### Table 2.1 Numerical convergence of the Multilevel scheme for the Laplace equation with dirichlet boundary conditions. Timings were obtained on IBM RS6000 workstation.

### Table 2: Comparaison of the methods OE and GOE. In conclusion we have examined numerical schemes for integrating the system (1:7) used for nding the rst m Lyapunov exponents of a dynamical system. An arbitrary convergent numerical scheme was modi ed to preserve orthonor- mality and convergence proved with explicit Euler apos;s method as the arbitrary scheme. This led to examining a more general system of which (1:7) is an ex- ample. Arbitrary convergent schemes were modi ed to preserve orthonormality and convergence proved in the general case. Some numerical results were then presented.

"... In PAGE 22: ... Indeed if the system were non{ regular it is exceedingly unlikely that any reasonable numerical results would have been found [5]. In Table2 we compare OE and GOE for di erent step sizes, integrating for a xed time of T = 1000. We know the second exponent should be zero since the chaotic trajectory is stable and a zero exponent corresponds to perturbations... ..."

### Table 5: Convergence results for different distortion parameters.

"... In PAGE 22: ...5. The numerical results presented in Table5 show that the convergence rate of the nonlinear FV scheme... ..."

### Table 1. Stability and convergence rates of rotational schemes: The first (resp. second) number in the parenthesis is the convergence rate for the velocity in the L2-norm (resp. the velocity in the H1-norm and the pressure in the L2-norm); s is the regularity index of the Stokes operator; the symbol star means numerical evidences only.

### Table 1 Convergence order for box scheme. h err p

"... In PAGE 10: ...ig. 2. Solution of (18) (left) and lt;(z) and =(z) transformed to [0; 100] (right). classic convergence order of the global error. In Table1 , we give the empirical convergence order of the numerical solutions computed from the solutions for three consecutive step-sizes h in the discretization, see Table 1. Table 1 Convergence order for box scheme.... ..."

### Table 1: Ideal 2-Grid convergence factors. One can notice that the results are much better in the case of agglome- ration (i). This is the option we keep in the sequel. 5.2 2-D advection-di usion equation: some numerical experiments Let us consider now the problem ? quot; u + div(V u) = 1 with homogeneous Dirichlet boundary conditions. We propose to solve this problem with a V{ cycle MG scheme where two iterations of the basic iterative method (one pre- and post-smoothing) are done on each level at each cycle. For most of the following numerical experiments, this number of smoothing sweeps per level corresponds with the optimal CPU time. The convergence of the MG scheme is studied for three types of meshes: a) a structured uniform ne mesh,

1993

"... In PAGE 16: ... The smoother used in our ideal 2-G algorithm is the classical point Gauss{ Seidel relaxation. The mean reduction factor of the residual and the number of ideal 2-G cycles needed to reach the convergence level of 10?6 are given in Table1 for the previous agglomeration procedures (i) and (ii). If conv denotes the total number of ideal 2-G cycles leading to convergence, then is de ned by = conv qRes( conv) : These notations will be kept in all the numerical tests that follow.... ..."

### Table 1 Stability and convergence rates of rotational schemesa

"... In PAGE 33: ... In short, for all three classes of projection schemes, their rotational versions should always be preferred over the stan- dard versions. We now summarize in Table1 the main results related to the rotational forms of the pressure-correction methods, velocity-correction methods, and consistent splitting methods. References [1] Y.... ..."

### Table 1. L1{error and EOC rate for subsequent re nement levels of the grid.

"... In PAGE 31: ...ith subsequently ner grid size. We notice that the numerical approximations improve as the grid is re ned. This indicates convergence of the scheme. The convergence of the scheme can be seen in Table1 where we list the L1{error of the numerical approx- -1.5 -1 -0.... ..."

### Table 3. Monotone scheme on smooth solution test problems.

1995

"... In PAGE 16: ... Next, we describe our numerical results. In Table3 , we show our results for the smooth solution test problems. We see that in each of the three problems, the monotone scheme converges linearly, as expected, and that the computational... ..."

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