| dc.description.abstract | In this thesis we study the approximate inverse based multigrid algorithm FAPIN
for the solution of large sparse linear systems of equations.
This algorithm, which is closely related to the well known multigrid V-cycle, has
proven successful in the numerical solution of several second order boundary value problems.
Here we are mainly concerned with its application to fourth order problems. In
particular, we demonstrate good multigrid performance with discrete problems arising
from the beam equation and the biharmonic (plate) equation. The work presented also
represents new experience with FAPIN using cubic B-spline, bicubic B-spline and piecewise
bicubic Hermite basis functions. We recast a convergence proof in matrix notation
for the nonsingular case.
Central to our development are the concepts of an approximate inverse and an
approximate pseudo-inverse of a matrix. In particular, we use least squares approximate
inverses (and related approximate pseudo-inverses) found by solving a Frobenius
matrix norm minimization problem. These approximate inverses are used in the multigrid
smoothers of our FAPIN algorithms. | |
