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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 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 • 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
Characteristics of Neumann Boundary Values • Solvability not every system of equation has a solution. • Unique if u is a solution, so is u + c.
Is there anything we can do? Let’s assume A is non-singular FIRST. • Direct solver • Iterative solver Krylov Subspace Methods.
Krylov Subspace Methods… • Conjugate Gradient (CG) symmetric positive definite matrix • Generalized Minimal Residual (GMRES) non-symmetric indefinite matrix
1.Solvability 2.Unique Mean zero Least square solution If A is singular …
If … Re-orthogonlization To assure there exists a solution.
Strategy I: Fix one point Spectrum shift
Spectrum shift by one point fixing… You are solving an approximate problem !!!
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,
If you want to try it… … I am happy to help you …