I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE ON COMPUTABLE ESTIMATES OF THE DISTANCE TO THE EXACT SOLUTION OF PARABOLIC PROBLEMS BASED ON LOCAL POINCAR´ E TYPE INEQUALITIES Svetlana Matculevich ∗† , Sergey Repin †∗ , and Pekka Neittaanm¨ aki ∗ ∗ Dept. of Mathematical Information Technology, University, of Jyv¨ askyl¨ a, Finland † St. Petersburg Dept. of V.A. Steklov Institute of Mathematics of RAS, Russia TIEJ601: Postgraduate Seminar in Information Technology January 20, 2015 1/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE B ACKGROUND Jun. 2010, BSc in Applied Mathematics and Informatics , SPbSPU, Russia. Mar. 2012, MSc in Information Technology , University of Jyv¨ askyl¨ a, Finland. Jun. 2012, MSc in Mathematical Modeling and Informatics , SPbSPU, Russia. 2/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE P H D TERMS AND DEFENSE PhD started on June 1, 2012. The defense is planned in Autumn, 2015. The thesis structure is the collection of papers. 3/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE S UPERVISION Prof. Sergey Repin St.Petersburg Department of Steklov Mathematical Institute RAS, Russia MIT, University of Jyv¨ askyl¨ a, Finland Prof. Pekka Neittaanm¨ aki MIT, University of Jyv¨ askyl¨ a, Finland 4/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE T HE PROGRESS 69 ECTs completed 1 + 3 journal paper accepted: S. Matculevich and S.Repin, Computable bounds of the distance to the exact solution of parabolic problems based on Poincar´ e type inequalities , Zap. Nauchn. Sem. S.-Peterburg Otdel Mat. Inst. Steklov (POMI), 425(1), 7–34, 2014. S. Matculevich, P.Neitaanm¨ aki, and S.Repin, A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne–Weinberger inequality , Discrete and Continuous Dynamical Systems - Series A, AIMS, 35(6), 2659–2677, 2015. S. Matculevich and S.Repin, Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation , Applied Mathematics and Computation, Elsevier, 247, 329–347, 2014. 5/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE S TAGES OF MATHEMATICAL MODELING Creating mathematical model of a certain phenomenon := 1 described by the system of partial differential equation (PDE’s). Solving the system := constructing numerical 2 representation of the key characteristics of phenomena. Analyze the results := comparison of the numerical data 3 with physical properties of the object. Conclusion. 4 6/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE S TAGE OF THE reliable MATHEMATICAL EXPERIMENT Mathematical model is correct. ⇒ Existence theory of PDEs : existence and stability of exact solution. Numerical method is correct. ⇒ Approximation theory of PDEs : convergence, stability, and ect. Errors (of approximation or numerical method) are explicitly controlled. ⇒ A posteriori error control theory for PDEs . 7/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE A PPLICATIONS OF EVOLUTIONARY REACTION - DIFFUSION EQUATIONS Heat conduction (transfer) in thermodynamics: modeling thermal energy storage, heat transfer in human body. 8/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE A PPLICATIONS OF EVOLUTIONARY REACTION - DIFFUSION EQUATIONS Heat conduction (transfer) in thermodynamics: modeling thermal energy storage, heat transfer in human body. Diffusion-convection-reaction in chemistry and biology: population dynamics, e.i. predator-prey system, ecological dynamics. 