Construction of the linear

The multigrid method has been established as perhaps the most efficient
method for solving the linear systems that arise from the discretization
of differential equations. The popularity of this method can be attributed to
the fact that it is optimal in the sense that one multigrid cycle can
reduce the norm of the error by a factor that is bounded away from 1
independent of *N*, the size of the linear system, while using only
O(*N*) operations.

The *h*-hierarchical basis multigrid method uses both the nodal basis
and the *h*-hierarchical basis. It can be applied to grids that
are uniform or adaptive in *h*, but must be uniform in *p*.
The usual nodal basis for the space of piecewise linear functions on a
given grid is defined by

In contrast, the *h*-hierarchical basis is defined using the family of nested grids
from the refinement process. The *h*-hierarchical basis begins with the nodal
basis on the initial grid. As refinement proceeds, with each element
division one or more new nodes are added, and for each node a new basis
function is defined as a nodal basis on the current grid,
but the existing basis functions remain unchanged.
A 2-level *h*-hierarchical basis for a grid can also be defined by starting
with the nodal basis for the next-to-last refinement of the grid, and defining the higher level basis
functions as above. It can be shown that conversion between
the nodal basis and the 2-level *h*-hierarchical basis forms of coefficient vectors
and finite element stiffness matrices is easily performed with a small
number of operations per node.

The *h*-hierarchical basis multigrid algorithm is based on performing a
sequence of transformations between nodal and 2-level *h*-hierarchical bases.
In the 2-level stiffness matrix, the block corresponding to the coarse
grid points is the nodal basis matrix of the coarse grid. The off-diagonal
blocks provide a means of implicitly moving the residual between fine and
coarse grids. Red-black Gauss-Seidel is used for the relaxation operator.
It can be shown that with a uniform grid on a square domain and linear
elements, this method
is mathematically equivalent to a standard geometric multigrid method.
It immediately extends the standard method to arbitrary domains, adaptive
grids, and high order elements.

The *h*-hierarchical basis multigrid method cannot be used with nonuniform
*p*, but it provides the motivation for the *hp*-hierarchical
basis multigrid method for *p*-adaptive and *hp*-adaptive grids. In
this method, the different values of *p* are considered to be the
sequence of nested grids with the linear basis functions forming the "coarsest"
grid. A V-cycle consists of performing a relaxation sweep with the entire
matrix, followed by a relaxation sweep with the equations corresponding to
the largest *p* omitted, and then with the two largest *p*
omitted, etc. The
"coarsest grid problem", i.e. the linear basis functions on the whole
*h* grid, is solved by the *h*-hierarchical basis multigrid method.
Relaxation sweeps are then performed while increasing the maximum value of
*p* included until the entire matrix is used again.
Numerical experiments with the method for Poisson's equation show that
the rate of convergence is independent of both *h* and *p*.

degrees of | max h | min p | max p | contraction |
---|---|---|---|---|

freedom | refinement | factor | ||

291 | 1 | 1 | 3 | .46 |

2320 | 7 | 1 | 5 | .42 |

18561 | 11 | 2 | 7 | .44 |

148480 | 15 | 3 | 10 | .44 |

1187894 | 19 | 3 | 12 | .40 |

4751622 | 22 | 3 | 14 | .41 |

Contraction factors for the

**Publications**

Mitchell, W.F., ** The hp-Multigrid Method Applied to hp-Adaptive
Refinement of Triangular Grids**,

Mitchell, W.F., **Optimal multilevel iterative methods for adaptive
grids**, *SIAM J. Sci. Statist. Comput.* 13 (1992),
pp. 146-167.

Mitchell, W.F., **Unified multilevel adaptive finite element
methods for elliptic problems**, Ph.D. thesis, Technical report
UIUCDCS-R-88-1436, Department of Computer Science, University
of Illinois, Urbana, IL, 1988.
(
gzipped postscript, 194k)

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