ∇ ϕ ) T ≤ � K local conservativity of t h : ( f − ∇· t h , ϕ ) T = ( f − ∇· t h , ϕ − Π 0 ( ϕ )) T Poincaré inequality ( C P = 1 /π 2 ), energy norm definition: P h T �∇ ϕ � T ≤ C 1 / 2 1 h T 2 P � ϕ − Π 0 ( ϕ ) � T ≤ C ||| ϕ ||| T c 1 / 2 K , T notation: c K , T ( C K , T ) is the min (max) eigenvalue of K on T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Towards an a posteriori error estimate recall that the energy norm framework gives � � � ( f −∇· t h , ϕ ) − ( K ∇ u h + t h , ∇ ϕ ) � sup ||| u − u h ||| ≤ ||| u h − s h ||| + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 note that, by the Cauchy–Schwarz inequality: 2 ∇ u h + K − 1 1 2 t h � T ||| ϕ ||| T ( K ∇ u h + t h , ∇ ϕ ) T ≤ � K local conservativity of t h : ( f − ∇· t h , ϕ ) T = ( f − ∇· t h , ϕ − Π 0 ( ϕ )) T Poincaré inequality ( C P = 1 /π 2 ), energy norm definition: P h T �∇ ϕ � T ≤ C 1 / 2 1 h T 2 P � ϕ − Π 0 ( ϕ ) � T ≤ C ||| ϕ ||| T c 1 / 2 K , T notation: c K , T ( C K , T ) is the min (max) eigenvalue of K on T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Towards an a posteriori error estimate recall that the energy norm framework gives � � � ( f −∇· t h , ϕ ) − ( K ∇ u h + t h , ∇ ϕ ) � sup ||| u − u h ||| ≤ ||| u h − s h ||| + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 note that, by the Cauchy–Schwarz inequality: 2 ∇ u h + K − 1 1 2 t h � T ||| ϕ ||| T ( K ∇ u h + t h , ∇ ϕ ) T ≤ � K local conservativity of t h : ( f − ∇· t h , ϕ ) T = ( f − ∇· t h , ϕ − Π 0 ( ϕ )) T Poincaré inequality ( C P = 1 /π 2 ), energy norm definition: P h T �∇ ϕ � T ≤ C 1 / 2 1 h T 2 P � ϕ − Π 0 ( ϕ ) � T ≤ C ||| ϕ ||| T c 1 / 2 K , T notation: c K , T ( C K , T ) is the min (max) eigenvalue of K on T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Towards an a posteriori error estimate recall that the energy norm framework gives � � � ( f −∇· t h , ϕ ) − ( K ∇ u h + t h , ∇ ϕ ) � sup ||| u − u h ||| ≤ ||| u h − s h ||| + ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 note that, by the Cauchy–Schwarz inequality: 2 ∇ u h + K − 1 1 2 t h � T ||| ϕ ||| T ( K ∇ u h + t h , ∇ ϕ ) T ≤ � K local conservativity of t h : ( f − ∇· t h , ϕ ) T = ( f − ∇· t h , ϕ − Π 0 ( ϕ )) T Poincaré inequality ( C P = 1 /π 2 ), energy norm definition: P h T �∇ ϕ � T ≤ C 1 / 2 1 h T 2 P � ϕ − Π 0 ( ϕ ) � T ≤ C ||| ϕ ||| T c 1 / 2 K , T notation: c K , T ( C K , T ) is the min (max) eigenvalue of K on T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Theorem (A posteriori error estimate, pure diffusion case) There holds � NC , T + ( η R , T + η DF , T ) 2 � � ||| u − u h ||| 2 ≤ η 2 . T ∈T h nonconformity estimator η NC , T := ||| u h − I Os ( u h ) ||| T diffusive flux estimator 2 ∇ u h + K − 1 1 2 t h � T η DF , T := � K residual estimator η R , T := C 1 / 2 h T � f − Π l ( f ) � T P c 1 / 2 K , T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Theorem (A posteriori error estimate, pure diffusion case) There holds � NC , T + ( η R , T + η DF , T ) 2 � � ||| u − u h ||| 2 ≤ η 2 . T ∈T h nonconformity estimator η NC , T := ||| u h − I Os ( u h ) ||| T diffusive flux estimator 2 ∇ u h + K − 1 1 2 t h � T η DF , T := � K residual estimator η R , T := C 1 / 2 h T � f − Π l ( f ) � T P c 1 / 2 K , T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Theorem (A posteriori error estimate, pure diffusion case) There holds � NC , T + ( η R , T + η DF , T ) 2 � � ||| u − u h ||| 2 ≤ η 2 . T ∈T h nonconformity estimator η NC , T := ||| u h − I Os ( u h ) ||| T diffusive flux estimator 2 ∇ u h + K − 1 1 2 t h � T η DF , T := � K residual estimator η R , T := C 1 / 2 h T � f − Π l ( f ) � T P c 1 / 2 K , T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) = f Theorem (A posteriori error estimate, pure diffusion case) There holds � NC , T + ( η R , T + η DF , T ) 2 � � ||| u − u h ||| 2 ≤ η 2 . T ∈T h nonconformity estimator η NC , T := ||| u h − I Os ( u h ) ||| T diffusive flux estimator 2 ∇ u h + K − 1 1 2 t h � T η DF , T := � K residual estimator η R , T := C 1 / 2 h T � f − Π l ( f ) � T P c 1 / 2 K , T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Convective flux reconstruction Diffusive flux reconstruction t h ∈ RTN l ( T h ) , l = k or l = k − 1 as in the pure diffusion case Convective flux reconstr. q h ∈ RTN l ( T h ) , l = k or l = k − 1 normal components on each side: ∀ q h ∈ P l ( F ) , ( q h · n F , q h ) F = ( β · n F { { u h } } + γ β , F [ [ u h ] ] , q h ) F on each element (only for l ≥ 1): ∀ r h ∈ P d l − 1 ( T ) , ( q h , r h ) T = ( u h , β · r h ) T Crucial property ( ∇· t h + ∇· q h +( µ −∇· β ) u h , ξ h ) T = ( f , ξ h ) T ∀ T ∈ T h , ∀ ξ h ∈ P l ( T ) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Convective flux reconstruction Diffusive flux reconstruction t h ∈ RTN l ( T h ) , l = k or l = k − 1 as in the pure diffusion case Convective flux reconstr. q h ∈ RTN l ( T h ) , l = k or l = k − 1 normal components on each side: ∀ q h ∈ P l ( F ) , ( q h · n F , q h ) F = ( β · n F { { u h } } + γ β , F [ [ u h ] ] , q h ) F on each element (only for l ≥ 1): ∀ r h ∈ P d l − 1 ( T ) , ( q h , r h ) T = ( u h , β · r h ) T Crucial property ( ∇· t h + ∇· q h +( µ −∇· β ) u h , ξ h ) T = ( f , ξ h ) T ∀ T ∈ T h , ∀ ξ h ∈ P l ( T ) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Convective flux reconstruction Diffusive flux reconstruction t h ∈ RTN l ( T h ) , l = k or l = k − 1 as in the pure diffusion case Convective flux reconstr. q h ∈ RTN l ( T h ) , l = k or l = k − 1 normal components on each side: ∀ q h ∈ P l ( F ) , ( q h · n F , q h ) F = ( β · n F { { u h } } + γ β , F [ [ u h ] ] , q h ) F on each element (only for l ≥ 1): ∀ r h ∈ P d l − 1 ( T ) , ( q h , r h ) T = ( u h , β · r h ) T Crucial property ( ∇· t h + ∇· q h +( µ −∇· β ) u h , ξ h ) T = ( f , ξ h ) T ∀ T ∈ T h , ∀ ξ h ∈ P l ( T ) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) + β ·∇ u + µ u = f Theorem (A posteriori error estimate, energy norm) There holds ||| u − u h ||| ≤ η, �� � 1 / 2 �� � 1 2 η 2 ( η R , T + η DF , T + η C , 1 , T + η C , 2 , T + η U , T ) 2 η := + , NC , T T ∈T h T ∈T h where η NC , T = ||| u h − I Os ( u h ) ||| T (nonconformity), � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min (diffusive flux), DF , T η R , T = m T � f − ∇· t h − ∇· q h − ( µ − ∇· β ) u h � 0 , T (residual), η C , 1 , T = m T � ( Id − Π 0 )( ∇· ( q h − β s h )) � 0 , T (convection), � � η C , 2 , T = c − 1 / 2 � 1 � 2 ( ∇· β )( u h − s h ) 0 , T (convection), β ,µ, T η U , T = � F ∈F T m F � Π 0 , F (( q h − β s h ) · n F ) � F (upwinding). A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A post. estimate for −∇· ( K ∇ u ) + β ·∇ u + µ u = f Theorem (A posteriori error estimate, energy norm) There holds ||| u − u h ||| ≤ η, �� � 1 / 2 �� � 1 2 η 2 ( η R , T + η DF , T + η C , 1 , T + η C , 2 , T + η U , T ) 2 η := + , NC , T T ∈T h T ∈T h where η NC , T = ||| u h − I Os ( u h ) ||| T (nonconformity), � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min (diffusive flux), DF , T η R , T = m T � f − ∇· t h − ∇· q h − ( µ − ∇· β ) u h � 0 , T (residual), η C , 1 , T = m T � ( Id − Π 0 )( ∇· ( q h − β s h )) � 0 , T (convection), � � η C , 2 , T = c − 1 / 2 � 1 � 2 ( ∇· β )( u h − s h ) 0 , T (convection), β ,µ, T η U , T = � F ∈F T m F � Π 0 , F (( q h − β s h ) · n F ) � F (upwinding). A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Diffusive flux estimator η DF , T � � η ( 1 ) DF , T , η ( 2 ) η DF , T = min DF , T η ( 1 ) 1 2 ∇ u h + K − 1 2 t h � 0 , T DF , T = � K η ( 2 ) DF , T = m T � ( Id − Π 0 )( ∇· ( K ∇ u h + t h )) � 0 , T + � m 1 / 2 F ∈F T C 1 / 2 � t , T , F � ( K ∇ u h + t h ) · n F � F T cutoff fcts of local Péclet and Damköhler numbers in η ( 2 ) DF , T : m T := min { C 1 / 2 h T c − 1 / 2 K , T , c − 1 / 2 β ,µ, T } , P m T := min { ( C P + C 1 / 2 β ,µ, T + c − 1 / 2 β ,µ, T c − 1 / 2 ) h T c − 1 K , T , h − 1 T c − 1 K , T / 2 } � P η ( 1 ) DF , T alone cannot be shown semi-robust (Verfürth ’08) the idea of defining of η DF , T using a min has recently been proposed in Cheddadi, Fuˇ cík, Prieto, Vohralík ’08 in context of conforming FEM and reaction–diffusion