1 / 40

Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Prblems

Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Prblems. Lecture 4. Finite Difference Formulas Problem Definition. Given distinct points. , find the weights such. that. is of optimal order of accuracy.

yeardley
Download Presentation

Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Prblems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Finite Difference Discretization of Elliptic Equations: FD Formulas and Multidimensional Prblems Lecture 4

  2. Finite Difference FormulasProblem Definition Given distinct points , find the weights such that is of optimal order of accuracy. • Lagrange interpolation • undetermined coefficients Two approaches :

  3. Finite Difference FormulasLangrage interpolation Lagrange polynomials Lagrange interpolant

  4. Finite Difference FormulasLangrage interpolation Approximate Therefore,

  5. Finite Difference FormulasLangrage interpolation Example… Set Second order Lagrange interpolant

  6. Finite Difference FormulasLangrage interpolation …Example… Assuming a uniform grid (First derivative) Forward Centered Backward

  7. Finite Difference FormulasLangrage interpolation …Example (Second derivative) Centered

  8. Finite Difference FormulasUndetermined coefficients Start from Insert Taylor expansions for about Determine coefficients to maximize accuracy.

  9. Finite Difference FormulasUndetermined coefficients Example… (uniform spacing )

  10. Finite Difference FormulasUndetermined coefficients …Example Equating coefficients of Solve,

  11. Poisson Equation in 2D Definition in on

  12. Poisson Equation in 2D Discretization

  13. Poisson Equation in 2D Approximation For example … for small

  14. Poisson Equation in 2D Equations suggests ….

  15. Poisson Equation in 2D Equations Example…

  16. Poisson Equation in 2D Equations …Example

  17. Poisson Equation in 2D Equations Numbering becomes component

  18. Poisson Equation in 2D Equations Block Matrix… Block tridiagonal matrix

  19. Poisson Equation in 2D Equations …Block Matrix Block Definitions has a banded structure Bandwidth :

  20. Poisson Equation in 2D Equations SPD Property ( is SPD) Hence for any : exists and is unique

  21. Poisson Equation in 2D Error Analysis Truncation Error For for all

  22. Poisson Equation in 2D Error Analysis Stability It can be shown that Ingredients: • Positivity of the coeffiicients of • Bound on the maximum row sum

  23. Poisson Equation in 2D Error Analysis Stability Error equation If

  24. Eigenvalue Problem in 2D Statement in on Assume (for simplicity) Solutions

  25. Eigenvalue Problem in 2D Exact Solution Eigenvalues Eigenvectors

  26. Eigenvalue Problem in 2D Discrete Problem Eigenvectors

  27. Eigenvalue Problem in 2D Discrete Problem Eigenvalues Low Modes High Modes ( , fixed) as

  28. Eigenvalue Problem in 2DCondition Number of A as If grows (in ) as number of grid points. (better than in 1D, relatively speaking !!)

  29. Eigenvalue Problem in 2DLink to Error Estimate Error equation

  30. Discrete Fourier Solution Poisson Problem 2D is SPD diagonal matrix of eigenvalues is matrix of eigenvectors

  31. Discrete Fourier Solution Poisson Problem 2D ALGORITHM Still cost is … But …

  32. Discrete Fourier Solution Poisson Problem 2D • Matrix multiplications can be reorganized • (tensor product evaluation) is a (Inverse) Discrete Fourier Transform Using FFT

  33. Discrete Fourier Solution Poisson Problem 2D We are interested in solving in on on and are given. where

  34. Non-Rectangular Domains Poisson Problem 2D Mapping Can we determine an equivalent problem to be solved on ?

  35. Non-Rectangular Domains Poisson Problem 2D Transformed equations How can we evaluate terms and ?

  36. Non-Rectangular Domains Poisson Problem 2D …Transformed equations…

  37. Non-Rectangular Domains Poisson Problem 2D …Transformed equations… and

  38. Non-Rectangular Domains Poisson Problem 2D …Transformed equations Finally becomes depend on the mapping and

  39. Non-Rectangular Domains Poisson Problem 2D Normal Derivatives… is parallel to (or ); e.g., on

  40. Non-Rectangular Domains Poisson Problem 2D Normal Derivatives… Thus, with

More Related