Two-level galerkin mixed finite element method for darcy-forchheimer model in porous media

A two-level mixed finite element method is developed and analyzed to solve the Darcy-Forchheimer equation modeling non-Darcy flows in porous media. Instead of solving a large nonlinear system of equations on a fine mesh, the two-level method provides the flexibility of solving a small nonlinear system of equations on a coarse mesh with mesh size H followed by solving one linear system of equations on a fine mesh with mesh size h. In constructing the two-level algorithm, we introduce a small positive constant ϵ to the original nonlinear term of the Darcy-Forchheimer equation to avoid difficulties associated with nondifferentiability of the Euclidean norm at zero. The priori error estimates for the velocity and pressure gradient are obtained in L2 and L3/2 discrete norms, respectively. These estimates state that if piecewise constant velocities and piecewise continuous, linear pressures are used, then the coarse and fine meshes are related by h = O(H2). Numerical examples are provided to show the efficiency and accuracy of the method. Numerical results show that the twolevel Galerkin mixed finite element method is computationally more cost-effective than the standard Galerkin mixed finite element and the rate of convergence also agrees with the theoretical results.