Mixed finite element methods for second order elliptic problems
Second order elliptic equations -div(a grad p) = f can be solved approximately for the scalar variable p directly. However the vector flux u = -a grad p is often the variable of interest. Mixed methods write the equation as a system of first order equations for p and u. Finite element solution provides accurate approximation of both variables. We describe the basic theory of mixed methods, including the need for an inf-sup condition and implementation in the hybrid form. We review existing families of mixed finite elements and discuss new families of finite elements that work well on quadrilaterals and cuboidal hexahedra. To show the richness of mixed method approaches, we then discuss several applications, including two-phase flow in a porous medium, variational multiscale techniques (i.e., multiscale finite elements), multiscale mortar methods, and mantle dynamics.