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

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96 C. W. OOSTERLEE AND T. WASHIO}r L := f L − A L u L ;f L−1 := R L−1 r L ;u L−1 := 0;u L−1 := MGF(A L−1 , f L−1 , u L−1 , ν 1 , ν 2 , ν 3 );u L := u L + P L u L−1 ;u L := smoother(A L , f L , u L , ν 2 );r L := f L − A L u L ;f L−1 := R L−1 r L ;u L−1 := 0;u L−1 := MGV(A L−1 , f L−1 , u L−1 , ν 1 , ν 2 , ν 3 );u L := u L + P L u L−1 ;u L := smoother(A L , f L , u L , ν 2 );All components in the code have been explained in this section.2.2.2. Multigrid based on the block LU decomposition. The third algorithm,MG3, evaluated as a solver and a preconditioner, was recently proposed byReusken ([12]). It originates from several publications in which the fine grid matrixA is decomposed ([10], [11]). In matrix A the fine grid points (excluding the coarsegrid points) are ordered in the first part, the coarse grid points in the second part.The coefficient matrix on the fine grid is now described as( )A11 A(39)A =12.A 21 A 22An approximation M of the above matrix based on the block LU decomposition ismade as follows:() ( )IM =1 0 A11 A(40)12−1.A 21 A 11 I 2 0 SIf S equals the Schur complement S A (= A 22 − A 21 A −1 11 A 12 ), M is equal to A.Multigrid methods based on the formulation and splitting from (40) lead to interestingproperties of coarse grid matrices. If the fine grid matrix is an M-matrix ([20]),S will also be an M-matrix. In [12], S is an approximate Schur complement constructedfrom the Schur complement of a modified matrix A. With the modificationbased on linear interpolation, A 11 and A 12 reduce to A 11 and A 12 , where A 11 isdiagonal:(41)( )A11 AA =12; AA 21 A 11 = diag(A 11 ), A 12 = A 12 + offdiag(A 11 )P 12 .22In (41), P 12 is an operator that interpolates fine grid elements bilinearly to adjacentcoarse grid points.A damping parameter ω is used to correct an incorrect scaling of S compared toS A . So(42)S = ω −1 (A 22 − A 21 A −111 A 12 ).Another choice for S based on lumping is found in [11].We construct multigrid cycles based on the Richardson iteration of the splittingof A = M + (A − M) as in (4).
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 );
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