 
Summary: A Spectral Method for Elliptic Equations:
The Neumann Problem
Kendall Atkinson
Departments of Mathematics & Computer Science
The University of Iowa
Olaf Hansen, David Chien
Department of Mathematics
California State University San Marcos
May 22, 2010
Abstract
Let be an open, simply connected, and bounded region in Rd
, d 2,
and assume its boundary @ is smooth. Consider solving the elliptic par
tial di¤erential equation u+ u = f over with a Neumann boundary
condition. The problem is converted to an equivalent elliptic problem over
the unit ball B, and then a spectral method is given that uses a special
polynomial basis. In the case the Neumann problem is uniquely solvable,
and with su¢ ciently smooth problem parameters, the method is shown to
have very rapid convergence. Numerical examples illustrate exponential
convergence.
