Finite difference method

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Finite difference method

• Poisson equation in 2D with Dirichelt BC
• Well-posedness
• Existence
• Uniqueness

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Finite difference method
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Finite difference method

• Finite difference approximation
  Dimension-by-dimension
• Computation error analysis
• Solve the linear system efficiently
• Error bound???

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Local truncation error
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Finite difference method

• Order of accuracy second order
• Error analysis -- maximum principle
• Proof Exercise!!
• Efficient solver
• Iterative solvers CG, PCG, Gauss-Seidel, SOR,
  ..
• Direct Poisson solver

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Linear system
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Linear system
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Matrix form
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Matrix form
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Matrix form

• Linear system
• Iterative solvers Rewrite the difference
  equations as

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SOR method An example
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Fast Poisson solver via DST

• For Poisson equation in 1D with Dirichlet BC
• Finite difference discretization
• Linear system

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Fast Poisson solver

• Homogeneous BC
• In PDE level sine transform

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Fast Poisson solver

• Exact solution

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Fast Poisson solver

• In discrtization level Discrete since transform
  (DST)

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Fast Poisson solver
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Algorithm for fast Poisson solver via DST
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Fast Poisson solver

• Inhomogeneous BC --- Homogenizing the BC
• Introducing
• Plugging into the difference equations
• With

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Algorithm for fast Poisson solver via DST
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Fast Poisson solver in 2D

• The equation
• The finite difference discretization

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Fast Poisson solver in 2D

• Discrete sine transform in 2D
• Plugging into the finite difference equations

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Algorithm for Fast Poisson solver in 2D
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Algorithm for Fast Poisson solver in 2D
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Comments on fast Poisson solver

• Advantages
• Direct solver give exact solution to the linear
  system
• Memory cost O(M) no extra memory is needed!!
• Computational cost O(M ln M)
• Very efficient in 2D 3D due to FST!!!
• Can be extended to Neumann BC or periodic BC
• Disadvantages
• The domain should be a rectangle in 2D box in
  3D
• Uniform mesh in each direction is needed!!
• The coefficients of the PDE must be constant!!
• BC much be the same type in opposite edges!!!

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Extension to Neumann BC

• The problem

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Extension to Neumann BC

• Discretization at shifted grids points by half
  grid
• Direct Poisson solver via discrete cosine
  transform (DCT)
• Exercise!!
• Extension to periodic BC discrete Fourier
  transform (DFT)

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Poisson equation in 2D on a disk or shell

• The problem
• The ideas Domain mapping or variable transform

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The discretization
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The discretization for a shell
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The discretization for a disk
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The discretization

• Local truncation error
• Order of accuracy second order
• Error estimate -- maximum principle
• Solution of the linear system
• Fast Poisson solver
• Discrete Fourier transform in transverse
  direction
• Solve a linear system with tri-diagonal
  coefficient matrix in r-direction

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Extension

• 3D Poisson equation
• In a box, sphere, shell, cylindrical cylinder,
  etc.
• General linear elliptic equation
• Elliptic system
• Navier system,

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More topics

• Compact scheme --- high order methods with less
  grid points
• 4th (6th ,) order methods need 5 (7, ) grid
  points in 1D
• Need use ghost points near boundary
• Question Can we design high order methods using
  less grid points???
• An example in 1D
• Extension to 2D 3D Exercise!!!!

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More topics

• General geometry in 2D 3D
• FEM or FVM or Boundary integral method
• Adaptive mesh refinement (AMR)
• Solution has sharp change locally
• Refine the mesh adaptively based on the
  approximation
• Nonlinear problem
• Discretization, solve nonlinear system
• ..