An approximate inverse based multigrid approach to the biharmonic problem

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Abstract
The approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations is examined. This algorithm has proven successful in the numerical solution of several second order boundary value problems. Here we are concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using bicubic Hermite basis functions. Central to our development is the concept of an approximate inverse of a matrix. In particular, we use a least squares approximate inverse found by solving a Frobenius matrix norm minimization problem. This approximate inverse is used in the multigrid smoothers of our algorithm FAPIN. The algorithms presented are well suited for implementation on hypercube multiprocessors.

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