Dual formulations of mixed finite element methods with applications

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

► Development of finite element techniques for polyhedral meshes. ► Definition of novel scalar and vector interpolation functions. ► Introduction of a sparse inverse discrete Hodge star. ► Demonstration of how our approach can result in better conditioned linear systems. ► Applications to magnetostatics and Darcy flow problems are provided.