We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues $\lambda(n)$ of shapes with $n$ edges that are of the form $\lambda(n) \sim x\sum_{k=0}^{\infty} \frac{C_k(x)}{n^k}$ where $x$ is the limiting eigenvalue for $n\rightarrow \infty$. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary condition and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order $C_k(x)$ and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons and star shapes with sinusoidal boundary).

翻译：我们用任意精确算法,以二维形状用二维对称法计算最低的Laplacian egen 值。 我们的方法基于域分解的特殊解决方案方法。 我们特别感兴趣的是, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以二维形状计算最低的 Laplacecian egen值。 我们的方法是以 $x 来限制 $n\rightrow\ infty$ 的 egen值 。 我们特别感兴趣的是, 以美元为单位, 以美元为单位, 用美元为单位, 用美元为单位, 以美元为单位, 用美元为单位, 美元为单位, 美元为单位, 美元为单位。 这种形式的扩展仅以具有drichlet 边界条件的常规多边形( 令人惊讶的是), 涉及 Riemann zeta 值和 单值 倍值的多个zeta 值的形状, 这使它们值得研究。 我们为更高级的封闭式表达 $_k (x) 提供了数字证据,, 并举更多例子说明这些恒星系的形状的形状,, 和恒星系为星系的星系,,, 与 常规 等为星系为星系 。