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How To Solve Poisson Equation with Neumann Boundary Values

How To Solve Poisson Equation with Neumann Boundary Values. Jin Chen CPPG. F equation in M3D (Auxiliary quantities related to perturbed toroidal flux). Background. Outlines. Characteristics of Neumann Boundary Values Numerical singularity of such Boundary Values

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How To Solve Poisson Equation with Neumann Boundary Values

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  1. How To Solve Poisson Equation with Neumann Boundary Values Jin Chen CPPG

  2. F equation in M3D (Auxiliary quantities related to perturbed toroidal flux) Background

  3. Outlines • Characteristics of Neumann Boundary Values • Numerical singularity of such Boundary Values • Null Space method for such singularity • CG and GMRES for non-singular linear equation • Null Space based CG and GMRES for singular linear equation • Application to eigenvalue problem • Application to M3D

  4. Characteristics of Neumann Boundary Values • Solvability not every system of equation has a solution. • Unique if u is a solution, so is u + c.

  5. Eigenvalues of the Poisson Equation with Neumann BV

  6. Is there anything we can do? Let’s assume A is non-singular FIRST. • Direct solver • Iterative solver Krylov Subspace Methods.

  7. Iterative solver

  8. Krylov Subspace Methods… • Conjugate Gradient (CG) symmetric positive definite matrix • Generalized Minimal Residual (GMRES) non-symmetric indefinite matrix

  9. CG

  10. GMRES

  11. Numerical Complexity

  12. 1.Solvability 2.Unique Mean zero Least square solution If A is singular …

  13. Definition of Null Space

  14. Null Space

  15. CG for singular systems

  16. If

  17. If …

  18. If …

  19. If …

  20. If … Re-orthogonlization To assure there exists a solution.

  21. GMRES for singular systems

  22. GMRES for singular systems …

  23. Numerical Experiment

  24. Matrix Structure

  25. Singular and Solvability Check

  26. Spectrum with a zero eigenvalue

  27. Strategy I: Fix one point Spectrum shift

  28. Spectrum shift by one point fixing… You are solving an approximate problem !!!

  29. Eigenmode k=2, l=0

  30. Strategy II: null space

  31. Eigenmode k=2 l=0

  32. Eigenmode k=3, l=0

  33. Application in M3D • F equation, • Singular check: Ae=0, • Solvability check: (b,e)=0, • Re-orthogonalization: b=b-(b,e)/(e,e), • Uniqueness check: (x,e)=0, • CG with nullspace, • GMRES with nullspace,

  34. If you want to try it… … I am happy to help you …

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