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An Introduction to Coupling Conditions Homogeneous Heterogeneous - PowerPoint PPT Presentation

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Heterogeneous DD Martin J. Gander An Introduction to Coupling Conditions Homogeneous Heterogeneous Domain Decomposition Problems Heterogeneous Problems with a New Approach based on Factorization Best Coupling Conditions Advection-

  1. Heterogeneous DD Martin J. Gander An Introduction to Coupling Conditions Homogeneous Heterogeneous Domain Decomposition Problems Heterogeneous Problems with a New Approach based on Factorization Best Coupling Conditions Advection- Diffusion Model Problem Martin J. Gander Factorization Stationary Case martin.gander@unige.ch Algorithm Error Estimates Numerical Experiments University of Geneva Evolution Case Algorithm September 2016 Error Estimates Numerical Experiments Conclusions Joint work with Summary References Laurence Halpern and Veronique Martin

  2. Heterogeneous DD Coupling Conditions for Homogeneous Problems Martin J. Gander L u := ( η − ∆) u = f , in a domain Ω Coupling Conditions Homogeneous Problems = ⇒ Solution u we want to compute is well defined ! Heterogeneous Problems Best Coupling Conditions Advection- Diffusion Model Problem Factorization Stationary Case Algorithm Error Estimates Numerical Experiments Evolution Case Algorithm Error Estimates Numerical Experiments Conclusions Summary References

  3. Heterogeneous DD Coupling Conditions for Homogeneous Problems Martin J. Gander L u := ( η − ∆) u = f , in a domain Ω Coupling Conditions Homogeneous Problems = ⇒ Solution u we want to compute is well defined ! Heterogeneous Problems Best Coupling Conditions H.A. Schwarz 1869: Przemieniecki 1963: Advection- ¨ Uber einen Grenz¨ ubergang Matrix Structural Analysis Diffusion Model Problem durch alternierendes Verfahren of Substructures Factorization Stationary Case Algorithm Error Estimates Numerical Ω 1 Γ 2 Γ 1 Experiments Ω 2 Ω 1 Γ Ω 2 Evolution Case Algorithm Error Estimates ∂ Ω ∂ Ω Numerical Experiments Conclusions Summary L u 1 = f , in Ω 1 L u 1 = f , in Ω 1 References L u 2 = f , in Ω 2 L u 2 = f , in Ω 2 u 1 = u 2 , on Γ 1 u 1 = u 2 , on Γ u 2 = u 1 , on Γ 2 ∂ n u 2 = ∂ n u 1 , on Γ

  4. Heterogeneous DD Coupling Conditions for Heterogeneous Problems Martin J. Gander 1) Physics are different in different regions: Coupling ������������������ ������������������ ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Conditions ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Homogeneous ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� ������������������ ������������������ Problems ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Structure Fluid Heterogeneous Problems Best Coupling Solution u we want to compute is well defined Conditions = ⇒ coupling conditions are given by the physics Advection- Diffusion Model Problem Factorization Stationary Case Algorithm Error Estimates Numerical Experiments Evolution Case Algorithm Error Estimates Numerical Experiments Conclusions Summary References

  5. Heterogeneous DD Coupling Conditions for Heterogeneous Problems Martin J. Gander 1) Physics are different in different regions: Coupling ������������������ ������������������ ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Conditions ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Homogeneous ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� ������������������ ������������������ Problems ������������������� ������������������� ������������������ ������������������ ������������������� ������������������� ������������������� ������������������� Structure Fluid Heterogeneous Problems Best Coupling Solution u we want to compute is well defined Conditions = ⇒ coupling conditions are given by the physics Advection- Diffusion 2) Different models for computational savings: Model Problem ◮ Using an expensive model only where it is necessary Factorization Stationary Case Γ Algorithm airfoil Error Estimates Ω 2 Numerical Experiments Ω 1 Evolution Case ◮ Coupling across different dimensions Algorithm Error Estimates Blood Vessel Numerical Γ Ω 2 Experiments 1d PDE model Heart 3d PDE model Conclusions Ω 1 R C Summary Ω 2 References Γ Transistor Circuit ODE Ω 1 PDE = ⇒ coupling conditions are unknown !

  6. Heterogeneous DD Coupling Conditions for Computational Savings Martin J. Gander airfoil Ω Coupling Conditions Homogeneous L NS u = f in Ω Problems Heterogeneous Problems = ⇒ Solution u we want to compute is well defined ! Best Coupling Conditions Advection- Diffusion Model Problem Factorization Stationary Case Algorithm Error Estimates Numerical Experiments Evolution Case Algorithm Error Estimates Numerical Experiments Conclusions Summary References

  7. Heterogeneous DD Coupling Conditions for Computational Savings Martin J. Gander airfoil Ω Coupling Conditions Homogeneous L NS u = f in Ω Problems Heterogeneous Problems = ⇒ Solution u we want to compute is well defined ! Best Coupling Conditions Advection- Γ Diffusion Model Problem airfoil Factorization Ω 2 Stationary Case Ω 1 Algorithm Error Estimates Numerical Experiments L NS u 1 = f in Ω 1 L E u 2 = f in Ω 2 Evolution Case Algorithm Error Estimates Numerical Experiments Conclusions Summary References

  8. Heterogeneous DD Coupling Conditions for Computational Savings Martin J. Gander airfoil Ω Coupling Conditions Homogeneous L NS u = f in Ω Problems Heterogeneous Problems = ⇒ Solution u we want to compute is well defined ! Best Coupling Conditions Advection- Γ Diffusion Model Problem airfoil Factorization Ω 2 Stationary Case Ω 1 Algorithm Error Estimates Numerical Experiments L NS u 1 = f in Ω 1 L E u 2 = f in Ω 2 Evolution Case Algorithm Dubach (1993): “L’objectif est alors d’essayer des conditions de Error Estimates Numerical transmission ad´ equates ` a la fronti` ere de fa¸ con ` a minimser l’erreur entre Experiments Conclusions la solution du probl` eme de transmission et celle de Navier Stokes Summary complet dans tout le domaine.” References

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