  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).
