A Discontinuous Galerkin Method for the Stokes Equation by Divergence-free Patch Reconstruction
This work offers a novel approach to solving Stokes equations that simplifies the numerical treatment by avoiding saddle-point formulations, which is relevant for computational fluid dynamics researchers.
The paper proposes a discontinuous Galerkin method for Stokes flows that uses a locally divergence-free reconstruction space, enabling the Stokes equation to be solved as an elliptic system rather than a saddle-point problem. The method achieves the same number of degrees of freedom as the number of mesh elements, with error estimates verified by numerical examples.
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.