# FEAST for differential eigenvalue problems

## TL;DR

This paper introduces an operator analogue of the FEAST eigensolver for differential operators, enabling precise computation of targeted spectral regions, especially high-frequency modes, with robustness and structure preservation.

## Contribution

It develops a novel operator-based FEAST method with a rational filter for unbounded regions, preserving operator structure and achieving high-precision eigenvalue computation.

## Key findings

- Accurately computes eigenvalues in specified spectral regions.
- Handles unbounded search regions with a new rational filter.
- Preserves operator structure for normal or self-adjoint operators.

## Abstract

An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal or self-adjoint, then the operator analogue preserves that structure and robustly computes eigenvalues to near machine precision accuracy. The algorithm is particularly adept at computing high-frequency modes of differential operators that possess self-adjoint structure with respect to weighted Hilbert spaces.