Locally conservative discontinuous bubble scheme for Darcy flow and its application to Hele-Shaw equation based on structured grids

In this paper, we present an algorithm to solve the Darcy flow coupled with a transport for the interface tracking and apply the algorithm to solve a Hele-Shaw flow. The main challenge in the solution of the Hele-Shaw flow can be found at the change in the jump of the pressure along with the moving interface. We notice that such a challenge can be adequately handled by maintaining the conservation of the flux. Our algorithm employs the immersed finite element method equipped with the enrichment of piecewise constants to guarantee the conservative flux while the change of the jump condition for the pressure is handled via discontinuous bubble function, non-zero only near the interface. On the other hand, the interface motion is modeled and solved by the level set framework and WENO scheme. One important advantage of the proposed scheme is that the resulting algebraic system is efficiently handled by a proven-to-be fast and optimal algorithm in time evolution. A number of numerical tests are given to demonstrate the simplicity, efficiency and robustness of the proposed scheme.