Geometric Singularities for the Laplace Eigenvalue Problem

\[-\Delta u=\lambda u\mbox{ in }\Omega\quad,\quad \mbox{homogeneous (mixed) boundary conditions}\]

Many of the essential difficulties (singular eigenfunctions) are illustrated by considering a sector of the unit disk, where eigenvalues and eigenvectors are known explicitly. Fix \(\alpha\in[1/2,1)\) and let \(\Omega\subset \mathbb{R}^2\) be the sector of the unit disk for which \( 0 < r < 1 \) and \( 0 < \theta < \pi/\alpha \); when \(\alpha=1/2\), \(\Omega\) is the unit disk with a slit along the positive \(x\)-axis.
\(\alpha=4/7\) \(\alpha=1/2\)
Disk Sector Slit Disk
We recall the first-kind Bessel functions \[ J_\sigma(z)=\left(\frac{z}{2}\right)^\sigma\sum_{k=0}^\infty\frac{(-1)^k}{k!\Gamma(k+\sigma+1)}\left(\frac{z}{2}\right)^{2k} \quad,\quad j_m(\sigma)=m^{th}\mbox{ positive root of }J_\sigma\quad,\quad J_\sigma(z)\sim \frac{1}{\Gamma(\sigma+1)}\left(\frac{z}{2}\right)^\sigma\mbox{ for }z\mbox{ near } 0 \] We note that some values of \(\sigma\) yield more recognizable forms; for example, \[ J_{1/2}(z)=\sqrt{\frac{2}{\pi z}}\,\sin(z)~. \]

Dirichlet condition on \(r=1\), Dirichlet conditions on \(\theta=0\) and \(\theta=\pi/\alpha\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\sin(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=k\alpha\quad,\quad k\geq 1\;,\; m\geq 1 \] The strongest singularity, \(r^{\alpha}\), occurs for eigenvectors associated with the first eigenvalue \(\lambda_1=\lambda_{1,1}\), and its latter occurrences depend on how the roots of \(J_{\sigma_1}\) are distributed in relation to those of \(J_{\sigma_k}\) for \(k\geq 1\).

For example, when \(\alpha=1/2\), the first several eigenvalues are given below in a scatter plot, organized as follows: for each \(m\) indicated on the x-axis, we see the eigenvalues \([j_m(k/2)]^2\) for increasing \(k\) stacked above it. The eigenvalues \([j_m(1/2)]^2=(m\pi)^2\) whose eigenvectors have an \(r^{1/2}\)-singularity at the origin, are marked in red; and the dashed lines help indicate how the gap between where these occur in the spectrum increases.
Dirichlet Slit Eigenvectors
More specifically, the first eight occurrences of a \(r^{1/2}\)-type singularity are associated with the eigenvalues: \[ \begin{array}{ll} \lambda_1=\lambda_{1,1}=\pi^2&\lambda_6=\lambda_{1,2}= 4\pi^2\\ \lambda_{17}=\lambda_{1,3}=9\pi^2&\lambda_{32}=\lambda_{1,4}= 16\pi^2\\ \lambda_{53}=\lambda_{1,5}=25\pi^2&\lambda_{76}=\lambda_{1,6}= 36\pi^2\\ \lambda_{107}=\lambda_{1,7}=49\pi^2&\lambda_{143}=\lambda_{1,8}= 64\pi^2 \end{array} \] Employing a sequence of hp-adapted meshes obtained using the Discontinuous Galerkin approach described in The final mesh (top) is given together with a history of the errors and error estimates (lower left), and the corresponding effectivities (lower right),
EFF = (error estimate)/(true error),
for the smallest eigenvalue when \(\alpha=1/2\).
Dirichlet Slit Mesh Dirichlet Slit Eig Error Dirichlet Slit Eig Effectivity
We note that this is not an indefinite problem, but the algorithm described in the paper above can be applied.

Dirichlet condition on \(r=1\), Neumann conditions on \(\theta=0\) and \(\theta=\alpha\pi\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\cos(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=k\alpha\quad,\quad k\geq 0\;,\; m\geq 1 \] Here, the eigenmode associated with the smallest eigenvalue \(\lambda_1=\lambda_{0,1}\) are analytic, and the strongest singularity, \(r^\alpha\), first occurs in eigenmodes associated with the second eigenvalue, \(\lambda_2=\lambda_{1,1}\). For example, when when \(\alpha=1/2\), \(\lambda_1=\lambda_{0,1}\approx 5.78318596294678452117599575846\), \(\lambda_2=\lambda_{1,1}=\pi^2\), and the next occurrence of a \(r^{1/2}\)-type singularity is for the eigth eigenvalue \(\lambda_8=\lambda_{1,2}=4\pi^2 \).

Dirichlet condition on \(r=1\), Dirichlet condition on \(\theta=0\) and Neumann condition on \(\theta=\alpha\pi\)

All eigenpairs doubly-indexed, \( (u_{km},\lambda_{km})\): \[ u_{km}=J_{\sigma_k}(j_m(\sigma_k)\,r)\,\sin(\sigma_k\,\theta)\quad,\quad \lambda_{km}=[j_m(\sigma_k)]^2\quad,\quad \sigma_k=\frac{(2k+1)\alpha}{2}\quad,\quad k\geq 0\;,\; m\geq 1 \] As in the Dirichlet-Dirichlet case, the strongest singularity, which is \(r^{\alpha/2}\) in this case, occurs in the eigenmode associated with the smallest eigenvalue \(\lambda_1=\lambda_{0,1}\). For \(\alpha=1/2\), the first two occurrences of the strongest singularity, \(r^{1/4}\) happen for \(\lambda_1=\lambda_{0,1}\approx 7.73333653346596686390263803337\) and \(\lambda_6=\lambda_{0,2}\approx 34.8825215790904790430911907100\). In contrast, when \(\alpha=2/3\), the first two occurrences of the strongest singularity, \(r^{1/3}\) happen for \(\lambda_1=\lambda_{0,1}\approx 8.42500692949919857451071877294 \) and \(\lambda_5=\lambda_{0,2}\approx 36.3940370569496758450772289141\). Employing a sequence of hp-adapted meshes obtained using the Discontinuous Galerkin approach described in The final mesh (top) is given together with a history of the errors and error estimates (lower left), and the corresponding effectivities (lower right),
EFF = (error estimate)/(true error),
for the smallest eigenvalue when \(\alpha=1/2\).
Dirichlet Neumann Slit Mesh Dirichlet Neumann Slit Eig Error Dirichlet Neumann Slit Eig Effectivity
We note that this is not an indefinite problem, but the algorithm described in the paper above can be applied.

The same experimental results are reported for the "Pac-man" shape \(\alpha=2/3\), using instead the Continuous Galerkin approach described in which can certainly be applied to problems which are positive definite.
Dirichlet Neumann Mesh Dirichlet Neumann Eig Error Dirichlet Neumann Eig Effectivity