Numerical Experiments with Isometric Mapping and Back-Projection for Conditioning Families in a Simple Sobolev Space
Department of Mathematical Sciences
Numerically stable Galerkin procedures can be constructed by ensuring that the families of trial- and test-functions are well conditioned in the respective Hilbert spaces between which the operator is an (appropriate) isomorphism. We explain an idea for constructing a family that may be well conditioned in a given Sobolev space of nonzero fractional order from a family that is well conditioned in the corresponding zero-order space, by using a naturally occurring isometric operator followed by projection back onto the original subspace. Effectively, the construction results in "preconditioning matrices," to be used in transforming the original Galerkin matrix to produce new ones which may be of much smaller condition number. The underlying geometric setting must be sufficiently simple, so that the Sobolev structures can me "manipulated numerically." While we have not yet proven the well-conditioning of the constructed families, the use of the scheme is illustrated numerically in applications to the approximate solution of a first-kind integral equation arising in two-dimensional acoustic scattering, where a pronounced stabilizing effect is observed.