problems A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Upwinding estimator η U , T η U , T = � F ∈F T m F � Π 0 , F (( q h − β s h ) · n F ) � F cutoff function of local Péclet and Damköhler numbers: � � � �� � | F | h 2 | F | m 2 F = min max T , max C F , T , F | T | c K , T | T | c β ,µ, T T ∈T F T ∈T F A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Individual estimators Upwinding estimator η U , T η U , T = � F ∈F T m F � Π 0 , F (( q h − β s h ) · n F ) � F cutoff function of local Péclet and Damköhler numbers: � � � �� � | F | h 2 | F | m 2 F = min max T , max C F , T , F | T | c K , T | T | c β ,µ, T T ∈T F T ∈T F A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Properties of the estimate Principal properties guaranteed upper bound no constants in principal estimators, known constants in the other ones cutoff functions of local Péclet ( h T � β � ∞ , T c − 1 K , T ) and Damköhler ( h 2 T c β ,µ, T c − 1 K , T ) numbers (here c β ,µ, T is the (essential) minimum of ( µ − 1 2 ∇· β ) ) explicit dependence on the mesh and data valid for arbitrary polynomial degree and data nonmatching meshes residual estimator η R , T is a higher-order term (data oscillation) A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Loc. efficiency for −∇· ( K ∇ u ) + β ·∇ u + µ u = f Theorem (Local efficiency, energy norm) There holds η NC , T + η DF , T + η R , T + η C , 1 , T + η C , 2 , T + η U , T ≤ C eff , T ||| u − u h ||| ∗ , � E T . Properties the estimates are locally efficient only semi-robustness: overestimation is a function of local Péclet and Damköhler numbers A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Loc. efficiency for −∇· ( K ∇ u ) + β ·∇ u + µ u = f Theorem (Local efficiency, energy norm) There holds η NC , T + η DF , T + η R , T + η C , 1 , T + η C , 2 , T + η U , T ≤ C eff , T ||| u − u h ||| ∗ , � E T . Properties the estimates are locally efficient only semi-robustness: overestimation is a function of local Péclet and Damköhler numbers A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Loc. efficiency for −∇· ( K ∇ u ) + β ·∇ u + µ u = f Theorem (Local efficiency, energy norm) There holds η NC , T + η DF , T + η R , T + η C , 1 , T + η C , 2 , T + η U , T ≤ C eff , T ||| u − u h ||| ∗ , � E T . Properties the estimates are locally efficient only semi-robustness: overestimation is a function of local Péclet and Damköhler numbers A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. A dual norm augmented by the convective derivative define � B D ( u , v ) := − ( β · n F [ [ u ] ] , { { Π 0 v } } ) F . F ∈F h introduce the augmented norm sup ||| v ||| ⊕ := ||| v ||| + {B A ( v , ϕ ) + B D ( v , ϕ ) } ϕ ∈ H 1 0 (Ω) , ||| ϕ ||| = 1 when �∇· β � ∞ , T is controlled by ( µ − 1 2 ∇· β ) on T for all T and when v ∈ H 1 0 (Ω) , recover the augmented norm introduced by Verfürth ’05 B D contribution is new and specific to the nonconforming case A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm a posteriori error estimate Estimator � � � 1 / 2 η U , T ) 2 η := 2 η + ( η R , T + η DF , T + � η C , 1 , T + � � T ∈T h η , η R , T , and η DF , T defined previously for the energy norm η C , 1 , T and � η U , T – slight modifications of η C , 1 , T and η U , T � Global jump seminorm define � � � 1 c K , T ||| v ||| 2 α F γ K , F h − 1 ] � 2 # , F h = F � [ [ v ] F #( T F ) c K , T T T ∈T h F ∈ F T � ∞ , T T h − 1 ] � 2 F + m 2 T T � β � 2 ] � 2 + c β ,µ, T h F � [ [ v ] F � [ [ v ] , 0 , F F ∩ F T the first two terms are natural for DG methods the third term at least contains the cutoff factor m T T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm a posteriori error estimate Estimator � � � 1 / 2 η U , T ) 2 η := 2 η + ( η R , T + η DF , T + � η C , 1 , T + � � T ∈T h η , η R , T , and η DF , T defined previously for the energy norm η C , 1 , T and � η U , T – slight modifications of η C , 1 , T and η U , T � Global jump seminorm define � � � 1 c K , T ||| v ||| 2 α F γ K , F h − 1 ] � 2 # , F h = F � [ [ v ] F #( T F ) c K , T T T ∈T h F ∈ F T � ∞ , T T h − 1 ] � 2 F + m 2 T T � β � 2 ] � 2 + c β ,µ, T h F � [ [ v ] F � [ [ v ] , 0 , F F ∩ F T the first two terms are natural for DG methods the third term at least contains the cutoff factor m T T A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm estimate and its efficiency Theorem (Fully robust a posteriori estimate) There holds ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ≤ � η + ||| u h ||| # , F h ≤ ˜ C ( ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ) . fully robust with respect to convection- or reaction dominance sharper than Schötzau & Zhu ’08 because of the cutoff factor in the jump seminorm only global efficiency the norm ||| · ||| ⊕ is a dual norm and is difficult to evaluate A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm estimate and its efficiency Theorem (Fully robust a posteriori estimate) There holds ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ≤ � η + ||| u h ||| # , F h ≤ ˜ C ( ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ) . fully robust with respect to convection- or reaction dominance sharper than Schötzau & Zhu ’08 because of the cutoff factor in the jump seminorm only global efficiency the norm ||| · ||| ⊕ is a dual norm and is difficult to evaluate A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm estimate and its efficiency Theorem (Fully robust a posteriori estimate) There holds ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ≤ � η + ||| u h ||| # , F h ≤ ˜ C ( ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ) . fully robust with respect to convection- or reaction dominance sharper than Schötzau & Zhu ’08 because of the cutoff factor in the jump seminorm only global efficiency the norm ||| · ||| ⊕ is a dual norm and is difficult to evaluate A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm estimate and its efficiency Theorem (Fully robust a posteriori estimate) There holds ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ≤ � η + ||| u h ||| # , F h ≤ ˜ C ( ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ) . fully robust with respect to convection- or reaction dominance sharper than Schötzau & Zhu ’08 because of the cutoff factor in the jump seminorm only global efficiency the norm ||| · ||| ⊕ is a dual norm and is difficult to evaluate A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Augmented norm estimate and its efficiency Theorem (Fully robust a posteriori estimate) There holds ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ≤ � η + ||| u h ||| # , F h ≤ ˜ C ( ||| u − u h ||| ⊕ + ||| u − u h ||| # , F h ) . fully robust with respect to convection- or reaction dominance sharper than Schötzau & Zhu ’08 because of the cutoff factor in the jump seminorm only global efficiency the norm ||| · ||| ⊕ is a dual norm and is difficult to evaluate A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Outline Introduction and motivation 1 Abstract framework 2 Optimal energy norm abstract framework A first computable estimate Optimal augmented norm abstract framework A posteriori error estimates 3 Flux reconstructions Energy norm error estimates Augmented norm error estimates Nonmatching grids Numerical experiments Concluding remarks and future work 4 A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems

I Abstract framework A posteriori estimates C Flux. rec. En. norm Aug. norm Nonm. grids Num. exp. Nonmatching grids Oswald interpolate on nonmatching grids consider a matching simplicial submesh � T h of T h consider u h ∈ P k ( T h ) as function in P k ( � T h ) take I Os ( u h ) on � T h Reconstruction of t h by direct prescription directly prescribe t h ∈ RTN l ( � T h ) by the values of u h this gives ( ∇· t h , ξ h ) T = ( f , ξ h ) T for all T ∈ T h and all ξ h ∈ P l ( T ) Reconstruction of t h by solving local linear systems consider the simplicial submesh R T of each T solve a local minimization problem (local linear system) on each T get in particular ∇· t h = � Π l f A. Ern, A. F. Stephansen & M. Vohralík Guaranteed and robust estimates for DG and CRD problems