A Spectral Method for Elliptic Equations: The Neumann Problem

Abstract

Let $\Omega$ be an open, simply connected, and bounded region in ${\mathbb{R}}^d$, $d\ge 2$, 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 $\le n$ 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.

23 pages, 11 figures

References in corpus (2)

• A Spectral Method for Elliptic Equations: The Neumann Problem
• A spectral method for elliptic equations: the Dirichlet problem

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