Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

Abstract

Abstract. We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic the different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L 2 -norm.Key words. Darcy's problem, stabilized finite element methods, characteristic length scale, orthogonal subgrid scales AMS subject classifications. 65N30, 35Q301. Introduction. Darcy's problem governs the flow of an incompressible fluid through a porous medium. It is composed by the Darcy law that relates the fluid velocity (the flux) and the pressure gradient and the mass conservation equation. In flow in porous media, a proper functional setting for this problem is to consider the flux in H(div, Ω) and the pressure in L 2 (Ω). This yields a saddle-point problem that is well posed due to inf-sup conditions known to hold at the continuous level, and that allow one to obtain stability estimates for the pressure and the velocity divergence.The Galerkin approximation of this indefinite system is a difficult task, because the continuous inf-sup conditions are not naturally inherited by most finite element (FE) velocitypressure spaces. We can avoid these problems by invoking the Darcy law in the mass conservation equation, getting a pressure Poisson problem; this is an elliptic problem that can be easily approximated by the Galerkin technique and Lagrangian elements. The fluxes can be obtained as a postprocess by using a L 2 -projection. This approach is computationally appealing because pressure and velocity computations are decoupled and the implementation is easy. Unfortunately, this approach has two drawbacks: the loss of accuracy for the velocity and the very weak enforcement of the mass conservation equation. Improved post-processing techniques that reduce these problems can be found e.g. in [15,17]. This approach has been restricted to continuous (H 1 -conforming) pressure FE spaces. However, the continuous pressure admits discontinuities, e.g. in regions with jumps of the physical properties (conductivity), and this approach leads to poor accuracy in the vicinity of these regions.Th...

Stokes, Maxwell and Darcy: A single finite element approximation for three model problems

“…We refer to [4,5] for the proof of these results. Let us discuss the implications of the choice of ℓ in view of the working norm in (34) and the error function in (35).…”

“…In fact, the length scale in τ p could be different from that in τ u , but we have taken them equal to simplify the discussion. Using an approximate Fourier analysis as in [5], the stabilization parameters are found, now depending on scaling coefficients µ u and µ p . In turn, these scaling coefficients depend on a length scale of the problem that may be taken as L 0 or h. This is what determines the choice for ℓ.…”

“…Many years later, these ideas were extended to the Darcy problem in primal form in [20]. Then, a stabilized FE technique for the dual Darcy problem that exhibits the same convergence rates as inf-sup stable FEs was proposed in [4,5]. Very recently, a stabilized FE formulation for the Maxwell problems that allows to use Lagrange finite element methods and converge also to singular solutions has been designed in [3].…”

On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

“…We refer to [3] for dG finite element approximations to Darcy's problem including stabilization terms.…”

Stress-accurate Mixed FEM for soil failure under shallow foundations involving strain localization in plasticity

“…Mixed formulations in terms of both the pressure and the flow velocity are classical in the numerical solution of Darcy's equation [7,8,9,10,11,12], where the focus is placed in achieving enhanced accuracy in the velocity. The mathematical structure of Darcy's and Cauchy's problems is analogous, with the pressure and velocity fields in the first one corresponding to the displacement and stress fields in the second one.…”