How To Solve Poisson Equation with Neumann Boundary Values

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How To Solve Poisson Equation with Neumann
Boundary Values

• Jin Chen
• CPPG

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Background
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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

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Characteristics of Neumann Boundary Values

• Solvability
• not every system of equation has a solution.
• Unique
• if u is a solution, so is u c.

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Eigenvalues of the Poisson Equation with Neumann
BV
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Is there anything we can do?

• Lets assume A is non-singular FIRST.
• Direct solver
• Iterative solver
• Krylov Subspace Methods.

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Iterative solver
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Krylov Subspace Methods

• Conjugate Gradient (CG)
• symmetric positive definite matrix
• Generalized Minimal Residual (GMRES)
• non-symmetric indefinite matrix

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CG
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GMRES
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Numerical Complexity
Methods Inner Product SAXPY Matrix-Vector Product
CG 2 3 1
GMRES i1 i1 1
Methods Storage requirements
CG Matrix6n
GMRES Matrix(i5)n
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If A is singular
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Definition of Null Space
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Null Space
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CG for singular systems
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If
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If
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If
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If
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If
Re-orthogonlization
To assure there exists a solution.
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GMRES for singular systems
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GMRES for singular systems
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Numerical Experiment
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Matrix Structure
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Singular and Solvability Check
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Spectrum with a zero eigenvalue
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Strategy I Fix one point
Spectrum shift
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Spectrum shift by one point fixing
You are solving an approximate problem !!!
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Eigenmode k2, l0
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Strategy II null space
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Eigenmode k2 l0
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Eigenmode k3, l0
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Application in M3D

• F equation,
• Singular check Ae0,
• Solvability check (b,e)0,
• Re-orthogonalization bb-(b,e)/(e,e),
• Uniqueness check (x,e)0,
• CG with nullspace,
• GMRES with nullspace,

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If you want to try it
I am happy to help you
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Eigenvalues for Poisson Equation
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Dirichlet Boundary Values
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Neumann Boundary Values
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Krylov Subspace Methods
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Matrix Representation of Krylov Subspace Methods
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Krylov Subspace Methods

• CG
• symmetric positive definite matrix
• GMRES
• non-symmetric indefinite matrix

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How to interpret poisson equation
Neumann correspond to a solid boundary which
fluid cannot penetrate. www.math.umn.edu/olver/a
ppl_/
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Matrix Structure
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Eigenvalues via grid resolution
You are solving a different problem !!!