Unconditionally stable mixed finite element methods for Darcy flow

https://doi.org/10.1016/j.cma.2007.11.025Get rights and content

Abstract

Unconditionally stable finite element methods for Darcy flow are derived by adding least squares residual forms of the governing equations to the classical mixed formulations. The proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous Lagrangian finite element spaces of any order for the velocity and the potential. Stability, convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence.

Introduction

The flow of an incompressible homogeneous fluid in a rigid saturated porous media leads to the classical Darcy 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. A standard way to solve this system of partial differential equations is based on the second order elliptic equation obtained by substituting Darcy’s law in the mass conservation equation leading to a single field problem in potential only with velocity calculated by taking the gradient of the solution multiplied by the hydraulic conductivity. Constructing finite element methods based on this kind of formulation is straightforward. However, this direct approach leads to lower-order approximations for velocity compared to potential and, additionally, the corresponding balance equation is satisfied in an extremely weak sense. Alternative formulations have been employed to enhance the velocity approximation like mixed methods [1], [2] and post-processing techniques [3], [4], [5], [6].

Mixed methods are based on the simultaneous approximation of potential and velocity fields using different spaces for velocity and potential to satisfy a compatibility between the finite element spaces, which reduces the flexibility in constructing stable finite approximations (LBB condition – [2], [7]). One well known successful approach is the Dual mixed formulation developed by Raviart and Thomas [1] 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 [8], [9], [10], [11], [12], [13], [24], [26], [14]. In [9] stabilized mixed Petrov–Galerkin methods are proposed for a heat transfer problem identical to Darcy’s problem, where the original saddle point associated with the classical mixed formulation is converted into a minimization problem. Non-symmetrical stabilized mixed formulations for Darcy flow are presented in [11] by adding an adjoint residual form of Darcy’s law to the mixed Galerkin formulation. More recently Petrov–Galerkin enriched methods [14] as well as stabilized mixed formulations associated with discontinuous Galerkin methods [12], [13] have been proposed and analyzed for the same problem.

In this work we present a stabilized mixed finite element methods derived by adding to the classical Dual mixed formulation least square residuals of the Darcy’s law, the mass balance equation and the curl of Darcy’s law. For sufficiently smooth hydraulic conductivity, an unconditionally stable formulation is presented with optimal rates of convergence in [H1(Ω)]n×H1(Ω) norm.

Some other possible stabilizations are commented, such as a Stokes compatible [H1(Ω)]n×L2(Ω) stabilization as well as H(div,Ω)×H1(Ω) and [L2(Ω)]n×H1(Ω) stable methods. The equivalence between a symmetric [L2(Ω)]n×H1(Ω) stable method and the non symmetric method proposed in [11] is observed. The present stabilized methods are free of mesh dependent parameters and all of them, except Stokes compatible mixed method, allow the use of the classical continuous Lagrangian finite element spaces of any order for velocity and potential, including equal order interpolations. The error estimates are numerically verified, demonstrating the flexibility in the choice of the finite element spaces, derived by the proposed stabilizations.

An outline of the paper follows. In Section 2 we present the model problem, introduce some notations and recall standard single field and mixed Galerkin methods. In Section 3 we present and analyze the [H1(Ω)]n×H1(Ω) stable method. In Section 4 we comment on some alternative stabilized formulations. Numerical experiments illustrating the performance of the methods are reported in Section 5 and in Section 6 we draw some conclusions.

Section snippets

Model problem and Galerkin approximations

In this section we present our model problem, introduce some notations and recall the standard single field and mixed Galerkin methods as the starting point to the stabilized formulations discussed next. Let Ω, a bounded, open, connected domain in Rn (n=2 or n=3) with Lipschitz boundary Γ=Ω and outward unit normal vector n, be the domain of a rigid porous media saturated with an incompressible homogeneous fluid. Our model problem is formulated as follows: Find the velocity u and the hydraulic

Unconditionally stable formulation

In this Section we present a stabilized formulation for Darcy problem, unconditionally stable in [H1(Ω)]n×H1(Ω) and with no mesh dependent parameters, such that any conforming finite element approximation is stable. Though the formulations presented here are clearly applicable to more general situations, in what follows we will consider only isotropic media, i.e., we admit that the hydraulic conductivity K(x) and the hydraulic resistivity Λ(x) are functions only of the position given byK(x)=K(x)

Alternative formulations

The analysis of Problem CGLS indicates many other possibilities of deriving unconditionally stable finite element methods for Darcy flow by adequately including least square residuals. In this section we comment some of these possibilities, concerning C0(Ω) Lagrangian spaces.

Numerical experiments

In this Section we present two numerical studies to confirm the predicted rates of convergence and to demonstrate the flexibility and the robustness of the proposed methods.

Concluding remarks

Unconditionally stable mixed finite element methods for Darcy flow were derived by combining least squares residual forms of the governing equations with the classical mixed formulations as in the mixed Petrov–Galerkin method or Galerkin Least Squares formulations. Considering continuous Lagrangian interpolation of equal order for velocity and potential fields, numerical analysis of the proposed formulations lead the following conclusions:

  • Subtracting to the Primal mixed formulation only the

References (33)

  • C. Cordes et al.

    Continuous groundwater velocity fields and path lines in linear, bilinear and trilinear finite elements

    Water Resources Res.

    (1992)
  • L.J. Durlofsky

    Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities

    Water Resources Res.

    (1994)
  • F. Brezzi et al.

    uniqueness and approximation of saddle point problems arising from Lagrange multipliers

    Analyse numérique/Numerical Analysis (RAIRO)

    (1974)
  • A.F.D. Loula, E.M. Toledo, Dual and primal mixed Petrov–Galerkin finite element methods in heat transfer problems,...
  • L.P. Franca et al.

    A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov–Galerkin method

    Numer. Math.

    (1988)
  • F. Brezzi et al.

    A mixed discontinuous Galerkin method for Darcy flow

    SIAM J. Sci. Comput.

    (2005)
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