A Spectral Method for Elliptic Equations: The Neumann Problem

TL;DR

This paper introduces a spectral Galerkin method for solving elliptic PDEs with Neumann boundary conditions by transforming the domain to a unit ball and employing multivariate polynomials, achieving rapid convergence.

Contribution

The paper develops a spectral method for elliptic equations with Neumann conditions using domain transformation and polynomial approximation, demonstrating exponential convergence rates.

Findings

Method converges faster than any polynomial rate for smooth problems.

Numerical experiments confirm exponential convergence in 2D and 3D.

Transformation to the unit ball is essential for spectral approximation.

Abstract

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.