Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity
Introduction
For convenience of presentation, we concentrate on the two-dimensional Darcy equation, which is usually formulated as where is a bounded polygonal domain, p the unknown pressure, u the Darcy velocity, K a permeability tensor that is uniformly symmetric positive-definite, f a source term, are respectively Dirichlet and Neumann boundary data, n the unit outward normal vector on ∂Ω, which has a nonoverlapping decomposition . When , the problem has a unique solution.
Here is a list of commonly used finite element methods for solving the Darcy equation, or more generally, elliptic boundary value problems.
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Continuous Galerkin (CG) FEMs [7]. There are different opinions on whether CGFEMs are not locally conservative [21]. CGFEMs generally have smaller numbers of unknowns. CGFEMs assume continuous (polynomial) approximants and are not flexible for certain problems that have solutions with low regularity.
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Discontinuous Galerkin (DG) FEMs [2] (and the references therein). DGFEMs utilize discontinuous shape functions that exhibit great flexibility in handling complicated domain geometry and problems with low regularity. DGFEMs are parallelizable by design. But for the symmetric formulations, one has to choose sufficiently large penalty factors to ensure stability of the numerical methods [18]. Choosing appropriate penalty factors is problem-dependent and hence a drawback in applications of DGFEMs.
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DGFEMs with weak overpenalization (WOP) [6]. The inconvenience of choosing penalty factors motivates the weakly over-penalized DGFEMs, which avoid choosing penalty factors by means of weak over-penalization. However, these methods result in large condition numbers, even though simple block-diagonal preconditioners are available [6].
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Mixed FEMs [9]. The mixed methods solve for the pressure and Darcy velocity simultaneously. But it is well known that the mixed FEMs result in saddle-point problems, whose solvers are more complicated than those for definite systems [5].
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Hybridized FEMs [13] (and the references therein). Hybridization introduces new unknowns (Lagrangian multipliers) and relax the continuity constraints across element interfaces. Hybridization proliferates (conceptually) the numbers of unknowns, although degrees of freedom are actually smaller after certain variables are eliminated.
The weak Galerkin finite element methods introduced in [34] adopt a completely different approach. They rely on novel concepts such as weak functions, the weak gradient operator, discrete weak functions, and discrete weak gradients. As is well known, the variational formulation of a second order elliptic problem relies on the duality of the classical gradient operator. Locality of a finite element space implies relative independence of elementwise shape functions and local conservation. However, shape functions in element interiors could be related to their values on the element interfaces through integration by parts. The weak gradient operator characterizes exactly this connection, see Eq. (3) in Section 2. Discrete weak gradient operators inherit the connection for finite dimensional (Galerkin type) polynomial approximation subspaces, see Eq. (5) in Section 2.
The Darcy's law has its fundamental importance in flow and transport in porous media [12], [15], [16], [17], [19], [23], [24], [27]. Although Darcy's law can be essentially treated as an elliptic problem, the quantities of interest are pressure and velocity (flux) and the focus is placed on flow features. In this paper, we investigate the performance of the novel WGFEMs on handling the numerical difficulties brought up by anisotropy and heterogeneity in permeability.
The rest of this paper is organized as follows. In Section 2, we explain a line of new concepts such as weak functions, weak gradients, discrete weak functions, and discrete weak gradients. Based on these, we develop weak Galerkin finite element methods for Darcy flow. Section 3 addresses implementation issues of the WGFEMs. Section 4 briefly reviews other existing numerical methods for Darcy flow. Section 5 presents numerical simulations of the WGFEMs on several benchmark problems for Darcy flow. Section 6 concludes the paper with some remarks.
Section snippets
WGFEMs for Darcy flow
The weak Galerkin finite element methods were first introduced in [34] by using a weakly defined gradient operator over discontinuous functions. The concept of weak gradient and its approximations result in discrete weak gradients, which will play an important role in the WGFEMs for Darcy flow.
Implementation of WGFEMs for Darcy flow
For WGFEM formulations, it is assumed that permeability is in . For implementation, it is convenient to assume the permeability is a constant scalar or a constant SPD matrix on each element of a given mesh. In general, permeability could vary spatially within a cell/element. If the changes are not dramatic, then an average of the permeability scalar/matrix could be used. For dramatics changes, multiscale methods could be used [17].
For convenience, we assume the meshes are either triangular or
Other numerical methods for Darcy flow
In this section, we briefly review other commonly used numerical methods for Darcy flow. Due to limitation of space, we discuss mainly three types of methods: the cell-centered finite difference method [19], the classical mixed finite element methods [1], [3], [9], [28], and some control-volume finite volume methods [15], [16], [30].
Numerical experiments
In this section, we conduct numerical experiments to examine performance of the weak Galerkin finite element methods on solving the Darcy equation. For comparison, we also present numerical results of other commonly used numerical methods, e.g., MFEM and the cell-centered finite difference method on some benchmark problems. Some mesh data structures recommended in the iFEM package [11] are used in our testing code. Some techniques for the Matlab implementations of the lowest order
Concluding remarks
It has been observed in our numerical experiments that WGFEMs and MFEMs produce very close numerical results in pressure and flux, when permeability is piecewise constant, even though these two types of methods are established on totally different concepts. There actually exists some “equivalence” between WGFEMs and MFEMs, which can be established via hybridized MFEMs [20].
We examine the equivalence for a simple case : Let be a triangular mesh. Note that the MFEM
Acknowledgements
G. Lin would like to acknowledge support by the Applied Mathematics program of the US DOE Office of Advanced Scientific Computing Research and Pacific Northwest National Laboratory's Carbon Sequestration Initiative, which is part of the Laboratory Directed Research and Development Program. J. Liu was partially supported by the National Science Foundation under Grant No. DMS-1419077. X. Ye was supported in part by the National Science Foundation under Grant No. DMS-1115097. Computations were
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