A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\mathrm{\Omega }$ be an open, simply connected, and bounded region in ${\mathbb{R}}^d$, $d\ge 2$, and assume its boundary $\mathrm{\partial }\mathrm{\Omega }$ is smooth. Consider solving an elliptic partial differential equation $-\mathrm{\Delta }u+\gamma u=f$ over $\mathrm{\Omega }$ 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 $u_n$ of degree $\le n$ that is convergent to $u$. The transformation from $\mathrm{\Omega }$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$ and assuming $\mathrm{\partial }\mathrm{\Omega }$ is a $C^{\mathrm{\infty }}$ boundary, the convergence of $\mathrm{\parallel }u-u_n{\mathrm{\parallel }}_{H^1}$ to zero is faster than any power of $1\mathrm{/}n$. Numerical examples in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ show experimentally an exponential rate of convergence.