Abstract: This paper presents a weak Galerkin (WG) finite element solver for Darcy flow and its implementation on the \texttt{deal.II} platform. The solver works for quadrilateral and hexahedral meshes in a unified way. It approximates pressure by $ Q $-type degree $ k (\ge 0) $ polynomials separately in element interiors and on edges/faces. Numerical velocity is obtained in the unmapped Raviart-Thomas space $ RT_{[k]} $ via postprocessing based on the novel concepts of discrete weak gradients. The solver is locally mass-conservative and produces continuous normal fluxes. It is implemented in \texttt{deal.II} in the dimension-independent paradigm and allows polynomial degrees up to $ 5 $. Numerical experiments show that our new WG solver performs better than the classical mixed finite element methods.

Abstract: We study a mathematical model and its finite element approximation for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic material. The free fluid flow is governed by the Stokes equations, while the poroelastic material is modeled using the Biot system of poroelasticity. The model is based on a mixed stress-displacement-rotation elasticity formulation and mixed velocity-pressure Darcy and Stokes formulations. The mixed finite element approximation provides local mass and momentum conservation in the poroelastic media. We discuss stability, accuracy, and robustness of the method. Applications to flows in fractured poroelastic media and arterial flows are presented.

Abstract: The velocity, coupling term in the flow and transport problems, is important in the accurate numerical simulation or in the posteriori error analysis for adaptive mesh refinement. We consider Enhanced Velocity Mixed Finite Element Method for the incompressible Darcy flow. In this paper, our aim to study the improvement of velocity at interface to achieve the better approximation of velocity between subdomains. We propose the reconstruction of velocity at interface by using the post-processed pressure. Numerical results at the interface show improvement on convergence rate.

Abstract: In this work we propose a new combination of finite element methods to solve incompressible miscible displacements in heterogeneous media formed by the coupling of the free-fluid with the porous medium employing the stabilized hybrid mixed finite element method developed and analyzed by Igreja and Loula in \cite{Igreja:2018} and the classical Streamline Upwind Petrov--Galerkin (SUPG) method presented and analyzed by Brooks and Hughes in \cite{brooks-hughes:82}. The hydrodynamic problem is governed by the Stokes and Darcy systems coupled by Beavers-Joseph-Saffman interface conditions. To approximate the Stokes-Darcy coupled system we apply the stabilized hybrid mixed method, characterized by the introduction of the Lagrange multiplier associated with the velocity field in both domains. This choice naturally imposes the Beavers-Joseph-Saffman interface conditions on the interface between Stokes and Darcy domains. Thus, the global system is assembled involving only the degrees of freedom associated with the multipliers and the variables of interest can be solved at the element level. Considering the velocity fields given by the hybrid method we adopted the SUPG method combined with an implicit finite difference scheme to solve the transport equation associated with miscible displacements. Numerical studies are presented to illustrate the flexibility and robustness of the hybrid formulation. To verify the efficiency of the combination of hybrid and SUPG methods, computer simulations are also presented for the recovery hydrological flow problems in heterogeneous porous media, such as continuous injection.