Robust a posteriori error control and adaptivity for multiscale, - PowerPoint PPT Presentation

Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling Martin Vohralík Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris 6) joint work with Gergina Pencheva, Mary Wheeler, Tim Wildey (CSM, ICES, Austin) Linz, October 3, 2011

I Estimates Efficiency Application Simplif. Num. exp. C Outline Introduction 1 A posteriori error estimates 2 A general framework Discrete setting Potential and flux reconstructions Local efficiency 3 Application to different numerical methods 4 Multi-scale mortar mixed finite element method Multi-scale mortar discontinuous Galerkin method Multi-scale mortar coupled DG–MFEM A simplification without flux reconstruction 5 Numerical experiments 6 Mortar coupling Multiscale Multinumerics and adaptivity 7 Conclusions and future work M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Outline Introduction 1 A posteriori error estimates 2 A general framework Discrete setting Potential and flux reconstructions Local efficiency 3 Application to different numerical methods 4 Multi-scale mortar mixed finite element method Multi-scale mortar discontinuous Galerkin method Multi-scale mortar coupled DG–MFEM A simplification without flux reconstruction 5 Numerical experiments 6 Mortar coupling Multiscale Multinumerics and adaptivity 7 Conclusions and future work M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Multiscale Multiscale subdomain meshes of size h (low order polynomials) interface meshes of size H (high order polynomials) M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Multinumerics Multinumerics different numerical methods in different subdomains M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Mortar coupling T h Ω 2 G H Ω 3 Ω 1 Ω 4 Nonmatching subd. grids Interface grid Mortar coupling mortars used to enforce weakly mass conservation over the interface grid effective parallel implementation: independent local subd. problems, only the mortar unknowns globally coupled M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Aims of this work Aims of this work derive guaranteed a posteriori error estimates � p − p h � ≤ η ( p h ) ensure their local efficiency η T ≤ C � p − p h � neighbors of T look for robustness with respect to the ratio H / h (the constant C is independent of the ratio H / h ) bound separately the subdomain and interface errors propose an adaptive strategy which balances the subdomain and interface errors develop a unified setting encompassing different numerical methods M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Previous works Multiscale/multinumerics/mortars Arbogast, Pencheva, Wheeler, Yotov (2007) (multiscale mortar mixed finite element method) Girault, Sun, Wheeler, Yotov (2008) (coupling DG and MFE by mortars) A posteriori error estimates Prager and Synge (1947) (error equality) Ladevèze and Leguillon (1983) and Repin (1997) (application to a posteriori error estimation) Wohlmuth (1999) / Bernardi and Hecht (2002) (mortars) Wheeler and Yotov (2005) (mortar MFE) Aarnes and Efendiev (2006) / Larson and Målqvist (2007) (multiscale) Ainsworth / Kim / Ern, Nicaise, Vohralík (2007) (DG) Vohralík (2007, 2010) (MFE) Creusé and Nicaise (2008) (multinumerics) M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Previous works Multiscale/multinumerics/mortars Arbogast, Pencheva, Wheeler, Yotov (2007) (multiscale mortar mixed finite element method) Girault, Sun, Wheeler, Yotov (2008) (coupling DG and MFE by mortars) A posteriori error estimates Prager and Synge (1947) (error equality) Ladevèze and Leguillon (1983) and Repin (1997) (application to a posteriori error estimation) Wohlmuth (1999) / Bernardi and Hecht (2002) (mortars) Wheeler and Yotov (2005) (mortar MFE) Aarnes and Efendiev (2006) / Larson and Målqvist (2007) (multiscale) Ainsworth / Kim / Ern, Nicaise, Vohralík (2007) (DG) Vohralík (2007, 2010) (MFE) Creusé and Nicaise (2008) (multinumerics) M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Setting Model problem −∇· ( K ∇ p ) = f in Ω , p = 0 on ∂ Ω Ω ⊂ R d , d = 2 , 3, polygonal K is symmetric, bounded, and uniformly positive definite f ∈ L 2 (Ω) Potential and flux p : potential (pressure head); p ∈ H 1 0 (Ω) u := − K ∇ p : flux (Darcy velocity); u ∈ H ( div , Ω) , ∇· u = f Energy (semi-)norms � � ||| ϕ ||| 2 := 1 � 2 , ϕ ∈ H 1 ( T h ) � K 2 ∇ ϕ � � � K − 1 � 2 , v ∈ L 2 (Ω) ||| v ||| 2 2 v ∗ := M. Vohralík A posteriori control for multiscale, multinumerics, and mortars

I Estimates Efficiency Application Simplif. Num. exp. C Setting Model problem −∇· ( K ∇ p ) = f in Ω , p = 0 on ∂ Ω Ω ⊂ R d , d = 2 , 3, polygonal K is symmetric, bounded, and uniformly positive definite f ∈ L 2 (Ω) Potential and flux p : potential (pressure head); p ∈ H 1 0 (Ω) u := − K ∇ p : flux (Darcy velocity); u ∈ H ( div , Ω) , ∇· u = f Energy (semi-)norms � � ||| ϕ ||| 2 := 1 � 2 , ϕ ∈ H 1 ( T h ) � K 2 ∇ ϕ � � � K − 1 � 2 , v ∈ L 2 (Ω) ||| v ||| 2 2 v ∗ := M. Vohralík A posteriori control for multiscale, multinumerics, and mortars