A Spectral Method for Elliptic Equations: The Neumann Problem - Mathematics > Numerical AnalysisReport as inadecuate




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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 $-\Delta u+\gammau=f$ over $\Omega$ with a Neumann boundary condition. The problem is convertedto an equivalent elliptic problem over the unit ball $B$, and then a spectralGalerkin method is used to create a convergent sequence of multivariatepolynomials $u {n}$ of degree $\leq n$ that is convergent to $u$. Thetransformation from $\Omega$ to $B$ requires a special analytical calculationfor its implementation. With sufficiently smooth problem parameters, the methodis 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 experimentallyan exponential rate of convergence.



Author: Kendall Atkinson, David Chien, Olaf Hansen

Source: https://arxiv.org/







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