Summing Logarithmic Expansions for Singularly Perturbed Eigenvalue Problems

Source abstract

Strong localized perturbations of linear and nonlinear eigenvalue problems in a bounded two-dimensional domain D are considered. The effects on an eigenvalue $\lambda _0$ of the Lapla-cian, and on the fold point $\lambda _{c0}$ of a nonlinear eigenvalue problem, of removing a small subdomain $D_\epsilon$, of “radius” $\epsilon$, from D and imposing a condition on the boundary of the resulting hole, are determined. Using the method of matched asymptotic expansions, it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start with infinite series in powers of $( - 1/\log [ \epsilon d( \kappa ) ] )$. Here $d( \kappa )$ is a constant that depends on the shape of $D_\epsilon$ and on the precise form of the boundary condition on the hole. In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of the hole. This related problem is not stiff and can be solved numerically in a straightforward way. Thus a hybrid asymptotic-numerical method is obtained, which has the effect of summing these infinite logarithmic expansions. Results obtained from the hybrid formulation are shown to be in close agreement with full numerical solutions to the perturbed problems. The hybrid method is then used to determine the absorption time distribution for a particle performing Brownian motion in a domain with reflecting walls containing several small absorbing obstacles.