Error bounds in electromagnetic and acoustic wave scattering.

摘要

This thesis deals with electromagnetic and acoustic wave scattering; its primary goals are to derive a guaranteed error bound for certain scattering problems, and subsequently to utilize it in a variational argument to derive a solution that minimizes that bound.;The following particular examples are considered: transverse magnetic and transverse electric waves incident on a conducting strip, acoustic wave incident on a soft square membrane. Each problem is posed in the form of a Fredholm integral equation or integrodifferential equation of the first kind.;We first utilize the Petrov-Galerkin method (method of moments with identical sets for basis functions and testing functions) to derive an approximate solution to the equation. Then we compute a posteriori error bounds---in various norms---for that approximate solution.;Subsequently, we use the error bounds to approach the problems from a variational principle, a least-square method, designed to minimize the bounds. The resulting solutions are compared with the Petrov-Galerkin solutions.;Error bounds and least-square methods require the use of Sobolev spaces of fractional orders and their associated norms and duality pairings. In particular, our scattering problems will involve H1/2 ( R ), H-1/2 ( R ), H1/2 ( R2 ) and H-1/2 ( R2 ), the spaces of order 1/2, and of order -1/2 on the real line and the Euclidean plane. We shall also make use of similar spaces on open subsets O of R and R2 . Finally, for further ease of computations, we shall also consider norms in the Sobolev spaces of (integer) order one.