Multigrid method by convection dominated currence

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Multigrid method by convection dominated currence

• By
• Jakob Angele

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Model Problem

• Aproximation with
• finite difference method
• n subintervalls length h1/n
• n-1 linear equations
• Properties of A
• Symmetric
• Block tridiagonal
• Spare
• Diagonally dominant

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Basic Iterative MethodsJakobi Method

• Model problem
• First iteration
• Splitting matrix A
• In matrix-form
• 2n storage locations are needed

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Gaus-Seidel Method

• Components of a approximation are used as soon as
  they are computed. n storage locations.
• In matrix-form
• The order of updating is significant
• Ascending and descending order
• Red-black Gause-Seidel method

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Weighted Jakobi Method

• Weighted Jakobi iteration
• Eigenvalues
• Eigenvectors
• Wave numbers greater then n cannot be represented
  correctly on the grid

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Decreasing of the Error

• Eigenvectors

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Calculation of Eigenvalues

• Weighted Jakobi iteration
• Eigenvalues of
• with
• The eigenvectors of A are the same as the
  eigenvectors of

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Calculation of Eingenvalues

• Eigenvalueproblem
• The j-th line
• This equals
• By using the identity
• one can get
• Finally by using
• one can get

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The Error

• Each method discussed so far may be present in
  the form
• All iterative method are designed so that the
  following equation is true
• Error done by approximation
• Furthermore, the error can be written as
• This leads to an Error after m iterations

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The Error

• Significance of the weighting factor

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The Error

• If the eigenvalue is small, the error decreasing
  effectively with each iteration.
• If the eignevalue is close or equal to 1 the
  error does not decrease significantly.
• We choose . All the eigenvalues
  with an are smaller then .
• These components of the error are dumped
  effectively.
• They correspond to the oscillatory eigenvectors.

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The Error

• Smooth modes with are eliminated slowly.
• Oscillatory modes with are eliminated
  quickly.

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Convergence of different modes

• Convergence slows down and eventually seems to
  stall.

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Convergence of different modes

• Efficient elimination of the oscillatory modes
• It cannot eliminate the oscillatory modes
  effectively!

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The Multigrid

• For the approximation different grids are used.
• Coarser grids are introduced. Each new grid has
  half as many grid points as the previous one.
• Different components of the error are can be
  approximated effectively on different grids.
• The values calculated on one grid can be
  interpolated from one to another grid and so
  enhance the values there.

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Multigrids

• What does smooth wave look like under a coarser
  grid?
• Grid transformation
• Grid points of the grid are the even numbered
  grid points of the old one
• The eigenvectors are transformed
• A smooth mode appears on a coarser grid as a more
  oscillatory one.
• There, it can be approximated effectively with
  same iteration method.

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Multigrids

• A wave with wavenumber k6 on Oh (n12 points is
  projected ont O2h (n6 points)
• The coarser grid sees the wave as more
  oscillatory on the grid then on the fine grid

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Linear Interpolation

• The easiest method to transfer values from a
  courser grid to a finer grid is through linear
  interpolation.
• This method is most of the time sufficient
  enough.
• For the case on n8 the linear interpolation
  operator has the following form

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Linear Interpolation

• How well does linear interpolation work?
• The exakt error is smooth. When the coarsegrid
  approximation is interpolated to the fine grid,
  the interpolation is also smooth.
• Therefore the linear approximation is very
  accurate.
• (b) The exakt error is oscillatory. But the
  interpolation to the fine grid is still smooth.
  Therefore the approximation is not very
  effective.

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Linear Interpolation

• Linear interpolation works best on smooth errors.
• Relaxation methods, like Gause-Seidel, work best
  on osscillatory error.
• Algorithm
• Error is made smooth by a relaxation mehtod.
• Smooth error is interpolated to a coarser grid.
• The error appears more oscillatory.
• The error made smooth again through relaxation.

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Intergrid Transfer

• The second class of linear transfer is moving
  from a fine grid to a coarse grid
• These operators are called restriction operators.
• Two important restriction operators
• Injection
• Full weighting

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Injection

• Injection takes its values directly from the fine
  grid.
• It is definded simply by
• For the case n8 the injection operator has the
  form

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Full weighting

• Full weighting is defined by
• Again, on a grid with n8 the full weighting
  operator is represented by
• An important property of the full weighting
  operator
• It is the transposing of the linear interpolation
  operator.

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Multigrid Problems with k-d equations

• The smoothing through Gause-Seidel or other
  approximation becomes inefficient.
• If the order of relaxation is not in direction of
  the current the approximation substantional
  improves only to the next grid point.
• The numbering of the grid points has to be
  adjusted to the current.
• Stability of coarser grid
• Due to the strong current, the second coarser
  grid is loosing its symmetric dominant property
• The Gause-Seidel method is a lot more inefficient
• Errors occur through interpolation of the
  solutions from one grid to another
• Main Problem The coarser grids have to be chosen
  so that
• Diffusion may not be overrepresented.

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Ideal Coarse Grid

• Ordinary gird Through approximation the
  diffusion is overrepresented
• The ideal grid approximates the current a lot
  more precise