椭圆方程谱方法：诺伊曼问题
A Spectral Method for Elliptic Equations: The Neumann Problem
AI总结：
针对诺伊曼边界条件的椭圆方程，通过变换到单位球并使用谱伽辽金方法，实现了快速收敛的多项式逼近。
摘要：
设$Ω$是$\mathbb{R}^{d}$（$d\geq2$）中一个开、单连通且有界区域，且其边界$\partialΩ$光滑。考虑在$Ω$上求解带有诺伊曼边界条件的椭圆偏微分方程$-Δu+γu=f$。该问题被转化为单位球$B$上的等价椭圆问题，然后使用谱伽辽金方法构造一个次数$\leq n$的多项式序列$u_{n}$，该序列收敛于$u$。从$Ω$到$B$的变换需要特殊的解析计算来实现。在问题参数足够光滑的情况下，该方法被证明是快速收敛的。对于$u\in C^{\infty}(\overlineΩ)$且假设$\partialΩ$是$C^{\infty}$边界，$\Vert u-u_{n}\Vert_{H^{1}}$收敛到零的速度快于$1/n$的任何幂次。$\mathbb{R}^{2}$和$\mathbb{R}^{3}$中的数值例子实验性地展示了指数收敛速度。
英文摘要：
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.
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