A generalization for stable mixed finite elements

Mixed finite element methods solve a PDE involving two or more variables. In typical problems from electromagnetics and electrodiffusion, the degrees of freedom associated to the different variables are stored on both primal and dual domain meshes and a discrete Hodge star is used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the model and numerical stability of a finite element method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods.