A spectral method for elliptic equations: the Neumann problem

Let Omega be an open, simply connected, and bounded region in a"e (d) , d a parts per thousand yenaEuro parts per thousand 2, and assume its boundary a,Omega is smooth. Consider solving the elliptic partial differential equation -aEuro parts per thousand I"u + gamma u = f over Omega 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 sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.