### Table 1: The number of iterations of the parallel domain decomposition algorithm required to solve a typical three-dimensional convection-diffusion problem in [12].

"... In PAGE 5: ... Furthermore, it is applied to a class of convection- diffusion equations in three dimensions that is not covered by the underlying theory in [3]. Nevertheless, it proves to be surprisingly robust, as illustrated by the iteration counts shown in Table1 that are typical of the results in [12]. Furthermore, very creditable parallel performances are recorded, including parallel speed-ups in excess of 12 when using locally refined ... In PAGE 5: ...Table 1: The number of iterations of the parallel domain decomposition algorithm required to solve a typical three-dimensional convection-diffusion problem in [12]. The iteration counts shown in Table1 illustrate that the number of iterations of the parallel solver that are required to obtain a converged solution is essentially independent of the level of the finest mesh and the number of subdomains used. Hence, provided the sequential solver used on each processor (at step 4 of the algorithm in Figure 4) has a computational cost of O(N), the total cost of the parallel algorithm will also be approximately proportional to N.... ..."

### Table 2 Results of SPARK3 on the convection-diffusion equations (34).

2000

"... In PAGE 17: ... We have taken the absolute and relative error tolerances for each component equal to the same error tolerance TOL. We give some statistics obtained with the code SPARK3 on this problem in Table2 . Since this problem is linear, approximately only one Newton iteration per timestep was taken.... ..."

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### Table 3: Number of iterations (CPU time) for a convection-di usion equation (central di erences for convection terms). h?1

"... In PAGE 8: ... Tables 3 and 4 give equivalent results for the convection-di usion equation ? r2u + a:ru = g (7) with = 10?2 and a = (3; 4)T. In Table3 a central di erence approximation is used for the convective terms, while in Table 4 upwinding is used. Table 3 also gives the mesh Peclet number Peh for each test problem.... In PAGE 8: ... In Table 3 a central di erence approximation is used for the convective terms, while in Table 4 upwinding is used. Table3 also gives the mesh Peclet number Peh for each test problem. Note that for all but the last test the meshes are too coarse, and Peh gt; 2.... In PAGE 9: ... Here the advantage of the preconditioned iterative methods over both unpreconditioned and direct methods is still more marked. In Table3 , for example, PCGS is over 28 times faster than MA48 on the h = 1=128 mesh, while on the nest mesh the best preconditioned method is more than 20 times faster than the best unpreconditioned method. Note that unpreconditioned CGS, which has been robust for the previous test problems, runs into di culties for these convection-di usion problems irrespective of whether or not... ..."

### Table 2.1: Terms of the generic convection-diffusion equation for cylindrical coor- dinates system

### Table 1 lists likely physical parameters for Io apos;s asthenosphere, based on the model of [Segatz et al., 1988]. Constant viscosity is assumed in this study, both to allow higher Rayleigh number to be reached with available computational resources, and because the temperature differences (hence viscosity differences) in Io apos;s asthenosphere are arguably small, as discussed later. Because the Prandtl number of rocks is effectively infinite even for a viscosity as low as 108 Pa.s, the usual equations for Boussinesq, infinite- Prandtl number, internally-heated convection, using the standard thermal non-dimensionalization (e.g., [Parmentier et al., 1994; Travis et al., 1990]), are used:

"... In PAGE 2: ...0 (defined later). Note that due to the absence of a fixed temperature scale, T is nondimensionalized using a heating- derived scale: Tscale = D2 lt; Q gt; Cp (4) The key parameter in this scaling study is RaQ, the Rayleigh number for internal heating, given by: RaQ = g lt; Q gt; D5 2Cp (5) where the meaning of the symbols and representative values are given in Table1 . These representative values can be used to calculate bounds on RaQ.... ..."

### TABLE 4.3 Pre-eigenvalues,targets and computed eigenvaluesfor the convection-diffusion problem.

1999

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### TABLE 4.3 Pre-eigenvalues, targets and computed eigenvalues for the convection-diffusion problem.

1999

Cited by 3

### Table 4.3 Pre-eigenvalues, targets, and computed eigenvalues for the convection-diffusion problem.

1999

Cited by 3

### TABLE 4.3 Pre-eigenvalues, targets and computed eigenvalues for the convection-diffusion problem.

1999

Cited by 3