Tailored Finite Point Method for Numerical Solutions of Singular Perturbed Eigenvalue Problems

Abstract

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.

An Anisotropic Convection-Diffusion Model Using Tailored Finite Point Method for Image Denoising and Compression

“…Due to this special feature, many applications including convection-dominated convection-diffusion-reaction problems [19], first order wave equation [12] and steady magnetohydrodynamics duct flow problem [11] have been successfully implemented. More recently, TFPM demonstrates the effectiveness in solving singularly perturbed problems [9,10]. Here we will present an anisotropic CD (ACD) filter with the advantage of the TFPM for image denoising.…”