nonsymmetric dynamics

* Solve A k u = b (k is current grid level) */
proc multilevel(A k , b, u, k)
u = S 1 k (A k, b, u);
if ( k ≠ Nlevel − 1)
P k = determine interpolant( A k );
ˆr = P T k (b − A ku) ;
 k+1 = P T k A kP k ; v = 0;
multilevel(Âk+1, ˆr, v, k + 1);
u = u + P k v;
u = S 2 k (A k, b, u);
Figure 1: High level multigrid V cycle consisting of ‘Nlevel’ grids to solve (1), with A 0 = A.
are approximate solvers corresponding to k steps of pre and post smoothing, respectively.
These smoothers are discussed in Section 9. For now, it suffices to view them as basic iterative
methods (e.g. Gauss-Seidel) which effectively smooth out the error associated with
the current approximate solution. The P k ’s (interpolation operators that transfer solutions
from coarse grids to finer grids) are the key ingredient that are determined automatically by
the algebraic multigrid method 2 . For the purposes of this guide, it is important to understand
that when the multigrid method is used, a hierarchy of grids, grid transfer operators
(P k ), and coarse grid discretizations (A k ) are created. To complete the specification of the
multigrid method, smoothers must be supplied on each level. There are several smoothers
within ML or an iterative solver package can be used, or users can write their own smoother
(see Section 9).
4 Quick Start
This section is intended for the impatient user. It’s assumed that you’ve already have a local
copy of Trilinos 3 . Using the instructions here, your build of Trilinos will have the following
libraries: AztecOO, Epetra, EpetraExt, Ifpack, Loca, ML, new package,
NOX, Amesos and Teuchos.
1. cd into the Trilinos directory.
2. Make a build directory, e.g., mkdir sandbox.
3. cd sandbox.
4. Configure Trilinos:
(a) ../configure --enable-teuchos --enable-amesos --enable-aztecoo --enable-epetra
if you do not want to use MPI.
(b) ../configure --enable-teuchos
--enable-amesos --enable-aztecoo --enable-epetra --with-mpi-compilers=/usr/local
(or wherever your MPI compilers are located) to use MPI.
2 The P k ’s are usually determined as a preprocessing step and not computed within the iteration.
3 Please refer to the Trilinos web page, trilinos.sandia.gov, to obtain a copy of Trilinos.
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