Our goal is to either analytically solve, or numerically approximate, the 2nd Neumann eigenfunction of the Laplacian on a generic acute triangle 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 0 be the incenter of the triangle, and denote by Ω0 the interior of the incircle. This is tangent to the segments AB,BC and CA at F,D,E respectively. Denote by Γ10 the segment of the circle connecting E,F. Denote by Ω1 the sector AEF, and by Γ01 the arc connecting E,F. Note that Ω1 and Ω2 overlap. Repeat this process for the other three vertices. Let us denote the exact second Neumann eigenvalue as λ.
Now we'll proceed by iteration. At step n, suppose for i = 1,2,3 satisfies
and the Robin condition
The function solves
Now , λi) solve generalized eigenfunction problems on sectors of circles. If one knows on the curves Γ0i, then one can use local Fourier-Bessel expansions to get . Then one uses their traces onto Γ10, and has the correct data to solve the eigenvalue problem for on the disk. One can use Fourier-Bessel expansions to do this as well.
The claim is that as , the sequences converge to the restriction of the actual eigenfunction u on the sub-domains. Clearly we have to prescribe a starting guess for the iteration.
Solving on the wedge Ωi
We are at step n of the iteration. Suppose i = 1 is fixed for concreteness, and the traces of and on Γ01 are known.
We will try the method of particular solutions of Fox and Henrici, adapted to this problem. We know that since the opening angle is α, in Ω1 the functions will satisfy the Neumann conditions on the line segments AE,AF < math > ,aswellassatisfytheequation < math > − Δwk = λwk.
So, we suppose . We want to find the coefficients ck so as to satisfy the boundary condition on Γ01. Now, Γ01 is an arc of radius ρ1 = AE. Let (ρ1,θj) be 2M collocation points along this curve. At each point, we want to enforce
This is equivalent to solving the rectangular nonlinear system where
We find the solutions by looking for values of λ so that the smallest singular value of A(λ) approaches 0. This is the Moler approach to the original Fox-Henrici-Moler paper.
Once we locate the solution , we have the iterate . We do this same process for the other wedges as well.
Solving on the disk Ω0
We are at step n of the iteration, and have solved for the functions . We therefore have their traces on the arcs Γi0.
The function solves We shall again use a Fourier-Bessel ansatz: let zm(r,θ) = Jm(λr)eimθ, and assume . Repeat the process above of enforcing the (non-standard) boundary conditions at collocation points along Γi0.
At the end of the nth step, we have 4 functions: , i=0,1,2,3 and 4 eigenvalues. We repeat the process until the eigenvalues are all the same number.