Overlapping Schwarz

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Our goal is to either analytically solve, or numerically approximate, the 2nd Neumann eigenfunction of the Laplacian on a generic acute triangle [math]\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]0[/math] be the incenter of the triangle, and denote by [math]\Omega_0[/math] the interior of the incircle. This is tangent to the segments [math]AB, BC[/math] and [math]CA[/math] at [math]F,D, E[/math] respectively. Denote by [math]\Gamma_{10}[/math] the segment of the circle connecting [math]E,F[/math]. Denote by [math]\Omega_1[/math] the sector [math]AEF[/math], and by [math]\Gamma_{01}[/math] the arc connecting [math]E,F[/math]. Note that [math]\Omega_1[/math] and [math]\Omega_2[/math] overlap. Repeat this process for the other three vertices. Let us denote the exact second Neumann eigenvalue as [math]\lambda[/math].

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

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

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


Solving on the wedge [math]\Omega_i[/math]

We are at step [math]n[/math] of the iteration. Suppose [math]i=1[/math] is fixed for concreteness, and the traces of [math]u_0^{n-1}[/math] and [math]\frac{\partial u_0^{n-1}}{\partial \nu}[/math] on [math]\Gamma_{01}[/math] are known. [math] -\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]\alpha[/math], in [math]\Omega_1[/math] the functions [math] 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]AE, AF\ltmath\gt, as well as satisfy the equation \ltmath\gt-\Delta w_k =\lambda w_k[/math].

So, we suppose [math]u_1^n = \sum_{k=1}^M c_k w_k (r,\theta)[/math]. We want to find the coefficients [math]c_k[/math] so as to satisfy the boundary condition on [math]\Gamma_{01}[/math]. Now, [math]\Gamma_{01}[/math] is an arc of radius [math]\rho_1=AE[/math]. Let [math](\rho_1,\theta_j)[/math] be [math]2M[/math] collocation points along this curve. At each point, we want to enforce [math] 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]= 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] A(\lambda) \vec{c}=0[/math] where [math]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]\lambda[/math] so that the smallest singular value of [math]A(\lambda)[/math] approaches 0. This is the Moler approach to the original Fox-Henrici-Moler paper.

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

Solving on the disk [math]\Omega_0[/math]

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

The function [math]u_0^n[/math] solves [math] -\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]z_m(r,\theta) = J_m(\lambda r) e^{im\theta}[/math], and assume [math]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]\Gamma_{i0}[/math].


{Convergence?}

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