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5 . Solution of Linear Equation Systems
To derive the discretized equations on the coarser grid, we note that control volume around node I of the coarse grid consists of the whole control
volume around node i plus half of control volumes i - 1 and i + 1 of the
fine grid (see Fig. 5.2). This suggests that we add one half of equation (5.63)
with indices i - 1 and i + 1 to the full equation with index i, which leads to
(superscript n being omitted):
Using the relationship between the two grids ( A X = 2 Ax, see Fig. 5.2), this
is equivalent to the following equation on the coarse grid:
which also serves to define P I . The left hand side of this equation is the
standard approximation to the second derivative on the coarse grid, indicating
that the obvious discretization on the coarse grid is a reasonable one. The
right hand side is a smoothing or filtering of the fine grid forcing term and
provides the natural definition of the smoothing or restriction operation.
The simplest prolongation or interpolation of a quantity from the coarse
grid to the fine grid is linear interpolation. At coincident points of the two
grids, the value a t the coarse grid point is simply injected onto the corresponding fine grid point. At fine grid points that lie between the coarse grid
points, the injected value is the average of the neighboring coarse grid values.
A two-grid iterative method is thus:
On the fine grid, perform iterations with a method that gives a smooth
error;
Compute the residual on the fine grid;
Restrict the residual to the coarse grid;
Perform iterations of the correction equation on the coarse grid;
Interpolate the correction to the fine grid;
Update the solution on the fine grid;
Repeat the entire procedure until the residual is reduced to the desired
level.
It is natural to ask: why not use still coarser grids to further improve the
rate of convergence? This is a good idea. In fact, one should continue the
procedure until it becomes impossible to define a still coarser grid; on the
coarsest grid, the number of unknowns is so small that the equations can be
solved exactly a t a negligible cost.
Multigrid is more a strategy than a particular method. Within the framework just described there are many parameters that can be selected more
or less arbitrarily: the coarse grid structure, the smoother, the number of
iterations on each grid, the order in which the various grids are visited, and
5 . Solution of Linear Equation Systems
To derive the discretized equations on the coarser grid, we note that control volume around node I of the coarse grid consists of the whole control
volume around node i plus half of control volumes i - 1 and i + 1 of the
fine grid (see Fig. 5.2). This suggests that we add one half of equation (5.63)
with indices i - 1 and i + 1 to the full equation with index i, which leads to
(superscript n being omitted):
Using the relationship between the two grids ( A X = 2 Ax, see Fig. 5.2), this
is equivalent to the following equation on the coarse grid:
which also serves to define P I . The left hand side of this equation is the
standard approximation to the second derivative on the coarse grid, indicating
that the obvious discretization on the coarse grid is a reasonable one. The
right hand side is a smoothing or filtering of the fine grid forcing term and
provides the natural definition of the smoothing or restriction operation.
The simplest prolongation or interpolation of a quantity from the coarse
grid to the fine grid is linear interpolation. At coincident points of the two
grids, the value a t the coarse grid point is simply injected onto the corresponding fine grid point. At fine grid points that lie between the coarse grid
points, the injected value is the average of the neighboring coarse grid values.
A two-grid iterative method is thus:
On the fine grid, perform iterations with a method that gives a smooth
error;
Compute the residual on the fine grid;
Restrict the residual to the coarse grid;
Perform iterations of the correction equation on the coarse grid;
Interpolate the correction to the fine grid;
Update the solution on the fine grid;
Repeat the entire procedure until the residual is reduced to the desired
level.
It is natural to ask: why not use still coarser grids to further improve the
rate of convergence? This is a good idea. In fact, one should continue the
procedure until it becomes impossible to define a still coarser grid; on the
coarsest grid, the number of unknowns is so small that the equations can be
solved exactly a t a negligible cost.
Multigrid is more a strategy than a particular method. Within the framework just described there are many parameters that can be selected more
or less arbitrarily: the coarse grid structure, the smoother, the number of
iterations on each grid, the order in which the various grids are visited, and