APPROXIMATION BY COMPLETE AND INCOMPLETE SETS OF HARMONIC POLYNOMIALS.
Abstract
An empirical investigation is reported concerning approximate solution of the two-dimensional Dirichlet problem in a finite region R by linear combinations of harmonic polynomials selected from either complete or incomplete sets. The negative results of an earlier similar investigation of torsion problems by Poritsky and Danforth are traced to the collocation method they employed. An alternative method involving least squares fitting at boundary points in excess of the number required for collocation is found to yield more reliable results. An algorithm is given for the formal generation of complete sets of harmonic polynomials (of given maximum degree) from corresponding incomplete sets. The fitting capabilities of the latter are found to be at least comparable with those of complete sets, but to require trial-and-error adjustment of parameters of displacement, orientation, and scale of R. Extensions to more general partial differential equations and to higher dimensions are indicated. The present results encourage optimism regarding the usefulness of harmonic polynomials for approximation in a wide range of applications. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 15, 1967
- Accession Number
- AD0830061
Entities
People
- A. Wren
- T. E. Phipps Jr.
Organizations
- Naval Ordnance Laboratory