# A Discontinuous Galerkin Method for the Stokes Equation by Divergence-free Patch Reconstruction

## TL;DR

This paper introduces a discontinuous Galerkin method using divergence-free patch reconstruction for Stokes flows, transforming the problem into an elliptic system and providing error estimates validated by numerical tests.

## Contribution

The paper presents a novel divergence-free patch reconstruction approach within a DG framework that simplifies the Stokes problem into an elliptic system, reducing computational complexity.

## Key findings

- Achieves the same degrees of freedom as the mesh elements.
- Provides classical error estimates for the method.
- Numerical examples verify the theoretical error bounds.

## Abstract

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle-point problem due to such weak form. The number of degree of freedoms of our method is the same as the number of elements in the mesh for different order of accuracy. The error estimations of the proposed method are given in a classical style, which are then verified by some numerical examples.