Abstract Spectral penalty methods were originally introduced to deal with the stability of the spectral solution coupled with the boundary conditions for differential equations [Funaro 1986, Funaro and Gottlieb 1988]. Later the penalty method was used for spectral methods to be implemented in irregular domains in multiple dimensions. It has been also shown that there are close relations between discontinuous Galerkin methods and spectral penalty methods in multi-domain and element setting. In addition to stability, the penalty method provides a better accuracy because of its asymptotic behavior in the neighborhood of boundaries. The optimal value of the penalty parameter for accuracy has not been studied thoroughly for the exact form. In this short note, we consider a simple differential equation and study the optimal value of the penalty parameter by minimizing the error in maximum norm. We focus on the optimization for the case of Chebyshev spectral collocation method. We provide its exact form and verify it numerically.

中文翻译：

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微分方程谱惩罚法最优惩罚参数值的精确公式

摘要 谱惩罚方法最初被引入来处理谱解的稳定性以及微分方程的边界条件 [Funaro 1986, Funaro 和 Gottlieb 1988]。后来惩罚方法被用于在多维不规则域中实现的谱方法。还表明，在多域和元素设置中，不连续伽辽金方法与谱惩罚方法之间存在密切关系。除了稳定性之外，惩罚方法提供了更好的准确性，因为它在边界附近的渐近行为。精确形式的惩罚参数的最佳值尚未彻底研究。在这个简短的说明中，我们考虑一个简单的微分方程，并通过最小化最大范数的误差来研究惩罚参数的最优值。我们专注于对切比雪夫谱搭配方法的情况进行优化。我们提供其确切形式并通过数字对其进行验证。