Mixed finite elements for solving 2-D diffusion-type equations

Abstract

[1] Mixed finite elements are a numerical method becoming more and more popular in geosciences. This method is well suited for solving elliptic and parabolic partial differential equations, which are the mathematical representation of many problems, for instance, flow in porous media, diffusion/ dispersion of solutes, and heat transfer, among others. Mixed finite elements combine the advantages of finite elements by handling complex geometry domains with unstructured meshes and full tensor coefficients and advantages of finite volumes by ensuring mass conservation at the element level. In this work, a physically based presentation of mixed finite elements is given, and the main approximations or reformulations made to improve the efficiency of the method are detailed. These approximations or reformulations exhibit links with other numerical methods (nonconforming finite elements, finite differences, finite volumes, and multipoint flux methods). Some improvements of the mixed finite element method are suggested, especially to avoid oscillations for transient simulations and distorted quadrangular grids.

Identification of groundwater flow parameters using reciprocal data from hydraulic interference tests

“…Eq. (1) is numerically solved by way of a Mixed Hybrid Finite Element method (MHFE), discretizing the domain in triangular elements, the method being well-known and very accurate (Younes et al, 2010). The numerical code uses an implicit discretization in time and mass lumping to avoid spurious oscillations (Belfort et al, 2009).…”

Acid/base front propagation in saturated porous media: 2D laboratory experiments and modeling

“…TRACES is designed to calculate flow and reactive transport in saturated porous media. It handles transient or steady state computation in 2D or 3D heterogeneous domains and is based on mixed and discontinuous finite element methods (Siegel et al, 1997;Younes et al, 2010).…”

Efficiency of the Eulerian–Lagrangian localized adjoint method for solving advection–dispersion equations on highly heterogeneous media

“…The flow equation (10) is solved with the mixed finite element approximation detailed in [13]. Accurate fluxes at each element edge are required to track the particles.…”

“…Indeed, although the ELLAM can use a single strategic integration point per element, the total number of particles to be tracked increases drastically because it is proportional to the number of elements multiplied by The flow problem is solved with a lumped mixed hybrid finite element approximation [13]. Owing to the high contrast of permeability, a very fine mesh is required to obtain an accurate velocity field.…”