8/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE M OTIVATION To control of the quality of numerical computation for EVOLUTIONARY CLASS OF PROBLEMS ∂ u ∂ t = A u + f , t ∈ ( 0 , T ) , (1) u ( 0 ) = u 0 , 9/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE M OTIVATION To control of the quality of numerical computation for EVOLUTIONARY CLASS OF PROBLEMS ∂ u ∂ t = A u + f , t ∈ ( 0 , T ) , (1) u ( 0 ) = u 0 , by equipping numerical methods with error estimates M ( v , D ) ≤| | | u − v | | | ≤ M ( v , D ) , which must be (2) explicitly computable, universal, reliable, efficient. 9/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and S T := ∂ Ω × [ 0 , T ] = Γ D × [ 0 , T ] = S D , where Γ D is part of boundary with Dirichlet BC. 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and S T := ∂ Ω × [ 0 , T ] = Γ D × [ 0 , T ] = S D , where Γ D is part of boundary with Dirichlet BC. Find u ( x , t ) ∈ H 1 0 ( Q T ) satisfying the following system: u t − ∇ · A ∇ u + λ u = f ∈ L 2 ( Q T ) in Q T , u ( x , 0 ) = ϕ ∈ L 2 (Ω) in Ω , u = 0 on S D , where H 1 0 (Ω) subspace of H 1 (Ω) with functions satisfying the Dirichlet boundary condition. 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and S T := ∂ Ω × [ 0 , T ] = Γ D × [ 0 , T ] = S D , where Γ D is part of boundary with Dirichlet BC. Find u ( x , t ) ∈ H 1 0 ( Q T ) satisfying the following system: u t − ∇ · A ∇ u + λ u = f ∈ L 2 ( Q T ) in Q T , u ( x , 0 ) = ϕ ∈ L 2 (Ω) in Ω , u = 0 on S D , where H 1 0 (Ω) subspace of H 1 (Ω) with functions satisfying the Dirichlet boundary condition. 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and S T := ∂ Ω × [ 0 , T ] = Γ D × [ 0 , T ] = S D , where Γ D is part of boundary with Dirichlet BC. Find u ( x , t ) ∈ H 1 0 ( Q T ) satisfying the following system: u t − ∇ · A ∇ u + λ u = f ∈ L 2 ( Q T ) in Q T , u ( x , 0 ) = ϕ ∈ L 2 (Ω) in Ω , u = 0 on S D , where H 1 0 (Ω) subspace of H 1 (Ω) with functions satisfying the Dirichlet boundary condition. 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E VOLUTIONARY REACTION - DIFFUSION PROBLEM Ω ∈ R d is a bounded connected domain with Lipschitz boundary, Q T := Ω × ( 0 , T ) , T > 0, and S T := ∂ Ω × [ 0 , T ] = Γ D × [ 0 , T ] = S D , where Γ D is part of boundary with Dirichlet BC. Find u ( x , t ) ∈ H 1 0 ( Q T ) satisfying the following system: u t − ∇ · A ∇ u + λ u = f ∈ L 2 ( Q T ) in Q T , u ( x , 0 ) = ϕ ∈ L 2 (Ω) in Ω , u = 0 on S D , where H 1 0 (Ω) subspace of H 1 (Ω) with functions satisfying the Dirichlet boundary condition. 10/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE W EAK FORMULATION OF THE PROBLEM Find u ( x , t ) ∈ H 1 0 ( Q T ) [4, 5, 6, 15] satisfying the integral identity � � � � u ( x , T ) η ( x , T ) − u ( x , 0 ) η ( x , 0 ) d x − u η t d x d t + Ω Q T � � � A ∇ u · ∇ η d x d t + λ u η d x d t = f η d x d t , Q T Q T Q T ∀ η ∈ H 1 0 ( Q T ) . (3) [4] S. I. Repin , Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation, Rend. Mat. Acc. Lincei, 2002. [8] S. Repin and S. Sauter , Functional a posteriori estimates for the reaction-diffusion problem, C. R. Acad. Sci. Paris , 2006. 11/42
I NTRODUCTION G ENERAL ERROR ESTIMATES E XAMPLES A PPLICATION FOR ADVANCED PROBLEMS A DVANCED ERROR ESTIMATE E RROR ENERGY NORM Let u , v ∈ H 1 0 (Ω) , then e ( x , t ) = ( u − v )( x , t ) is measured by the norm | 2 A + θ � w ( θ, χ, λ ) e � 2 Q T + ζ � e ( x , T ) � 2 [ e ] ( ν,θ,ζ ) = ν | | |∇ e | | Ω , (4) where � | 2 | | |∇ e | | A := A ∇ e · ∇ e d x d t , Q T ν, θ, χ , and ζ are positive numbers, and w is positive function. 12/42
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