Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{ R}^{ d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ 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 Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{ n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ 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^{ \infty}(\overline{ \Omega}) $ and assuming $\partial\Omega$ is a $C^{ \infty}$ boundary, the convergence of $\Vert u-u_{ n}\Vert_{ H^{ 1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{ R}^{ 2}$ and $\mathbb{ R}^{ 3}$ show experimentally an exponential rate of convergence.
Categories: math.NA