A spectral method for elliptic equations: the Neumann problem

Let Ω be an open, simply connected, and bounded region in R, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving an elliptic partial differential equation −∆u + γu = f over Ω with a Neumann boundary condition. 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 Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For u ∈ C∞ `