# American Institute of Mathematical Sciences

June  2006, 5(2): 241-250. doi: 10.3934/cpaa.2006.5.241

## Defect correction for spectral computations for a singular integral operator

 1 LaMUSE - Laboratoire de Mathématiques de l'Université de Saint-Étienne (ex EA3058), Université de Saint-Étienne, 23 rue Paul Michelon, 42023 Saint-Étienne cedex 2, France, France 2 CMUP - Centro Matemática da Universidade Porto, rua do Campo Alegre 687, 4169-007 Porto, Portugal, Portugal

Received  March 2005 Revised  July 2005 Published  March 2006

We will consider the weakly singular Fredholm integral operator

$T:L^{1}([0,\tau^\star])\rightarrow L^{1}([0,\tau^\star]),\quad (T\varphi)(\tau)=\frac{\bar\omega}{2} \int_{0}^{\tau^\star}E_{1}(|\tau -\tau^'|)\varphi(\tau')\,d\tau',$

where $E_{1}$ denotes the first exponential integral function,

$E_{1}(\tau)=\int_{1}^{\infty}\frac{\exp(-\tau\mu)}{\mu}\mu,\quad\tau>0,$

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 $T$.

Citation: Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241
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