-<h2>Possible Extensions</h2>
+<h3>Possible Extensions</h3>
<a name="improved-solver">
-<h3>Improved linear solver in 3D</h3>
+<h4>Improved linear solver in 3D</h4>
</a>
We have seen in the section of computational results that the number of outer
unsatisfactory, if one compares with an elliptic problem of similar size.
<a name="improved-ilu">
-<h4>Better ILU decomposition by smart reordering</h4>
+<h5>Better ILU decomposition by smart reordering</h5>
</a>
A first attempt to improve the speed of the linear solution process is to choose
a dof reordering that makes the ILU being closer to a full LU decomposition, as
infeasible algorithm for the last cycle in 3D with 1.2 million
unknowns.
-<h4>Better preconditioner for the inner CG solver</h4>
+<h5>Better preconditioner for the inner CG solver</h5>
Another idea to improve the situation even more would be to choose a
preconditioner that makes CG for the (0,0) matrix $A$ converge in a
mesh-independent number of iterations, say 10 to 30. We have seen such a
canditate in @ref step_16 "step-16": multigrid.
-<h4>Block Schur complement preconditioner</h4>
+<h5>Block Schur complement preconditioner</h5>
But even with a good preconditioner for $A$ at hand, we still
need to solve of the same linear system repeatedly (with different
right hand sides, though). The approach we are going to discuss here is how this
Nonetheless, the improvement by a factor of 3-5 for moderate problem sizes
is quite impressive.
-<h4>Combining block preconditioner and multigrid</h4>
+<h5>Combining block preconditioner and multigrid</h5>
An ultimate linear solver for this problem could be imagined as a combination of
an optimal preconditioner for $A$ (e.g. multigrid) and the block preconditioner
described above.
-<h4>No block matrices and vectors</h4>
+<h5>No block matrices and vectors</h5>
Another possibility that can be taken into account is to not set up a block
system, but rather solve the system of velocity and pressure all at once. The
options are direct solve with UMFPACK (2D) or GMRES with ILU