Overlapping Schwarz

Our goal is to either analytically solve, or numerically approximate, the 2nd Neumann eigenfunction of the Laplacian on a generic acute triangle [math]\displaystyle{ \Omega \equiv }[/math]ABC (see figure). Since we don't have an analytic solution on this domain, but can solve problems on sectors or the circle, we propose a domain decomposition strategy in the spirit of the overlapping Schwarz iteration.

Let [math]\displaystyle{ 0 }[/math] be the incenter of the triangle, and denote by [math]\displaystyle{ \Omega_0 }[/math] the interior of the incircle. This is tangent to the segments [math]\displaystyle{ AB, BC }[/math] and [math]\displaystyle{ CA }[/math] at [math]\displaystyle{ F,D, E }[/math] respectively. Denote by [math]\displaystyle{ \Gamma_{10} }[/math] the segment of the circle connecting [math]\displaystyle{ E,F }[/math]. Denote by [math]\displaystyle{ \Omega_1 }[/math] the sector [math]\displaystyle{ AEF }[/math], and by [math]\displaystyle{ \Gamma_{01} }[/math] the arc connecting [math]\displaystyle{ E,F }[/math]. Note that [math]\displaystyle{ \Omega_1 }[/math] and [math]\displaystyle{ \Omega_2 }[/math] overlap. Repeat this process for the other three vertices. Let us denote the exact second Neumann eigenvalue as [math]\displaystyle{ \lambda }[/math].

Now we'll proceed by iteration. At step [math]\displaystyle{ n }[/math], suppose [math]\displaystyle{ u_i^n }[/math] for [math]\displaystyle{ i=1,2,3 }[/math] satisfies

[math]\displaystyle{ -\Delta u_i^n = \lambda^n_i u_i, x\in \Omega_i, \frac{\partial u_i^n}{\partial \nu}=0, x\in \partial \Omega_i \setminus{\Gamma_{0i}} }[/math] and the Robin condition [math]\displaystyle{ u_0^{n-1} \frac{\partial u_i^{n}}{\partial \nu} - \frac{\partial u_0^{n-1}}{\partial \nu} u_i^n =0, x\in \Gamma_{0i} }[/math]

The function [math]\displaystyle{ u_0^n }[/math] solves [math]\displaystyle{ -\Delta u_0^n = \lambda^n_0 u_0^n, x\in \Omega_0, u_i^{n} \frac{\partial u_0^{n}}{\partial \nu} - \frac{\partial u_i^{n}}{\partial \nu} u_0^n =0, x\in \Gamma_{i0}, }[/math]

Now [math]\displaystyle{ (u_{i}^n }[/math], [math]\displaystyle{ \lambda_i) }[/math] solve generalized eigenfunction problems on sectors of circles. If one knows [math]\displaystyle{ u_0^{n-1}, \frac{\partial u_0^{n-1}}{\partial \nu} }[/math] on the curves [math]\displaystyle{ \Gamma_{0i} }[/math], then one can use local Fourier-Bessel expansions to get [math]\displaystyle{ u_i^n }[/math]. Then one uses their traces onto [math]\displaystyle{ \Gamma_{10} }[/math], and has the correct data to solve the eigenvalue problem for [math]\displaystyle{ u_0^n }[/math] on the disk. One can use Fourier-Bessel expansions to do this as well.

The claim is that as [math]\displaystyle{ n\rightarrow \infty }[/math], the sequences [math]\displaystyle{ u_i^n }[/math] converge to the restriction of the actual eigenfunction [math]\displaystyle{ u }[/math] on the sub-domains. Clearly we have to prescribe a starting guess for the iteration.

\begin{figure}[htbp] % figure placement: here, top, bottom, or page

  \centering
  \includegraphics[width=2in]{ddmethod.eps} 
  \caption{Domains}
  \label{fig:example}

\end{figure}

Solving on the wedge [math]\displaystyle{ \Omega_i }[/math]

We are at step [math]\displaystyle{ n }[/math] of the iteration. Suppose [math]\displaystyle{ i=1 }[/math] is fixed for concreteness, and the traces of [math]\displaystyle{ u_0^{n-1} }[/math] and [math]\displaystyle{ \frac{\partial u_0^{n-1}}{\partial \nu} }[/math] on [math]\displaystyle{ \Gamma_{01} }[/math] are known. [math]\displaystyle{ -\Delta u_1^n = \lambda^n_1 u_1, x\in \Omega_1, \frac{\partial u_1^n}{\partial \nu}=0 x\in \partial \Omega_1\setminus{\Gamma_{01}}, u_0^{n-1} \frac{\partial u_1^{n}}{\partial \nu} - \frac{\partial u_0^{n-1}}{\partial \nu} u_1^n =0, x\in \Gamma_{01} }[/math]

We will try the method of particular solutions of Fox and Henrici, adapted to this problem. We know that since the opening angle is [math]\displaystyle{ \alpha }[/math], in [math]\displaystyle{ \Omega_1 }[/math] the functions [math]\displaystyle{ w_k(r,\theta):= J_{\frac{\pi k}{\alpha}} \sqrt{\lambda} r) \cos(\frac{\pi k}{\alpha} \theta) }[/math] will satisfy the Neumann conditions on the line segments [math]\displaystyle{ AE, AF\lt math\gt , as well as satisfy the equation \lt math\gt -\Delta w_k =\lambda w_k }[/math].

