SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3210-A3232, January 2020.
Recently, there has been interest in high precision approximations of the first eigenvalue of the Laplace--Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the three-dimensional Kreweras model that has been the object of previous efforts.

中文翻译：

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拉普拉斯特征值的紧紧外壳的计算

SIAM科学计算杂志，第42卷，第5期，第A3210-A3232页，2020年1月。
近年来，出于组合目的，人们对球形三角形上Laplace-Beltrami算子的第一特征值的高精度近似产生了兴趣。我们针对这些特征值计算经过改进和认证的外壳。这是通过以高精度应用特定解决方案的方法来实现的，通过间隔算术和泰勒模型的组合来获得外壳。通过利用特征值相对于域的单调性来证明特征值的索引。通过合并所有拐角处的展开和从内部点开始的展开来处理经典麻烦的奇异拐角情况。尤其是，这使我们能够为以前的研究对象的三维Kreweras模型计算100个基本特征值。