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Wavelet Methods for Elliptic Partial Differential Equations$

Karsten Urban

Print publication date: 2008

Print ISBN-13: 9780198526056

Published to British Academy Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780198526056.001.0001

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(p.xxvi) LIST OF TABLES

(p.xxvi) LIST OF TABLES

Source:
Wavelet Methods for Elliptic Partial Differential Equations
Publisher:
Oxford University Press

  1. Table 2.1 Approximation error for f ( x ) : = sin ( 2 π x ) ( f P j I f ) for the supremum and Euclidean norm.

  2. Table 2.2 Approximation error ( f J P j I f J ) for fJ in (2.9) for the supremum and Euclidean norm.

  3. Table 2.3 Approximation error for f ( x ) : = sin ( 2 π x ) ( f P j O f ) for the supremum and Euclidean norm.

  4. Table 2.4 Approximation error for fJ for the supremum and Euclidean norm.

  5. Table 2.5 Refinement coefficients of Daubechies scaling functions Nφ for N = 1, …, 10.

  6. Table 2.6 Refinement coefficients of B-spline scaling functions and dual scaling functions for d = 1,2 and some values of d̃.

  7. Table 2.7 Refinement coefficients of B-spline scaling functions and dual scaling functions for d = 3, 4 and some values of d̃.

  8. Table 4.1 Condition numbers for the stiffness matrix A j hat for homogeneous Dirichlet boundary conditions for different values of c and j.

  9. Table 4.2 Condition numbers for the stiffness matrix A j hat in the periodic case for different values of c and j. For c = 0, the matrix is singular.

  10. Table 4.3 L 2-errors and convergence factors in the three different cases.

  11. Table 4.4 H 1-errors and convergence factors for u 1 as solution for the homogeneous Dirchilet problem (3.2) and the periodic boundary value problem (3.1).

  12. Table 4.5 Condition numbers of the stiffness matrix A j for the case d = 2 and different values of c and j.

  13. Table 4.6 Factor between condition numbers of level j and j − 1 (d = 2).

  14. Table 4.7 Condition numbers of stiffness matrix Aj depending on d for fixed level j = 12. The last column labeled ‘∞’ contains the condition number of the mass matrix (without derivatives).

  15. Table 4.8 Comparison of CPU times for BPX and MultiGrid using c = 0.1, d = 3 (and d̃ = 5 for the restriction).

  16. Table 5.1 Sizes of the supports of semiorthogonal and biorthogonal spline wavelets.

  17. Table 5.2 Condition numbers for scaling functions and wavelets of Chui and Wang ( ρ Φ and ρ Ψ ) for L 2(ℝ), and of Chui and Quak ( ρ Φ j [ 0 , 1 ] and ρ Ψ j [ 0 , 1 ] ) for L 2([0, 1]) and j ≤ 11 in dependence of the spline order d. For comparison, we also display corresponding numbers for biorthogonal spline wavelets on ℝ from [71] and on [0, 1] from [35].

  18. Table 6.1 Condition numbers of wavelet-preconditioned stiffness matrix depending on d = d̃ = 2 for different values of c and j and factor between condition numbers of level j and j − 1.

  19. (p.xxvii)
  20. Table 6.2 Condition numbers of wavelet-preconditioned (diagonal scaling) stiffness matrix depending on d and d̃.

  21. Table 6.3 Comparison of CPU times for d = 3, d̃ = 5 and c = 0.1.

  22. Table 7.1 Sizes of the active wavelet sets, interior residual (on Λ) and global error.

  23. Table 7.2 Expected and observed slopes for the error of the adaptive approximate operator application in the periodic case for Example 7.1 (left) and 7.3 (right).

  24. Table 7.3 Coefficients γd,ℓ.

  25. Table 8.1 Minimal level depending on d and d̃.

  26. Table 8.2 Expected and observed slopes for the error of the adaptive approximate operator application.

  27. Table 8.3 Sobolev and Besov norms of the exact solution (8.106) to Example 8.20.

  28. Table 9.1 L -errors computed on a dyadic grid with mesh size h = 2−11.

  29. Table 9.2 Rate of convergence.

  30. Table 9.3 Comparision of one (left) and two (right) subdomains of the same domain Ω.

  31. Table 9.4 Comparison of Algorithm 7.9 (left) and Algorithm 7.6 (right).

  32. Table 9.5 Results for Example (I). Numbers of adaptively generated degrees of freedom, ratio to best N-term approximation and relative errors.

  33. Table 9.6 Results for Example (II). Numbers of adaptively generated degrees of freedom, ratio to best N-term approximation and relative error.

  34. Table 9.7 Minimal level j 0 and number of scaling functions N Φ on the minimal level for different order discretizations.

  35. Table 9.8 Results for the second example with piecewise linear trial functions for velocity and pressure. Note that in this case the number of degrees of freedom for the coarsest level is 243.

  36. Table 9.9 Numerical results for the experiment in Section 9.4.8.3 corresponding to UZAWAexact. Number of active coefficients, ratio of the error of the numerical approximation and the best N-term approximation and relative error for the first velocity component and the pressure. The results for the second velocity components are similar.

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