### Table 2: Results for the 2-dimensional incompressible Navier-Stokes example on an IBM SP2. The column heads are the same as

in A Note on the Consistent Initialization for Nonlinear Index-2 Differential-Algebraic Equations in

2004

### Table 3. Algorithmic scalability for Navier-Stokes optimal flow control problem on 64 and 128 processors of a Cray T3E for a doubling (roughly) of problem size.

2000

"... In PAGE 20: ... We re- fer to the latter as LRQN. In Table3 we quote a set of representative results from many we have obtained for up to 1.5 million state variables and 50,000 control variables on up to 256 processors.... ..."

Cited by 1

### Table 5 Comparison of convergences for di erent algorithms (NI = nearly incompressible, I = incompressible, C = compressible)

"... In PAGE 22: ... As already mentioned, the treatment of compressible problems with the pro- posed algorithms is not e cient. For nearly incompressible problems and in- compressible problems, the performance in terms of the number of iterations with di erent preconditioners can be found in Table5 . The tested problem is the die problem with n = 48 subdomains, h=H = 1=6 and for ~ = 0:3.... In PAGE 27: ... Concerning initialization phase, the cost in- creases from FETI-I, to FETI-DPI, and up to FETI-DPIA. To compare these approaches, let us recall the convergence results for 2D incompressible case, with compressible preconditioner, of Table5 . Table 6 evaluates the relative costs with the previous complexity analysis (n = 48, H = n 0:5, h=H = 1=6).... ..."

### Table 5.5 Results for Stokes problem.

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### Table 1: The governing PDE apos;s for incompressible ow with heat transfer are shown in dimensionless form, including the Navier-Stokes equations with the Boussinesq approximation, the continuity equation, and a heat balance. Momentum

2000

"... In PAGE 5: ... In addition, a heat equation with convective and conduction terms is solved. The equations are shown in Table1 and include the time dependent terms, which are important for the formulation of the stability (eigenvalue) calcu- lation.The computational domain is discretized using a mesh of 94656 hexehe- dral elements, which corresponds to 100043 nodes.... ..."

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### Table 2: Flow over M6 wing on SGI Origin; fixed-size mesh of 357,900 vertices (1,431,600 DOFs incompressible, 1,789,500 DOFs compressible).

1999

"... In PAGE 10: ... 5.4 Parallel Scalability across Flow Regimes Trans-Mach convergence comparisons of the same problem are given in Table2 . Here efficiencies are normalized by the number of timesteps, to factor convergence degrada- tion out of the performance picture and measure implementation factors alone (though convergence degradation with increasing granularity is modest).... ..."

Cited by 25

### Table 11. Stokes flow Example Problem 4, GMRES solver. one-level - preconditioner = ILUT(2.,.2), two-level smoother=Gauss-Seidel, coarse solver Superlu, fine/coarse mesh ratio = 64.

"... In PAGE 16: ... Clearly while the coarse grid solution is under-resolved, there is significant information about the fine grid solution structure for this problem. As is evident from the convergence results presented in Table11 and Table 12, optimal convergence rates are obtained along with faster solution times for the two-level method for sufficiently large coarse grids. The CPU time scaling is again non-optimal but still provides faster solutions than the corresponding one-level methods.... ..."

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### Table I. Comparison of vertical end displacements and deviation from incompressibility for explicitly meshed and enriched approximations to the fractured membrane problem.

Cited by 3

### Table 1: Computational performance of di erent preconditioners for the incompressible problem. CPU time in seconde.

### Table 6: The results, where is given in (4.1) For CGS the updated residual is such that kr583k2 10?10 whereas the norm of the exact residual is larger than 10?4; we consider this as a case of non- convergence. Note that in this problem GMRESR(10) is the best method. The following examples come from a discretization of the incompressible Navier Stokes equations. This discretization leads to two di erent linear systems, the momentum equations and the pressure equation (for a further description we refer to [19]). Here we consider a speci c test problem, which describes the ow through a curved channel. In the rst example the problem is discretized with 16 64 nite volumes. The pressure equations are solved with GMRES(m) and GMRESR(m). We start with x0 = 0 and stop when krkk2=kr0k2 10?6. This is essentially the same problem as the one for which results were reported in Section 2, only the dis- cretization is slightly di erent.

1994

Cited by 43