4.3 Eigen-Analysis of Spectral Derivative Matrices 203

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1 . Eigen-Analysis of Spectral Derivative Matrices 8 N = 8 N = N = 6 6 N = (Round off error) Fig... Legendre collocation first-derivative eigenvalues computed with 6-bit precision. Results contaminated by round-off error are indicated Fig... ǫ-pseudospectra, Λ ǫ, of Legendre collocation first-derivative matrix. Λ ǫ is plotted for ǫ =,.,..., for both N = 6 (left) and N = (right). The range of the isoline values are [, ] for both figures

2 . Algebraic Systems and Solution Techniques LC LG NI generalized LG NI Fig... Legendre first-derivative spectra and pseudospectra for N = 6. Topleft: spectra. Top-right: spectrum and pseudospectra of L coll (LC), bottom-left: spectrum and pseudospectra of K GNI (LG-NI), bottom-right: spectrum and pseudospectra of M GNIKGNI (generalized LG-NI). The range for isolines is [,] for the upper right figure, [., ] for the lower left figure, [ 7, ] for the lower right figure (u N x,v N ) N u N ()v N () = (f,v N ) N v N P N. (..) The associated (N + ) (N + ) matrix that represents the left-hand side of (..) for the nodal basis is K GNI = D T NM GNI diag{,,...,}, (..) where D N is the first-derivative matrix (..8) and M GNI = diag{w,..., w N } is the diagonal mass matrix of the LGL integration weights. Figures. and. illustrate the spectra and pseudospectra for N = 6 and N = 6, respectively, of the matrices for Legendre collocation (L coll ), Legendre G-NI (K GNI, see (..)), and generalized Legendre G-NI (M GNI K GNI) approximations. The spectra for the G-NI matrix are relatively insensitive to round-off errors, unlike the spectra for the other two

3 . Eigen-Analysis of Spectral Derivative Matrices LC LG NI generalized LG NI Fig... Legendre first-derivative spectra and pseudospectra for N = 6. Topleft: spectra. Top-right: spectrum and pseudospectra of L coll (LC), bottom-left: spectrum and pseudospectra of K GNI (LG-NI), bottom-right: spectrum and pseudospectra of M GNIKGNI (generalized LG-NI). The range for isolines is [-,] for the upper right figure, [-.,.7] for the lower left figure, [-6,] for the lower right figure matrices. The generalized G-NI matrix is even more sensitive than the collocation matrix. The extreme eigenvalues for these matrices, as computed in 6-bit arithmetic, are displayed in the left half of Fig... The abrupt slope changes in some of the curves for the extreme eigenvalues are produced by round-off error effects, as can be seen by careful comparison of Figs.. and.. The condition numbers κ (L) in the -norm for these matrices, again as computed in 6-bit arithmetic, are displayed in the right half of Fig... The condition numbers of both L coll and M GNI K GNI scale as O(N ), whereas those of K GNI scale sublinearly with N. The Fourier first-derivative matrix is skew-symmetric (see Sect...), and therefore is a normal matrix. Hence, the numerically computed eigenvalues of the Fourier collocation first-derivative matrix are not nearly so

4 8. Algebraic Systems and Solution Techniques ν d u dx + du dx =, < x <, u() =, u() =, (..7) this situation corresponds to a numerically unresolved boundary layer. The numerical solution of such an unresolved problem contains spurious oscillations, as illustrated in Fig..7. (See the theoretical discussion in Sect. 7.; in particular, see (7..) or (7..), and the discussion after (7..6).) Then, the regime of behavior of extreme eigenvalues is that of the pure-convection, first-order GNI matrix; a numerical stabilization (see Sect. 7..) should be used in order to get rid of potential instabilities Fig..8. Spectrum and pseudospectra of Legendre G-NI advection-diffusion matrices with N = for ν =. Stiffness matrix K GNI (left) and generalized matrix M GNIKGNI (right). The range for isolines is [.,] on the left, and [,] on the right To illustrate the sensitivity of the spectra to round-off errors, we furnish the pseudospectra in Figs..8 and.9 for ν = and ν =, respectively. For the advection-diffusion problem, as for the pure first-order problem, the generalized G-NI matrix is more sensitive to round-off error than the stiffness matrix. Perhaps surprisingly, there is greater sensitivity to round-off error for the ν = case than for the ν = one.. Preconditioning From the previous eigen-analysis it appears clear that spectral matrices ought to be preconditioned when solving the associated systems by iterative

5 . Preconditioning Fig..9. Spectrum and pseudospectra of Legendre G-NI advection-diffusion matrices with N = for ν =. Stiffness matrix K GNI (left) and generalized matrix M GNIKGNI (right). The range for isolines is [, ] on the left, and [,] on the right methods. We begin this section with an elementary discussion of iterative methods that serves to motivate the practical necessity for using preconditioning. Then we examine the basics of low-order finite-difference and finiteelement preconditioning for spectral discretizations by considering several one-dimensional model problems. Next, we survey the alternatives for efficient preconditioning in several dimensions. Finally, we summarize the use of spectral discretizations of constant-coefficient operators as preconditioners for variable-coefficient operators... Fundamentals of Iterative Methods for Spectral Discretizations The fundamentals of iterative methods for spectral equations, as well as the effect of preconditioning, are perhaps easiest to grasp for the simple onedimensional model problem d u = f in (,π), dx u π-periodic, (..)