We propose two variants of tailored finite point (TFP) methods for discretizing two dimensional
singular perturbed eigenvalue (SPE) problems. A continuation method and an iterative method
are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE.
We study the analytical solutions of two special cases of the SPE, and provide an asymptotic
analysis for the solutions. The theoretical results are verified in the numerical experiments. The numerical results
demonstrate that the proposed schemes effectively resolve the delta function like of the
eigenfunctions on relatively coarse grid.