AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

MULTIGRID AS A SOLVER AND A PRECONDITIONER 97An iteration with M in (40) is described as follows:1. r 1 = f 1 − A 11 u (k)1 − A 12 u (k)2 , r 2 = f 2 − A 21 u (k)1 − A 22 u (k)2 .2. Solve A 11 ∆u ∗ 1 = r 1 .3. r2 ∗ = r 2 − A 21 ∆u ∗ 1 = f 2 − A 21 (u (k)1 + ∆u ∗ 1) − A 22 u (k)2 .4. Solve S∆u (k)2 = r2.∗5. u (k+1)2 = u (k)2 + ∆u (k)2 .6. Solve A 11 ∆u (k)1 = r 1 − A 12 ∆u (k)2 .7. u (k+1)1 = u (k)1 + ∆u (k)1 .Vectors u, f, and r are also split according to (39); u ∗ 1 and r2 ∗ are intermediatevariables. Steps 6 and 7 are rewritten as follows:A 11 (∆u (k)1 − ∆u ∗ 1) = f 1 − A 11 (u (k)1 + ∆u ∗ 1) − A 12 (u (k)2 + ∆u (k)2 ),u (k+1)1 = (u (k)1 + ∆u ∗ 1) + (∆u (k)1 − ∆u ∗ 1).The algorithm from [12] is obtained if we putu ∗ 1 = u (k)1 + ∆u ∗ 1,∆u ∗∗1 = ∆u (k)1 − ∆u ∗ 1.The following equivalent process is obtained:1. r 1 = f 1 − A 11 u (k)1 − A 12 u (k)2 .2. Solve A 11 ∆u ∗ 1 = r 1 , u ∗ 1 = u (k)1 + ∆u ∗ 1.3. r2 ∗ = f 2 − A 21 u ∗ 1 − A 22 u (k)2 .4. Solve S∆u (k)2 = r2.∗5. u (k+1)2 = u (k)2 + ∆u (k)2 .6. r1 ∗ = f 1 − A 11 u ∗ 1 − A 12 u (k+1)2 .7. Solve A 11 ∆u ∗∗1 = r1, ∗ u (k+1)1 = u ∗ 1 + ∆u ∗∗1 .For solving the linear system A 11 , a certain number of iterations (ν) of an alternatingline Jacobi relaxation is performed on the fine grid points excluding the coarse gridpoints. This operation with right-hand side vector f 1 is described as(43)u 1 := J 11 (A 11 , f 1 , u 1 , ν).Rewriting the algorithm leads to the algorithm that has been implemented. Themultigrid V-, F-, and W-cycles are now constructed with recursive calls replacingstep 4 in the algorithms.The robustness of this algorithm has been presented in [12] by means of two-gridFourier analysis for the convection-diffusion equation. In the numerical results in [12]parameter ω was set to 0.7. For convenience the MGLU F-cycle is written out inmetalanguage.MGLU F-cycleMGLUF(A L , f L , u L , ν 1 , ν 2 , ν 3 ) {if (L = 1) {u 1 := smoother(A 1 , f 1 , u 1 , ν 3 );