Defect correction for spectral computations for a singular integral operator

Abstract

We will consider the weakly singular Fredholm integral operator <p align="center"> $T:L^{1}([0,\tau^\star])\rightarrow L^{1}([0,\tau^\star]),\quad (T\varphi)(\tau)=\frac{\bar\omega}{2} \int_{0}^{\tau^\star}E_{1}(|\tau -\tau^'|)\varphi(\tau')\,d\tau',$ <p align="left" class="times"> where $E_{1}$ denotes the first exponential integral function, <p align="center"> $E_{1}(\tau)=\int_{1}^{\infty}\frac{\exp(-\tau\mu)}{\mu}\mu,\quad\tau>0,$ <p align="left" class="times"> and $\bar\omega$ is a constant. The spectral elements of a matrix operator representing the discretization of the integral operator $T$ by a projection method on a subspace of dimension $n$ will be computed. These spectral elements will be refined iteratively, by a defect correction type formula to yield an approximation to the spectral elements of $T$.