# Solving Eigenvalue Problems in a Discontinuous Approximation Space by Patch Reconstruction

## TL;DR

This paper introduces a high-order discontinuous Galerkin method using patch reconstruction for solving elliptic eigenvalue problems, achieving high accuracy with fewer degrees of freedom and outperforming traditional methods in certain cases.

## Contribution

It extends a patch reconstructed DG method to eigenvalue problems, demonstrating high accuracy and efficiency with minimal degrees of freedom compared to classical methods.

## Key findings

- Higher order methods yield more reliable eigenvalues.
- The method attains similar accuracy with fewer degrees of freedom.
- Performance surpasses conforming finite element methods at higher polynomial orders.

## Abstract

We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in 2D and 3D. It is a direct extension of the method recently proposed to solve corresponding boundary value problems, and the optimal error estimates of the approximation to eigenfunctions and eigenvalues are instant consequences from existing results. The method enjoys the advantage that it uses only one degree of freedom on each element to achieve very high order accuracy, which is highly preferred for eigenvalue problems as implied by Zhang's recent study [J. Sci. Comput. 65(2), 2015]. By numerical results, we illustrate that higher order methods can provide much more reliable eigenvalues. To justify that our method is the right one for eigenvalue problems, we show that the patch reconstructed approximation space attains the same accuracy with fewer degrees of freedom than classical discontinuous Galerkin methods. With the increasing of the polynomial order, our method can even achieve a better performance than conforming finite element methods, such methods are traditionally the methods of choice to solve problems with high regularities.