 
Summary: A Spectral Method for Elliptic Equations:
The Dirichlet Problem
Kendall Atkinson
Departments of Mathematics & Computer Science
The University of Iowa
David Chien, Olaf Hansen
Department of Mathematics
California State University San Marcos
February 28, 2009
Abstract
Let be an open, simply connected, and bounded region in Rd
, d
2, and assume its boundary @ is smooth. Consider solving an elliptic
partial di¤erential equation Lu = f over with zero Dirichlet boundary
values. The problem is converted to an equivalent elliptic problem over
the unit ball B; and then a spectral Galerkin method is used to create
a convergent sequence of multivariate polynomials un of degree n that
is convergent to u. The transformation from to B requires a special
analytical calculation for its implementation. With su¢ ciently smooth
problem parameters, the method is shown to be rapidly convergent. For
