Numerical Analysis of Stabilized Finite Element Methods for Darcy Flow

The flow of an incompressible homogeneous fluid in a rigid saturated porous media leads to the classical Darcy’s problem which basically consists of the mass conservation equation plus Darcy’s law, that relates the average velocity of the fluid in a porous medium with the gradient of a potential field through the hydraulic conductivity tensor. Darcy flow is the starting point for more complex flow in porous media and despite its apparent simplicity it has some properties like incompressibility, anisotropy and heterogeneity of the medium that make the search of accurate numerical solutions an important and active research area of numerical analysis and computational modeling. Finite element formulations applied to this problem are essentially based on two different approaches: one involves a single-field formulation for potential [

1

] and the other employs a mixed formulation in potential and velocity fields. The main characteristic of the mixed finite element methods is the use of different spaces for velocity and potential, requiring a compatibility (LBB condition [

2

]) between the finite element spaces to ensure existence and uniqueness of solution, which reduces the flexibility in the choice of stable finite element spaces. One well known successful approach is the dual mixed formulation developed by Raviart and Thomas [

3

] using divergence based finite element spaces for the velocity field combined with discontinuous Lagrangian spaces for the potential. To overcome the compatibility condition, typical of mixed methods, stabilized finite element methods have been developed. A non-symmetrical stabilized mixed formulation for Darcy flow is presented in [

4

] by adding and adjoint residual form of Darcy’s law to the mixed Galerkin formulation of this problem.

We propose a symmetrical and stable mixed finite element method for Darcy’s problem by combining least-squares residual forms of the conservation of mass equation and Darcys law with the classical dual mixed formulation, so that equal-order classical Lagrangian finite element spaces can be adopted for both velocity and pressure fields with continuous or discontinuous pressure interpolations. Stability, convergence and error estimates are proved and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulation and to confirm the predicted rates of convergence.