Mixed finite elements for solving 2-d diffusion-type equations
Résumé
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.
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