Compatible discretizations of second-order elliptic problems

Differential forms provide a powerful abstraction tool to encode the structure of many partial differential equation problems. Discrete differential forms offer the same possibility with regard to compatible discretizations of these problems, i.e., for finite-dimensional models that exhibit similar conservation properties and invariants. We consider the application of a discrete exterior calculus to the approximation of second-order elliptic boundary-value problems. We show that there exist three possible discretization patterns. In the context of finite element methods, two of these lead to familiar classes of discrete problems, while the third offers a novel perspective about least-squares variational principles, namely how they can arise from particular choices for discrete Hodge–$*$ operators.