A Spectral Method for Elliptic Equations: The Neumann Problem
This provides a high-order numerical method for Neumann problems on smooth domains, but the approach is incremental as it extends existing spectral methods for Dirichlet problems to Neumann boundary conditions.
The authors develop a spectral Galerkin method for solving elliptic PDEs with Neumann boundary conditions on general smooth domains by transforming the problem to the unit ball. They prove convergence faster than any power of 1/n for smooth solutions and demonstrate exponential convergence in numerical examples in 2D and 3D.
Let $Ω$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partialΩ$ is smooth. Consider solving an elliptic partial differential equation $-Δu+γu=f$ 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 $u_{n}$ of degree $\leq n$ 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\in C^{\infty}(\overlineΩ) $ and assuming $\partialΩ$ 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.