Functional a posteriori error estimates for space-time isogeometric - PowerPoint PPT Presentation

Johann Radon Institute for Computational and Applied Mathematics Functional a posteriori error estimates for space-time isogeometric approximations of parabolic I-BVPs Svetlana Matculevich 1 joint work with U. Langer 2 and S. Repin 3 1 , 2 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Austria 3 St. Petersburg V.A. Steklov Institute of Mathematics, Russia; University of Jyväskylä, Finland Workshop 2: November 07-11, 2016 Space-Time Methods for PDEs 1/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Functional type error estimates For a class of parabolic I-BVP problems in Ω ⊂ R d , t ∈ ( 0 , T ) ∂ t u + L u = f , u ( 0 ) = u 0 , u = 0 on ∂ Ω , with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M M ( v , D (Ω , u 0 , f )) ≤ | | | u − v | | | ≤ M ( v , D (Ω , u 0 , f )) , V v minorant error majorant u universal for any v ∈ V , M computable , reliable , i.e., | | | u − v | | | ≤ M ( v , D ) , M realistic in comparison to error, i.e., I eff = | is close to 1, | | | u − v | | consistent , i.e., M ( v ) is continuous with respect to v and M ( u ) = 0, efficient for adaptive algorithms V h → V h ref . 2/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

Johann Radon Institute for Computational and Applied Mathematics Model I-BVP problem Find u : Q → R satisfying linear parabolic x ∈ Ω ⊂ R d , d = { 1 , 2 , 3 } , T > 0 initial-boundary value problem (I-BVP) ( x , t ) ∈ Q := Ω × ( 0 , T ) ( x , t ) ∈ ∂ Q := Σ ∪ Σ 0 ∪ Σ T Σ := ∂ Ω × ( 0 , T ) Σ 0 := Ω × { 0 } ∂ t u − div x · p = f in Q , Σ T := Ω × { T } p = ∇ x u [ 0 , T ] x 1 t u ( x , 0 ) = u 0 on Σ 0 , Ω t u = 0 on Σ , Σ T Ω Σ 0 where ∂ t denotes the time derivative, x 2 div x and ∇ x are divergence and gradient operators in space, respectively, u 0 ∈ H 1 0 (Σ 0 ) is a given initial state, � f is a source function in L 2 ( Q ) , with � u � L 2 ( Q ) = � u � Q induced by ( v , w ) Q =: Q v w d x d t . 3/34 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations