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Homogenisation Theory for PDEs

Discover the concept of homogenisation for advection-diffusion equations starting from various assumptions to derive the homogenised equation using Multiple Scale Method. Explore validity, existence, uniqueness, and convergence.

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Homogenisation Theory for PDEs

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  1. Homogenisation Theory for PDEs Homogenisation for Advection-Diffusion Equations

  2. Homogenisation Starting point ? Homogenised equation

  3. Review- Steady state heat conduction Starting point several assumptions… Goal: Homogenised equation

  4. Multiple Scale Method Assume the expansion Validity? macroscopic scale Independent Variables microscopic scale

  5. By inserting the expansion into , we obtain Solvability condition for :

  6. Instead of , we now have Cell problem: assume that Homogenised Equation , where

  7. Steady state heat conduction Starting point Homogenised equation

  8. Existence, uniqueness and convergence There exists a unique solution of There exists a unique solution of and weakly in .

  9. Remark (validity of the expansion) , solution of the homogenised problem cell problem higher order cell problem higher order cell problems Under certain conditions, we get the estimate

  10. Advection-Diffusion equations given, 1-periodic and sufficiently smooth incompressible: passive tracer

  11. Linear Transport equations ( ) Where is ?

  12. Rescaling New variables Goal: New formulation of the problem

  13. Multiple Scale Method By substituting the expansion in the equation, we obtain where Example: Problem!!!

  14. If Then indeed ergodic By computing we obtain the homogenised equation , where

  15. Advection-Diffusion equations ( ) Given, 1-periodic and sufficiently smooth incompressible: passive tracer

  16. Rescaling New variables Goal: New formulation of the problem

  17. Multiple Scale Method By substituting the expansion in the equation, we obtain where

  18. Solvability Condition Integrate over Y smooth 1-periodic function Solvability condition

  19. First step ( ) In fact,

  20. Second step ( ) Solvability condition Separation of variables Using this in , we obtain the cell problem

  21. Third step ( ) Solvability condition Leads to the homogenised equation Effective Diffusivity: matrix where

  22. Effective Diffusivity The homogenisation procedure enhances diffusion; the effective diffusivity is always greater than the molecular diffusivity in the following sense: For every vector we have

  23. Summary Steady state heat conduction (Review) Multiple Scale Method Existence, uniqueness and convergence Remark (validity of the expansion) Advection-Diffusion equations (linear transport equation)

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