So, we suppose [math]\displaystyle{ u_1^n = \sum_{k=1}^M c_k w_k (r,\theta) }[/math]. We want to find the coefficients [math]\displaystyle{ c_k }[/math] so as to satisfy the boundary condition on [math]\displaystyle{ \Gamma_{01} }[/math]. Now, [math]\displaystyle{ \Gamma_{01} }[/math] is an arc of radius [math]\displaystyle{ \rho_1=AE }[/math]. Let [math]\displaystyle{ (\rho_1,\theta_j) }[/math] be [math]\displaystyle{ 2M }[/math] collocation points along this curve. At each point, we want to enforce [math]\displaystyle{ 0= u_0^{n-1}(\rho_1,\theta_j) \frac{\partial u_1^{n}}{\partial \nu} (\rho_1,\theta_j)- \frac{\partial u_0^{n-1}}{\partial \nu}(\rho_1,\theta_j) u_1^n(\rho_1,\theta_j) }[/math] [math]\displaystyle{ = u_0^{n-1}(\rho_1,\theta_j) \sum_{k=1}^M c_k \frac{\partial}{\partial r} J_{\frac{\pi k}{\alpha}} (\sqrt{\lambda} \rho_1) \cos(\frac{\pi k}{\alpha} \theta_j) - \frac{\partial u_0^{n-1}}{\partial \nu}(\rho_1,\theta_j) \sum_{k=1}^M c_k J_{\frac{\pi k}{\alpha}} (\sqrt{\lambda} \rho_1) \cos(\frac{\pi k}{\alpha} \theta_j) . }[/math]

This is equivalent to solving the rectangular nonlinear system [math]\displaystyle{ A(\lambda) \vec{c}=0 }[/math] where [math]\displaystyle{ a_{jk}(\lambda) = u_0^{n-1}(\rho_1,\theta_j) \frac{\partial}{\partial r} J_{\frac{\pi k}{\alpha}} (\sqrt{\lambda} \rho_1) \cos(\frac{\pi k}{\alpha} \theta_j) - \frac{\partial u_0^{n-1}}{\partial \nu}(\rho_1,\theta_j)J_{\frac{\pi k}{\alpha}} (\sqrt{\lambda} \rho_1) \cos(\frac{\pi k}{\alpha} \theta_j) }[/math]

We find the solutions by looking for values of [math]\displaystyle{ \lambda }[/math] so that the smallest singular value of [math]\displaystyle{ A(\lambda) }[/math] approaches 0. This is the Moler approach to the original Fox-Henrici-Moler paper.

Once we locate the solution [math]\displaystyle{ \vec{c} }[/math], we have the iterate [math]\displaystyle{ u_1^n }[/math]. We do this same process for the other wedges as well.

Solving on the disk [math]\displaystyle{ \Omega_0 }[/math]

We are at step [math]\displaystyle{ n }[/math] of the iteration, and have solved for the functions [math]\displaystyle{ u_i^n }[/math]. We therefore have their traces on the arcs [math]\displaystyle{ \Gamma_{i0} }[/math].

The function [math]\displaystyle{ u_0^n }[/math] solves [math]\displaystyle{ -\Delta u_0^n = \lambda^n_0 u_0^n, x\in \Omega_0, u_i^{n} \frac{\partial u_0^{n}}{\partial \nu} - \frac{\partial u_i^{n}}{\partial \nu} u_0^n =0, x\in \Gamma_{i0} }[/math] We shall again use a Fourier-Bessel ansatz: let [math]\displaystyle{ z_m(r,\theta) = J_m(\lambda r) e^{im\theta} }[/math], and assume [math]\displaystyle{ u_0^n(r,\theta) = \sum_{k=0}^M d_m z_m(r,\theta) }[/math]. Repeat the process above of enforcing the (non-standard) boundary conditions at collocation points along [math]\displaystyle{ \Gamma_{i0} }[/math].

{Convergence?}

At the end of the nth step, we have 4 functions: [math]\displaystyle{ u_{i}^n }[/math], i=0,1,2,3 and 4 eigenvalues. We repeat the process until the eigenvalues are all the same number.