Scripps Institution of Oceanography

This technical report concerns the construction of a polynomial Galerkin basis, providing a powerful numerical technique for solving a host of applied mathematics problems. In a Galerkin basis set, each member has two fundamental properties: (i) it satisfies any given set of homogeneous linear boundary conditions up to some specified degree and (ii) it is orthogonal to all other basis functions.

The main bulk of the material is contained within the accompanying webpage, at present, archived in .zip format. The homepage that should be opened in a web-browser is Galerkin.html; the rest of the files contained within the archive are supplementary images and other files which are required by the website. When the webpage is opened, the left hand frame lists a range of boundary conditions for (a) orthogonality in Cartesian coordinates, (b) orthogonality in polar geometries and (c) more general orthogonality relations involving derivatives. Common and physically motivated boundary conditions are given explicitly, followed by more general boundary conditions as far as the extent of computer algebra allows. As will quickly become apparent, the formulae for the expressions are extremely lengthy, particularly for the most general cases considered. The user simply needs to copy and paste the expressions given in plain text into either a symbolic package such as Maple or Mathematica, or equally into a high-level language such as Matlab, C or Fortran. The accompanying .pdf file provides an overview of Galerkin schemes and a brief tour of the contents of the webpage.