# Analysis of FEAST spectral approximations using the DPG discretization

## TL;DR

This paper analyzes the FEAST spectral approximation method combined with DPG discretization for unbounded operators, demonstrating no spectral pollution and providing error bounds, with applications to optical fiber modes.

## Contribution

It introduces a theoretical framework showing FEAST with DPG avoids spectral pollution and offers error bounds, supported by numerical experiments and practical optical fiber applications.

## Key findings

- No spectral pollution when using DPG with FEAST.
- Provides bounds on discretization errors.
- Effective in computing optical fiber modes.

## Abstract

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as "FEAST", has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.