# A Discontinuous Galerkin Method by Patch Reconstruction for Elliptic Interface Problem on Unfitted Mesh

## TL;DR

This paper introduces a novel discontinuous Galerkin method with patch reconstruction for elliptic interface problems on unfitted meshes, achieving optimal error estimates and improved efficiency over conforming finite element methods.

## Contribution

The paper presents a new DG method using patch reconstruction with one degree of freedom per element, removing typical interface intersection constraints and enhancing efficiency.

## Key findings

- Achieves optimal error estimates in L2 and DG energy norms.
- Does not require constraints on interface intersection with mesh elements.
- Numerical examples demonstrate improved efficiency over conforming finite element methods.

## Abstract

We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constraints for DG method on how the interface intersects to the elements in the mesh. Other than enjoying the advantages of DG method, our method may achieve even better efficiency than the conforming finite element method, as illustrated by numerical examples.