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25, and coarse–fineGauss–Seidel as a smoother. The iterations were stopped when the relative residual was smaller than 10−8 . We also include operator complexity Cop , which is defined as the sum of the number of nonzeroes of all matrices Ak divided by the number of nonzeroes of the original matrix A = A1 . e. large operator complexities lead to large setup times, times per cycle and memory requirements. Table IV shows results for the 2D Poisson problem − u = f using a 5-point finite difference discretization and a 9-point finite element discretization.
C. c. 30(107) 79(256) 95(305) 113(357) 45(172) 96(330) 110(374) 123(408) 22(79) 47(152) 56(176) 62(193) 24(87) 59(196) 70(227) 82(263) 28(112) 70(254) 84(299) 100(347) Table XIII. Total times in seconds (number of iterations) for a 7-point 3D Laplace problem with 40×40×40 points per processor. 2. Three-dimensional structured problems We now consider 3D problems. Based on the sequential results in Section 6 we expect complexity reduction schemes to make a difference here. The first problem is a 7-point 3D Laplace problem on a structured cube with 40×40×40 unknowns per processor.
This requires an additional f 1 n 1 comparisons. The number of additions, multiplications and divisions can be determined similarly. For standard interpolation, at first the new stencil needs to be computed, leading to f 1 n 1 additions and multiplications and f 1 divisions. This can be done when setting up the data structure to avoid n 1 comparisons. After this one proceeds just as for direct interpolation with a much larger stencil of size n 1 +n 2 . The number of floating point additions, multiplications, and divisions to compute all interpolation weights for each F-point are given in Table I.