### Table 1 presents the computational costs for solving pressure equation, on the Cray J90 by Jacobi- PCG method, using P-EBE and an edge-based data structure. The problem analyzed is a quarter of a five- spot problem, in two and three-dimensions, discretized by linear triangles and tetrahedra. In the two- dimensional problem we have considered a heterogeneous medium, with a lognormal random permeability field, while the three-dimensional problem was homogeneous. The meshes comprise uniform cells, each one subdivided in two triangles or five tetrahedra, respectively for two and three dimensions.

"... In PAGE 5: ... Table1 . Computational data for pressure equation solving on a Cray J90 for the quarter of a five-spot problem.... ..."

### Table 1 shows the number of oating point operations required by SuperLU for various uniform meshes when using piecewise polynomials of degree 3. As can be seen the solver appears to have a complexity of approximately O(n1:8) for this class of problem (where n is the dimension of the linear system and the average value for the exponent is calculated from the three nest meshes), with a complexity of approximately O(n1:3) for subsequent right-hand sides. This second asymptotic rate is similar to that obtained by the multigrid approach in [23] however the cost of the sparse factorization is somewhat greater. This factorization cost may be contrasted with the number of operations required to solve nite element discretizations of Laplace apos;s equation on similar grids using SuperLU. These gures are given for piecewise cubic polynomial approximations in Table 2. For these problems we observe an approximate cost of O(n1:6) for the sparse factorization of a matrix of dimension n. It is this disparity that motivates the direct method that is described in the following section.

1999

"... In PAGE 7: ... Table1 : The number of oating point operations required to factorize and solve (forward and backward substitution) n n systems of the form (12) resulting from piecewise cubic nite element discretizations of (6) on uniform grids. Here f and s are the exponents in work estimates of the form Cn for the factorization and the solution steps respectively.... In PAGE 14: ... For second and subsequent right-hand sides the dominant terms will have a cost of O(ns b) O(n1:2 b ). Inspection of Table1 suggests that there is likely to be a signi cant saving when using this new algorithm in comparison to the application of a general sparse direct solver to the original system. 5 Numerical results In this section we report on the performance of the proposed algorithm when applied to linear systems of the form (14) obtained from the mixed nite element discretization of the biharmonic equation on unstructured grids (using SuperLU for the factorization and solution of systems in- volving Ki).... ..."

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### Table 5: Mesh features for the fracture problem.

in SUMMARY

"... In PAGE 10: ...2 Fracture problem In this section, a fracture problem is analyzed by taking four linear nite element meshes (see Figure 8). Its corresponding features are given in Table5 . Table 6 presents the results for SGE, CGGS, and some multigrid methods.... ..."

### Table 1: Number of QMR iterations for the solution of the linear problem with respect to di erent dimensions and parameters.

2006

"... In PAGE 6: ... More precisely, we propose to choose the preconditioner C in the form C 1 = 1 2 (A + M) 1 0 0 (A + M) 1 I I I I ; (15) where the inverse of A + M is approximated by a symmetric multigrid iteration. Numerical results show { also in the case of approximated inverse { the fast convergence of Krylov space iteration methods, independently of mesh size and parameters (see Table1 ). Moreover, we can prove the following result on a condition number bound by purely algebraic arguments: Lemma 1.... In PAGE 7: ... We solve the non-symmetric linear system (14) by the quasi-minimal residual method (QMR) [6] with the preconditioner (15). Table1 presents the number of steps needed to reach a relative accuracy 10 6 for di erent parameter settings and dimensions of the nite element space. All these examples are test cases of the shielding problem that is described in Section 6.... In PAGE 7: ... In all these cases, the reluctivity is set to Fe = 1 0 Fe = 1 4 10 4 8 102 m H in the iron domain and to 0 = 1 0 = 1 4 10 7 8 105 m H in the coil and the surrounding air. As can be seen in Table1 , the number of QMR iterations is independent of both dimension of the FE-space and choices of the parameters.... ..."

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### Table 5: Mesh features for the fracture problem. Mesh Nodes Elements Equations NCoefdir mdir NCoefite mite

"... In PAGE 10: ...2 Fracture problem In this section, a fracture problem is analyzed by taking four linear nite element meshes (see Figure 8). Its corresponding features are given in Table5 . Table 6 presents the results for SGE, CGGS, and some multigrid methods.... ..."

### Table 2: Solution of the linear system associated with the linear elasticity problem (P = M).

"... In PAGE 19: ... For P = 1 a sequential BiCGStab method preconditioned by one sequential multigrid V-cycle is used. The results are listed in Table2 where I denotes the number of used BiCGStab iterations. By comparison with the measurements in [4] we see that the overall performance has improved considerably, mainly due to the application of... ..."

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### Table 5. Failures on some di cult problems Problem SF, quadratic SF, linear LF, quadratic LF, linear

1996

"... In PAGE 16: ... Also, alternate techniques or improvements are in order for problems such as fpqp3, fphe1 or fppb1. Table5 , though somewhat negative, represents the experiments we have done by the writing of this report. It can be taken as a challenge for further innovation.... ..."

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### Table 5. Failures on some difficult problems Problem SF, quadratic SF, linear LF, quadratic LF, linear

1996

"... In PAGE 16: ... Also, alternate techniques or improvements are in order for problems such as fpqp3, fphe1 or fppb1. Table5 , though somewhat negative, represents the experiments we have done by the writing of this report. It can be taken as a challenge for further innovation.... ..."

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### Table 2: Iteration counts, using the modi ed algebraic preconditioner, for the sequences of matrices obtained from piecewise linear discretizations of Problems 1 and 2.

in A New Parallel Domain Decomposition Preconditioner II: Generalization to a Mesh-Free Parallel Solver

2000

"... In PAGE 13: ... Some signi cant improvements in performance may easily be obtained however. Table2 shows equivalent results to those in Ta- ble 1 but using a slightly di erent (less aggressive) coarsening algorithm. Here, step 6(a) has been modi ed to Set f := 2((`?1)=2) (where integer division is used in the exponent) and dmax has been doubled (to 16).... ..."

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### Table 1: Linear elasticity, di erent Poisson ratios, test1.i test1.i.

1996

"... In PAGE 29: ...001 amp; 1 amp; 1000g inner k value f.001 amp; 1 amp; 1000g no of grid levels 4 no of space dimensions 2 coarse partition [3,3] re nement [2,2] sweeps [2,2] element type ElmB4n2D basic method DDIter domain decomposition method Multigrid smoother basic method SOR Table1 0: Discontinuous coe cients resolved on the coarsest grid, test1.i either the solution inside or the solution outside of [1=3; 2=3]2 dominates, while the solution is almost constant elsewhere, see table 10, input le test1.... In PAGE 29: ...001 amp; 1000.g coarse partition f[2,2] amp; [4,4]g relative quadrature order 1 Table1 1: Discontinuous coe cients not aligned with / not resolved on the coarse grid, test2.i We can redo the computations for the case of a non-matching coarsest grid.... In PAGE 29: ...001 amp; 1000.g no of grid levels 3 no of space dimensions 3 coarse partition [3,3,3] re nement [2,2,2] element type ElmB8n3D Table1 2: Discontinuous coe cients aligned with the coarse grid, 3D, test3.i We can redo the computations for the three dimensional case on the unit cube.... ..."

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