# On symmetrizing the ultraspherical spectral method for self-adjoint problems

## TL;DR

This paper introduces a symmetrization technique for the ultraspherical spectral method applied to self-adjoint problems, resulting in symmetric, banded discretizations and an adaptive spectral decomposition algorithm.

## Contribution

It presents a novel symmetrization mechanism and an adaptive spectral decomposition algorithm for self-adjoint operators within the ultraspherical spectral framework.

## Key findings

- Discretizations become symmetric and banded after symmetrization
- The adaptive spectral decomposition effectively analyzes self-adjoint operators
- Applications demonstrate the advantages of the symmetrizer and decomposition methods

## Abstract

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.