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A Numerical study of the Dirichlet and Neumann eigenvalue problem of the Laplacian on cusp domains

机译:尖点域上Laplacian的Dirichlet和Neumann特征值问题的数值研究。

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Consider a planar annulus. A result of Ramm and Shivakumar states that as the inner circle moves toward the outer circle, the principal Dirichlet eigenvalue of the Laplacian decreases. Numerical experiments in that paper clearly verify this result. The purpose of this short note is to fill in a small gap in that paper: the numerical calculation of the principal eigenvalue when the two circles touch. This is a non-trivial numerical problem because the domain has a cusp which is a strong singularity. Adaptive finite element methods have difficulty converging in the presence of such singularities. Our method is to perform a transformation taking the domain to a rectangle, where it is relatively straightforward to compute the principal eigenvalue. We also calculate the minimal (non-zero) eigenvalue of the Neumann problem. Numerically, the Neumann eigenvalue has no such monotonicity property.
机译:考虑一个平面环。 Ramm和Shivakumar的结果表明,当内圆向外圆移动时,拉普拉斯算子的主要Dirichlet特征值减小。该论文中的数值实验清楚地证明了这一结果。本简短笔记的目的是填补该论文中的一个小空白:两个圆接触时本征值的数值计算。这是一个非平凡的数值问题,因为该域具有尖峰,该尖峰具有很强的奇异性。自适应有限元方法在存在此类奇异点时难以收敛。我们的方法是执行将域带到矩形的转换,在该矩形中计算主特征值相对简单。我们还计算了诺伊曼问题的最小(非零)特征值。在数值上,诺伊曼特征值不具有这种单调性。

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